title: Respiratory system

text: In mammals, inhalation at rest is primarily due to the contraction of the diaphragm. This is an upwardly domed sheet of muscle that separates the thoracic cavity from the abdominal cavity. When it contracts the sheet flattens, (i.e. moves downwards as shown in Fig. 7) increasing the volume of the thoracic cavity in the antero-posterior axis. The contracting diaphragm pushes the abdominal organs downwards. But because the pelvic floor prevents the lowermost abdominal organs from moving in that direction, the pliable abdominal contents cause the belly to bulge outwards to the front and sides, because the relaxed abdominal muscles do not resist this movement (Fig. 7). This entirely passive bulging (and shrinking during exhalation) of the abdomen during normal breathing is sometimes referred to as "abdominal breathing", although it is, in fact, "diaphragmatic breathing", which is not visible on the outside of the body. Mammals only use their abdominal muscles during forceful exhalation (see Fig. 8, and discussion below). Never during any form of inhalation.


title: Respiratory system

text: As the diaphragm contracts, the rib cage is simultaneously enlarged by the ribs being pulled upwards by the intercostal muscles as shown in Fig. 4. All the ribs slant downwards from the rear to the front (as shown in Fig. 4); but the lowermost ribs "also" slant downwards from the midline outwards (Fig. 5). Thus the rib cage's transverse diameter can be increased in the same way as the antero-posterior diameter is increased by the so-called pump handle movement shown in Fig. 4.


title: Respiratory system

text: The enlargement of the thoracic cavity's vertical dimension by the contraction of the diaphragm, and its two horizontal dimensions by the lifting of the front and sides of the ribs, causes the intrathoracic pressure to fall. The lungs interiors are open to the outside air and being elastic, therefore expand to fill the increased space, pleura fluid between double-layered pleura covering of lungs helps in reducing friction while lungs expansion and contraction. The inflow of air into the lungs occurs via the respiratory airways (Fig. 2). In a healthy person, these airways begin with the nose. (It is possible to begin with the mouth, which is the backup breathing system. However, chronic mouth breathing leads to, or is a sign of, illness.) It ends in the microscopic dead-end sacs called alveoli, which are always open, though the diameters of the various sections can be changed by the sympathetic and parasympathetic nervous systems. The alveolar air pressure is therefore always close to atmospheric air pressure (about 100 kPa at sea level) at rest, with the pressure gradients because of lungs contraction and expansion cause air to move in and out of the lungs during breathing rarely exceeding 2–3 kPa.


title: Respiratory system

text: During exhalation, the diaphragm and intercostal muscles relax. This returns the chest and abdomen to a position determined by their anatomical elasticity. This is the "resting mid-position" of the thorax and abdomen (Fig. 7) when the lungs contain their functional residual capacity of air (the light blue area in the right hand illustration of Fig. 7), which in the adult human has a volume of about 2.5–3.0 liters (Fig. 3). Resting exhalation lasts about twice as long as inhalation because the diaphragm relaxes passively more gently than it contracts actively during inhalation.


title: Respiratory system

text: The volume of air that moves in "or" out (at the nose or mouth) during a single breathing cycle is called the tidal volume. In a resting adult human it is about 500 ml per breath. At the end of exhalation, the airways contain about 150 ml of alveolar air which is the first air that is breathed back into the alveoli during inhalation. This volume air that is breathed out of the alveoli and back in again is known as dead space ventilation, which has the consequence that of the 500 ml breathed into the alveoli with each breath only 350 ml (500 ml - 150 ml = 350 ml) is fresh warm and moistened air. Since this 350 ml of fresh air is thoroughly mixed and diluted by the air that remains in the alveoli after a normal exhalation (i.e. the functional residual capacity of about 2.5–3.0 liters), it is clear that the composition of the alveolar air changes very little during the breathing cycle (see Fig. 9). The oxygen tension (or partial pressure) remains close to 13-14 kPa (about 100 mm Hg), and that of carbon dioxide very close to 5.3 kPa (or 40 mm Hg). This contrasts with composition of the dry outside air at sea level, where the partial pressure of oxygen is 21 kPa (or 160 mm Hg) and that of carbon dioxide 0.04 kPa (or 0.3 mmHg).


title: Respiratory system

text: During heavy breathing (hyperpnea), as, for instance, during exercise, inhalation is brought about by a more powerful and greater excursion of the contracting diaphragm than at rest (Fig. 8). In addition, the "accessory muscles of inhalation" exaggerate the actions of the intercostal muscles (Fig. 8). These accessory muscles of inhalation are muscles that extend from the cervical vertebrae and base of the skull to the upper ribs and sternum, sometimes through an intermediary attachment to the clavicles. When they contract the rib cage's internal volume is increased to a far greater extent than can be achieved by contraction of the intercostal muscles alone. Seen from outside the body the lifting of the clavicles during strenuous or labored inhalation is sometimes called clavicular breathing, seen especially during asthma attacks and in people with chronic obstructive pulmonary disease.


title: Respiratory system

text: During heavy breathing, exhalation is caused by relaxation of all the muscles of inhalation. But now, the abdominal muscles, instead of remaining relaxed (as they do at rest), contract forcibly pulling the lower edges of the rib cage downwards (front and sides) (Fig. 8). This not only drastically decreases the size of the rib cage, but also pushes the abdominal organs upwards against the diaphragm which consequently bulges deeply into the thorax (Fig. 8). The end-exhalatory lung volume is now well below the resting mid-position and contains far less air than the resting "functional residual capacity". However, in a normal mammal, the lungs cannot be emptied completely. In an adult human, there is always still at least 1 liter of residual air left in the lungs after maximum exhalation.


title: Respiratory system

text: The automatic rhythmical breathing in and out, can be interrupted by coughing, sneezing (forms of very forceful exhalation), by the expression of a wide range of emotions (laughing, sighing, crying out in pain, exasperated intakes of breath) and by such voluntary acts as speech, singing, whistling and the playing of wind instruments. All of these actions rely on the muscles described above, and their effects on the movement of air in and out of the lungs.


title: Respiratory system

text: Although not a form of breathing, the Valsalva maneuver involves the respiratory muscles. It is, in fact, a very forceful exhalatory effort against a tightly closed glottis, so that no air can escape from the lungs. Instead abdominal contents are evacuated in the opposite direction, through orifices in the pelvic floor. The abdominal muscles contract very powerfully, causing the pressure inside the abdomen and thorax to rise to extremely high levels. The Valsalva maneuver can be carried out voluntarily but is more generally a reflex elicited when attempting to empty the abdomen during, for instance, difficult defecation, or during childbirth. Breathing ceases during this maneuver.


title: Respiratory system

text: The primary purpose of the respiratory system is the equalizing of the partial pressures of the respiratory gases in the alveolar air with those in the pulmonary capillary blood (Fig. 11). This process occurs by simple diffusion, across a very thin membrane (known as the blood–air barrier), which forms the walls of the pulmonary alveoli (Fig. 10). It consists of the alveolar epithelial cells, their basement membranes and the endothelial cells of the alveolar capillaries (Fig. 10). This blood gas barrier is extremely thin (in humans, on average, 2.2 μm thick). It is folded into about 300 million small air sacs called alveoli (each between 75 and 300 µm in diameter) branching off from the respiratory bronchioles in the lungs, thus providing an extremely large surface area (approximately 145 m) for gas exchange to occur.


title: Respiratory system

text: The air contained within the alveoli has a semi-permanent volume of about 2.5-3.0 liters which completely surrounds the alveolar capillary blood (Fig. 12). This ensures that equilibration of the partial pressures of the gases in the two compartments is very efficient and occurs very quickly. The blood leaving the alveolar capillaries and is eventually distributed throughout the body therefore has a partial pressure of oxygen of 13-14 kPa (100 mmHg), and a partial pressure of carbon dioxide of 5.3 kPa (40 mmHg) (i.e. the same as the oxygen and carbon dioxide gas tensions as in the alveoli). As mentioned in the section above, the corresponding partial pressures of oxygen and carbon dioxide in the ambient (dry) air at sea level are 21 kPa (160 mmHg) and 0.04 kPa (0.3 mmHg) respectively.


title: Respiratory system

text: This marked difference between the composition of the alveolar air and that of the ambient air can be maintained because the functional residual capacity is contained in dead-end sacs connected to the outside air by fairly narrow and relatively long tubes (the airways: nose, pharynx, larynx, trachea, bronchi and their branches down to the bronchioles), through which the air has to be breathed both in and out (i.e. there is no unidirectional through-flow as there is in the bird lung). This typical mammalian anatomy combined with the fact that the lungs are not emptied and re-inflated with each breath (leaving a substantial volume of air, of about 2.5-3.0 liters, in the alveoli after exhalation), ensures that the composition of the alveolar air is only minimally disturbed when the 350 ml of fresh air is mixed into it with each inhalation. Thus the animal is provided with a very special "portable atmosphere", whose composition differs significantly from the present-day ambient air. It is this portable atmosphere (the functional residual capacity) to which the blood and therefore the body tissues are exposed – not to the outside air.


title: Respiratory system

text: The resulting arterial partial pressures of oxygen and carbon dioxide are homeostatically controlled. A rise in the arterial partial pressure of CO and, to a lesser extent, a fall in the arterial partial pressure of O, will reflexly cause deeper and faster breathing until the blood gas tensions in the lungs, and therefore the arterial blood, return to normal. The converse happens when the carbon dioxide tension falls, or, again to a lesser extent, the oxygen tension rises: the rate and depth of breathing are reduced until blood gas normality is restored.


title: Respiratory system

text: Since the blood arriving in the alveolar capillaries has a partial pressure of O of, on average, 6 kPa (45 mmHg), while the pressure in the alveolar air is 13-14 kPa (100 mmHg), there will be a net diffusion of oxygen into the capillary blood, changing the composition of the 3 liters of alveolar air slightly. Similarly, since the blood arriving in the alveolar capillaries has a partial pressure of CO of also about 6 kPa (45 mmHg), whereas that of the alveolar air is 5.3 kPa (40 mmHg), there is a net movement of carbon dioxide out of the capillaries into the alveoli. The changes brought about by these net flows of individual gases into and out of the alveolar air necessitate the replacement of about 15% of the alveolar air with ambient air every 5 seconds or so. This is very tightly controlled by the monitoring of the arterial blood gases (which accurately reflect composition of the alveolar air) by the aortic and carotid bodies, as well as by the blood gas and pH sensor on the anterior surface of the medulla oblongata in the brain. There are also oxygen and carbon dioxide sensors in the lungs, but they primarily determine the diameters of the bronchioles and pulmonary capillaries, and are therefore responsible for directing the flow of air and blood to different parts of the lungs.


title: Respiratory system

text: It is only as a result of accurately maintaining the composition of the 3 liters of alveolar air that with each breath some carbon dioxide is discharged into the atmosphere and some oxygen is taken up from the outside air. If more carbon dioxide than usual has been lost by a short period of hyperventilation, respiration will be slowed down or halted until the alveolar partial pressure of carbon dioxide has returned to 5.3 kPa (40 mmHg). It is therefore strictly speaking untrue that the primary function of the respiratory system is to rid the body of carbon dioxide “waste”. The carbon dioxide that is breathed out with each breath could probably be more correctly be seen as a byproduct of the body's extracellular fluid carbon dioxide and pH homeostats


title: Respiratory system

text: If these homeostats are compromised, then a respiratory acidosis, or a respiratory alkalosis will occur. In the long run these can be compensated by renal adjustments to the H and HCO concentrations in the plasma; but since this takes time, the hyperventilation syndrome can, for instance, occur when agitation or anxiety cause a person to breathe fast and deeply thus causing a distressing respiratory alkalosis through the blowing off of too much CO from the blood into the outside air.


title: Respiratory system

text: Oxygen has a very low solubility in water, and is therefore carried in the blood loosely combined with hemoglobin. The oxygen is held on the hemoglobin by four ferrous iron-containing heme groups per hemoglobin molecule. When all the heme groups carry one O molecule each the blood is said to be “saturated” with oxygen, and no further increase in the partial pressure of oxygen will meaningfully increase the oxygen concentration of the blood. Most of the carbon dioxide in the blood is carried as bicarbonate ions (HCO) in the plasma. However the conversion of dissolved CO into HCO (through the addition of water) is too slow for the rate at which the blood circulates through the tissues on the one hand, and through alveolar capillaries on the other. The reaction is therefore catalyzed by carbonic anhydrase, an enzyme inside the red blood cells. The reaction can go in both directions depending on the prevailing partial pressure of CO. A small amount of carbon dioxide is carried on the protein portion of the hemoglobin molecules as carbamino groups. The total concentration of carbon dioxide (in the form of bicarbonate ions, dissolved CO, and carbamino groups) in arterial blood (i.e. after it has equilibrated with the alveolar air) is about 26 mM (or 58 ml/100 ml), compared to the concentration of oxygen in saturated arterial blood of about 9 mM (or 20 ml/100 ml blood).


title: Respiratory system

text: Ventilation of the lungs in mammals occurs via the respiratory centers in the medulla oblongata and the pons of the brainstem. These areas form a series of neural pathways which receive information about the partial pressures of oxygen and carbon dioxide in the arterial blood. This information determines the average rate of ventilation of the alveoli of the lungs, to keep these pressures constant. The respiratory center does so via motor nerves which activate the diaphragm and other muscles of respiration.


title: Respiratory system

text: The breathing rate increases when the partial pressure of carbon dioxide in the blood increases. This is detected by central blood gas chemoreceptors on the anterior surface of the medulla oblongata. The aortic and carotid bodies, are the peripheral blood gas chemoreceptors which are particularly sensitive to the arterial partial pressure of O though they also respond, but less strongly, to the partial pressure of CO. At sea level, under normal circumstances, the breathing rate and depth, is determined primarily by the arterial partial pressure of carbon dioxide rather than by the arterial partial pressure of oxygen, which is allowed to vary within a fairly wide range before the respiratory centers in the medulla oblongata and pons respond to it to change the rate and depth of breathing.


title: Respiratory system

text: Exercise increases the breathing rate due to the extra carbon dioxide produced by the enhanced metabolism of the exercising muscles. In addition passive movements of the limbs also reflexively produce an increase in the breathing rate.


title: Respiratory system

text: Information received from stretch receptors in the lungs limits tidal volume (the depth of inhalation and exhalation).


title: Respiratory system

text: The alveoli are open (via the airways) to the atmosphere, with the result that alveolar air pressure is exactly the same as the ambient air pressure at sea level, at altitude, or in any artificial atmosphere (e.g. a diving chamber, or decompression chamber) in which the individual is breathing freely. With expansion of the lungs the alveolar air occupies a larger volume, and its pressure falls proportionally, causing air to flow in through the airways, until the pressure in the alveoli is again at the ambient air pressure. The reverse happens during exhalation. This "process" (of inhalation and exhalation) is exactly the same at sea level, as on top of Mt. Everest, or in a diving chamber or decompression chamber.


title: Respiratory system

text: However, as one rises above sea level the density of the air decreases exponentially (see Fig. 14), halving approximately with every 5500 m rise in altitude. Since the composition of the atmospheric air is almost constant below 80 km, as a result of the continuous mixing effect of the weather, the concentration of oxygen in the air (mmols O per liter of ambient air) decreases at the same rate as the fall in air pressure with altitude. Therefore, in order to breathe in the same amount of oxygen per minute, the person has to inhale a proportionately greater volume of air per minute at altitude than at sea level. This is achieved by breathing deeper and faster (i.e. hyperpnea) than at sea level (see below).


title: Respiratory system

text: There is, however, a complication that increases the volume of air that needs to be inhaled per minute (respiratory minute volume) to provide the same amount of oxygen to the lungs at altitude as at sea level. During inhalation the air is warmed and saturated with water vapor during its passage through the nose passages and pharynx. Saturated water vapor pressure is dependent only on temperature. At a body core temperature of 37 °C it is 6.3 kPa (47.0 mmHg), irrespective of any other influences, including altitude. Thus at sea level, where the ambient atmospheric pressure is about 100 kPa, the moistened air that flows into the lungs from the trachea consists of water vapor (6.3 kPa), nitrogen (74.0 kPa), oxygen (19.7 kPa) and trace amounts of carbon dioxide and other gases (a total of 100 kPa). In dry air the partial pressure of O at sea level is 21.0 kPa (i.e. 21% of 100 kPa), compared to the 19.7 kPa of oxygen entering the alveolar air. (The tracheal partial pressure of oxygen is 21% of [100 kPa – 6.3 kPa] = 19.7 kPa). At the summit of Mt. Everest (at an altitude of 8,848 m or 29,029 ft) the total atmospheric pressure is 33.7 kPa, of which 7.1 kPa (or 21%) is oxygen. The air entering the lungs also has a total pressure of 33.7 kPa, of which 6.3 kPa is, unavoidably, water vapor (as it is at sea level). This reduces the partial pressure of oxygen entering the alveoli to 5.8 kPa (or 21% of [33.7 kPa – 6.3 kPa] = 5.8 kPa). The reduction in the partial pressure of oxygen in the inhaled air is therefore substantially greater than the reduction of the total atmospheric pressure at altitude would suggest (on Mt Everest: 5.8 kPa "vs." 7.1 kPa).


title: Respiratory system

text: A further minor complication exists at altitude. If the volume of the lungs were to be instantaneously doubled at the beginning of inhalation, the air pressure inside the lungs would be halved. This happens regardless of altitude. Thus, halving of the sea level air pressure (100 kPa) results in an intrapulmonary air pressure of 50 kPa. Doing the same at 5500 m, where the atmospheric pressure is only 50 kPa, the intrapulmonary air pressure falls to 25 kPa. Therefore, the same change in lung volume at sea level results in a 50 kPa difference in pressure between the ambient air and the intrapulmonary air, whereas it result in a difference of only 25 kPa at 5500 m. The driving pressure forcing air into the lungs during inhalation is therefore halved at this altitude. The "rate" of inflow of air into the lungs during inhalation at sea level is therefore twice that which occurs at 5500 m. However, in reality, inhalation and exhalation occur far more gently and less abruptly than in the example given. The differences between the atmospheric and intrapulmonary pressures, driving air in and out of the lungs during the breathing cycle, are in the region of only 2–3 kPa. A doubling or more of these small pressure differences could be achieved only by very major changes in the breathing effort at high altitudes.


title: Respiratory system

text: All of the above influences of low atmospheric pressures on breathing are accommodated primarily by breathing deeper and faster (hyperpnea). The exact degree of hyperpnea is determined by the blood gas homeostat, which regulates the partial pressures of oxygen and carbon dioxide in the arterial blood. This homeostat prioritizes the regulation of the arterial partial pressure of carbon dioxide over that of oxygen at sea level. That is to say, at sea level the arterial partial pressure of CO is maintained at very close to 5.3 kPa (or 40 mmHg) under a wide range of circumstances, at the expense of the arterial partial pressure of O, which is allowed to vary within a very wide range of values, before eliciting a corrective ventilatory response. However, when the atmospheric pressure (and therefore the partial pressure of O in the ambient air) falls to below 50-75% of its value at sea level, oxygen homeostasis is given priority over carbon dioxide homeostasis. This switch-over occurs at an elevation of about 2500 m (or about 8000 ft). If this switch occurs relatively abruptly, the hyperpnea at high altitude will cause a severe fall in the arterial partial pressure of carbon dioxide, with a consequent rise in the pH of the arterial plasma. This is one contributor to high altitude sickness. On the other hand, if the switch to oxygen homeostasis is incomplete, then hypoxia may complicate the clinical picture with potentially fatal results.


title: Respiratory system

text: There are oxygen sensors in the smaller bronchi and bronchioles. In response to low partial pressures of oxygen in the inhaled air these sensors reflexively cause the pulmonary arterioles to constrict. (This is the exact opposite of the corresponding reflex in the tissues, where low arterial partial pressures of O cause arteriolar vasodilation.) At altitude this causes the pulmonary arterial pressure to rise resulting in a much more even distribution of blood flow to the lungs than occurs at sea level. At sea level the pulmonary arterial pressure is very low, with the result that the tops of the lungs receive far less blood than the bases, which are relatively over-perfused with blood. It is only in the middle of the lungs that the blood and air flow to the alveoli are ideally matched. At altitude this variation in the ventilation/perfusion ratio of alveoli from the tops of the lungs to the bottoms is eliminated, with all the alveoli perfused and ventilated in more or less the physiologically ideal manner. This is a further important contributor to the acclimatatization to high altitudes and low oxygen pressures.


title: Respiratory system

text: The kidneys measure the oxygen "content" (mmol O/liter blood, rather than the partial pressure of O) of the arterial blood. When the oxygen content of the blood is chronically low, as at high altitude, the oxygen-sensitive kidney cells secrete erythropoietin (EPO) into the blood. This hormone stimulates the red bone marrow to increase its rate of red cell production, which leads to an increase in the hematocrit of the blood, and a consequent increase in its oxygen carrying capacity (due to the now high hemoglobin content of the blood). In other words, at the same arterial partial pressure of O, a person with a high hematocrit carries more oxygen per liter of blood than a person with a lower hematocrit does. High altitude dwellers therefore have higher hematocrits than sea-level residents.


title: Respiratory system

text: Irritation of nerve endings within the nasal passages or airways, can induce a cough reflex and sneezing. These responses cause air to be expelled forcefully from the trachea or nose, respectively. In this manner, irritants caught in the mucus which lines the respiratory tract are expelled or moved to the mouth where they can be swallowed. During coughing, contraction of the smooth muscle in the airway walls narrows the trachea by pulling the ends of the cartilage plates together and by pushing soft tissue into the lumen. This increases the expired airflow rate to dislodge and remove any irritant particle or mucus.


title: Respiratory system

text: Respiratory epithelium can secrete a variety of molecules that aid in the defense of the lungs. These include secretory immunoglobulins (IgA), collectins, defensins and other peptides and proteases, reactive oxygen species, and reactive nitrogen species. These secretions can act directly as antimicrobials to help keep the airway free of infection. A variety of chemokines and cytokines are also secreted that recruit the traditional immune cells and others to the site of infections.


title: Respiratory system

text: Surfactant immune function is primarily attributed to two proteins: SP-A and SP-D. These proteins can bind to sugars on the surface of pathogens and thereby opsonize them for uptake by phagocytes. It also regulates inflammatory responses and interacts with the adaptive immune response. Surfactant degradation or inactivation may contribute to enhanced susceptibility to lung inflammation and infection.


title: Respiratory system

text: Most of the respiratory system is lined with mucous membranes that contain mucosa-associated lymphoid tissue, which produces white blood cells such as lymphocytes.


title: Respiratory system

text: The lungs make a surfactant, a surface-active lipoprotein complex (phospholipoprotein) formed by type II alveolar cells. It floats on the surface of the thin watery layer which lines the insides of the alveoli, reducing the water's surface tension.


title: Respiratory system

text: The surface tension of a watery surface (the water-air interface) tends to make that surface shrink. When that surface is curved as it is in the alveoli of the lungs, the shrinkage of the surface decreases the diameter of the alveoli. The more acute the curvature of the water-air interface the greater the tendency for the alveolus to collapse. This has three effects. Firstly the surface tension inside the alveoli resists expansion of the alveoli during inhalation (i.e. it makes the lung stiff, or non-compliant). Surfactant reduces the surface tension and therefore makes the lungs more compliant, or less stiff, than if it were not there. Secondly, the diameters of the alveoli increase and decrease during the breathing cycle. This means that the alveoli have a greater tendency to collapse (i.e. cause atelectasis) at the end of exhalation that at the end of inhalation. Since surfactant floats on the watery surface, its molecules are more tightly packed together when the alveoli shrink during exhalation. This causes them to have a greater surface tension-lowering effect when the alveoli are small than when they are large (as at the end of inhalation, when the surfactant molecules are more widely spaced). The tendency for the alveoli to collapse is therefore almost the same at the end of exhalation as at the end of inhalation. Thirdly, the surface tension of the curved watery layer lining the alveoli tends to draw water from the lung tissues into the alveoli. Surfactant reduces this danger to negligible levels, and keeps the alveoli dry.


title: Respiratory system

text: Pre-term babies who are unable to manufacture surfactant have lungs that tend to collapse each time they breathe out. Unless treated, this condition, called respiratory distress syndrome, is fatal. Basic scientific experiments, carried out using cells from chicken lungs, support the potential for using steroids as a means of furthering the development of type II alveolar cells. In fact, once a premature birth is threatened, every effort is made to delay the birth, and a series of steroid injections is frequently administered to the mother during this delay in an effort to promote lung maturation.


title: Respiratory system

text: The lung vessels contain a fibrinolytic system that dissolves clots that may have arrived in the pulmonary circulation by embolism, often from the deep veins in the legs. They also release a variety of substances that enter the systemic arterial blood, and they remove other substances from the systemic venous blood that reach them via the pulmonary artery. Some prostaglandins are removed from the circulation, while others are synthesized in the lungs and released into the blood when lung tissue is stretched.


title: Respiratory system

text: The lungs activate one hormone. The physiologically inactive decapeptide angiotensin I is converted to the aldosterone-releasing octapeptide, angiotensin II, in the pulmonary circulation. The reaction occurs in other tissues as well, but it is particularly prominent in the lungs. Angiotensin II also has a direct effect on arteriolar walls, causing arteriolar vasoconstriction, and consequently a rise in arterial blood pressure. Large amounts of the angiotensin-converting enzyme responsible for this activation are located on the surfaces of the endothelial cells of the alveolar capillaries. The converting enzyme also inactivates bradykinin. Circulation time through the alveolar capillaries is less than one second, yet 70% of the angiotensin I reaching the lungs is converted to angiotensin II in a single trip through the capillaries. Four other peptidases have been identified on the surface of the pulmonary endothelial cells.


title: Respiratory system

text: The movement of gas through the larynx, pharynx and mouth allows humans to speak, or "phonate". Vocalization, or singing, in birds occurs via the syrinx, an organ located at the base of the trachea. The vibration of air flowing across the larynx (vocal cords), in humans, and the syrinx, in birds, results in sound. Because of this, gas movement is vital for communication purposes.


title: Respiratory system

text: Panting in dogs, cats, birds and some other animals provides a means of reducing body temperature, by evaporating saliva in the mouth (instead of evaporating sweat on the skin).


title: Respiratory system

text: Disorders of the respiratory system are usually treated by a pulmonologist and respiratory therapist.


title: Respiratory system

text: Where there is an inability to breathe or insufficiency in breathing a medical ventilator may be used.


title: Respiratory system

text: Horses are obligate nasal breathers which means that they are different from many other mammals because they do not have the option of breathing through their mouths and must take in air through their noses.


title: Respiratory system

text: The elephant is the only mammal known to have no pleural space. Rather, the parietal and visceral pleura are both composed of dense connective tissue and joined to each other via loose connective tissue. This lack of a pleural space, along with an unusually thick diaphragm, are thought to be evolutionary adaptations allowing the elephant to remain underwater for long periods of time while breathing through its trunk which emerges as a snorkel.


title: Respiratory system

text: In the elephant the lungs are attached to the diaphragm and breathing relies mainly on the diaphragm rather than the expansion of the ribcage.


title: Respiratory system

text: The respiratory system of birds differs significantly from that found in mammals. Firstly, they have rigid lungs which do not expand and contract during the breathing cycle. Instead an extensive system of air sacs (Fig. 15) distributed throughout their bodies act as the bellows drawing environmental air into the sacs, and expelling the spent air after it has passed through the lungs (Fig. 18). Birds also do not have diaphragms or pleural cavities.


title: Respiratory system

text: Bird lungs are smaller than those in mammals of comparable size, but the air sacs account for 15% of the total body volume, compared to the 7% devoted to the alveoli which act as the bellows in mammals.


title: Respiratory system

text: Inhalation and exhalation are brought about by alternately increasing and decreasing the volume of the entire thoraco-abdominal cavity (or coelom) using both their abdominal and costal muscles. During inhalation the muscles attached to the vertebral ribs (Fig. 17) contract angling them forwards and outwards. This pushes the sternal ribs, to which they are attached at almost right angles, downwards and forwards, taking the sternum (with its prominent keel) in the same direction (Fig. 17). This increases both the vertical and transverse diameters of thoracic portion of the trunk. The forward and downward movement of, particularly, the posterior end of the sternum pulls the abdominal wall downwards, increasing the volume of that region of the trunk as well. The increase in volume of the entire trunk cavity reduces the air pressure in all the thoraco-abdominal air sacs, causing them to fill with air as described below.


title: Respiratory system

text: During exhalation the external oblique muscle which is attached to the sternum and vertebral ribs anteriorly, and to the pelvis (pubis and ilium in Fig. 17) posteriorly (forming part of the abdominal wall) reverses the inhalatory movement, while compressing the abdominal contents, thus increasing the pressure in all the air sacs. Air is therefore expelled from the respiratory system in the act of exhalation.


title: Respiratory system

text: During inhalation air enters the trachea via the nostrils and mouth, and continues to just beyond the syrinx at which point the trachea branches into two primary bronchi, going to the two lungs (Fig. 16). The primary bronchi enter the lungs to become the intrapulmonary bronchi, which give off a set of parallel branches called ventrobronchi and, a little further on, an equivalent set of dorsobronchi (Fig. 16). The ends of the intrapulmonary bronchi discharge air into the posterior air sacs at the caudal end of the bird. Each pair of dorso-ventrobronchi is connected by a large number of parallel microscopic air capillaries (or parabronchi) where gas exchange occurs (Fig. 16). As the bird inhales, tracheal air flows through the intrapulmonary bronchi into the posterior air sacs, as well as into the "dorso"bronchi, but not into the "ventro"bronchi (Fig. 18). This is due to the bronchial architecture which directs the inhaled air away from the openings of the ventrobronchi, into the continuation of the intrapulmonary bronchus towards the dorsobronchi and posterior air sacs. From the dorsobronchi the inhaled air flows through the parabronchi (and therefore the gas exchanger) to the ventrobronchi from where the air can only escape into the expanding anterior air sacs. So, during inhalation, both the posterior and anterior air sacs expand, the posterior air sacs filling with fresh inhaled air, while the anterior air sacs fill with "spent" (oxygen-poor) air that has just passed through the lungs.


title: Respiratory system

text: During exhalation the pressure in the posterior air sacs (which were filled with fresh air during inhalation) increases due to the contraction of the oblique muscle described above. The aerodynamics of the interconnecting openings from the posterior air sacs to the dorsobronchi and intrapulmonary bronchi ensures that the air leaves these sacs in the direction of the lungs (via the dorsobronchi), rather than returning down the intrapulmonary bronchi (Fig. 18). From the dorsobronchi the fresh air from the posterior air sacs flows through the parabronchi (in the same direction as occurred during inhalation) into ventrobronchi. The air passages connecting the ventrobronchi and anterior air sacs to the intrapulmonary bronchi direct the "spent", oxygen poor air from these two organs to the trachea from where it escapes to the exterior. Oxygenated air therefore flows constantly (during the entire breathing cycle) in a single direction through the parabronchi.


title: Respiratory system

text: The blood flow through the bird lung is at right angles to the flow of air through the parabronchi, forming a cross-current flow exchange system (Fig. 19). The partial pressure of oxygen in the parabronchi declines along their lengths as O diffuses into the blood. The blood capillaries leaving the exchanger near the entrance of airflow take up more O than do the capillaries leaving near the exit end of the parabronchi. When the contents of all capillaries mix, the final partial pressure of oxygen of the mixed pulmonary venous blood is higher than that of the exhaled air, but is nevertheless less than half that of the inhaled air, thus achieving roughly the same systemic arterial blood partial pressure of oxygen as mammals do with their bellows-type lungs.


title: Respiratory system

text: The trachea is an area of dead space: the oxygen-poor air it contains at the end of exhalation is the first air to re-enter the posterior air sacs and lungs. In comparison to the mammalian respiratory tract, the dead space volume in a bird is, on average, 4.5 times greater than it is in mammals of the same size. Birds with long necks will inevitably have long tracheae, and must therefore take deeper breaths than mammals do to make allowances for their greater dead space volumes. In some birds (e.g. the whooper swan, "Cygnus cygnus", the white spoonbill, "Platalea leucorodia", the whooping crane, "Grus americana", and the helmeted curassow, "Pauxi pauxi") the trachea, which some cranes can be 1.5 m long, is coiled back and forth within the body, drastically increasing the dead space ventilation. The purpose of this extraordinary feature is unknown.


title: Respiratory system

text: The anatomical structure of the lungs is less complex in reptiles than in mammals, with reptiles lacking the very extensive airway tree structure found in mammalian lungs. Gas exchange in reptiles still occurs in alveoli however. Reptiles do not possess a diaphragm. Thus, breathing occurs via a change in the volume of the body cavity which is controlled by contraction of intercostal muscles in all reptiles except turtles. In turtles, contraction of specific pairs of flank muscles governs inhalation and exhalation.


title: Respiratory system

text: Both the lungs and the skin serve as respiratory organs in amphibians. The ventilation of the lungs in amphibians relies on positive pressure ventilation. Muscles lower the floor of the oral cavity, enlarging it and drawing in air through the nostrils into the oral cavity. With the nostrils and mouth closed, the floor of the oral cavity is then pushed up, which forces air down the trachea into the lungs. The skin of these animals is highly vascularized and moist, with moisture maintained via secretion of mucus from specialised cells, and is involved in cutaneous respiration. While the lungs are of primary organs for gas exchange between the blood and the environmental air (when out of the water), the skin's unique properties aid rapid gas exchange when amphibians are submerged in oxygen-rich water.


title: Respiratory system

text: Some amphibians have gills, either in the early stages of their development (e.g. tadpoles of frogs), while others retain them into adulthood (e.g. some salamanders).


title: Respiratory system

text: Oxygen is poorly soluble in water. Fully aerated fresh water therefore contains only 8–10 ml O/liter compared to the O concentration of 210 ml/liter in the air at sea level. Furthermore, the coefficient of diffusion (i.e. the rate at which a substances diffuses from a region of high concentration to one of low concentration, under standard conditions) of the respiratory gases is typically 10,000 faster in air than in water. Thus oxygen, for instance, has a diffusion coefficient of 17.6 mm/s in air, but only 0.0021 mm/s in water. The corresponding values for carbon dioxide are 16 mm/s in air and 0.0016 mm/s in water. This means that when oxygen is taken up from the water in contact with a gas exchanger, it is replaced considerably more slowly by the oxygen from the oxygen-rich regions small distances away from the exchanger than would have occurred in air. Fish have developed gills deal with these problems. Gills are specialized organs containing filaments, which further divide into lamellae. The lamellae contain a dense thin walled capillary network that exposes a large gas exchange surface area to the very large volumes of water passing over them.


title: Respiratory system

text: Gills use a countercurrent exchange system that increases the efficiency of oxygen-uptake from the water. Fresh oxygenated water taken in through the mouth is uninterruptedly "pumped" through the gills in one direction, while the blood in the lamellae flows in the opposite direction, creating the countercurrent blood and water flow (Fig. 22), on which the fish's survival depends.


title: Respiratory system

text: Water is drawn in through the mouth by closing the operculum (gill cover), and enlarging the mouth cavity (Fig. 23). Simultaneously the gill chambers enlarge, producing a lower pressure there than in the mouth causing water to flow over the gills. The mouth cavity then contracts, inducing the closure of the passive oral valves, thereby preventing the back-flow of water from the mouth (Fig. 23). The water in the mouth is, instead, forced over the gills, while the gill chambers contract emptying the water they contain through the opercular openings (Fig. 23). Back-flow into the gill chamber during the inhalatory phase is prevented by a membrane along the ventroposterior border of the operculum (diagram on the left in Fig. 23). Thus the mouth cavity and gill chambers act alternately as suction pump and pressure pump to maintain a steady flow of water over the gills in one direction. Since the blood in the lamellar capillaries flows in the opposite direction to that of the water, the consequent countercurrent flow of blood and water maintains steep concentration gradients for oxygen and carbon dioxide along the entire length of each capillary (lower diagram in Fig. 22). Oxygen is, therefore, able to continually diffuse down its gradient into the blood, and the carbon dioxide down its gradient into the water. Although countercurrent exchange systems theoretically allow an almost complete transfer of a respiratory gas from one side of the exchanger to the other, in fish less than 80% of the oxygen in the water flowing over the gills is generally transferred to the blood.


title: Respiratory system

text: In certain active pelagic sharks, water passes through the mouth and over the gills while they are moving, in a process known as "ram ventilation". While at rest, most sharks pump water over their gills, as most bony fish do, to ensure that oxygenated water continues to flow over their gills. But a small number of species have lost the ability to pump water through their gills and must swim without rest. These species are "obligate ram ventilators" and would presumably asphyxiate if unable to move. Obligate ram ventilation is also true of some pelagic bony fish species.


title: Respiratory system

text: There are a few fish that can obtain oxygen for brief periods of time from air swallowed from above the surface of the water. Thus lungfish possess one or two lungs, and the labyrinth fish have developed a special "labyrinth organ", which characterizes this suborder of fish. The labyrinth organ is a much-folded suprabranchial accessory breathing organ. It is formed by a vascularized expansion of the epibranchial bone of the first gill arch, and is used for respiration in air. This organ allows labyrinth fish to take in oxygen directly from the air, instead of taking it from the water in which they reside through the use of gills. The labyrinth organ helps the oxygen in the inhaled air to be absorbed into the bloodstream. As a result, labyrinth fish can survive for a short period of time out of water, as they can inhale the air around them, provided they stay moist. Labyrinth fish are not born with functional labyrinth organs. The development of the organ is gradual and most juvenile labyrinth fish breathe entirely with their gills and develop the labyrinth organs when they grow older.


title: Respiratory system

text: Some species of crab use a respiratory organ called a branchiostegal lung. Its gill-like structure increases the surface area for gas exchange which is more suited to taking oxygen from the air than from water. Some of the smallest spiders and mites can breathe simply by exchanging gas through the surface of the body. Larger spiders, scorpions and other arthropods use a primitive book lung.


title: Respiratory system

text: Most insects breath passively through their spiracles (special openings in the exoskeleton) and the air reaches every part of the body by means of a series of smaller and smaller tubes called 'trachaea' when their diameters are relatively large, and 'tracheoles' when their diameters are very small. The tracheoles make contact with individual cells throughout the body. They are partially filled with fluid, which can be withdrawn from the individual tracheoles when the tissues, such as muscles, are active and have a high demand for oxygen, bringing the air closer to the active cells. This is probably brought about by the buildup of lactic acid in the active muscles causing an osmotic gradient, moving the water out of the tracheoles and into the active cells. Diffusion of gases is effective over small distances but not over larger ones, this is one of the reasons insects are all relatively small. Insects which do not have spiracles and trachaea, such as some Collembola, breathe directly through their skins, also by diffusion of gases.


title: Respiratory system

text: The number of spiracles an insect has is variable between species, however, they always come in pairs, one on each side of the body, and usually one pair per segment. Some of the Diplura have eleven, with four pairs on the thorax, but in most of the ancient forms of insects, such as Dragonflies and Grasshoppers there are two thoracic and eight abdominal spiracles. However, in most of the remaining insects, there are fewer. It is at the level of the tracheoles that oxygen is delivered to the cells for respiration.


title: Respiratory system

text: Insects were once believed to exchange gases with the environment continuously by the simple diffusion of gases into the tracheal system. More recently, however, large variation in insect ventilatory patterns has been documented and insect respiration appears to be highly variable. Some small insects do not demonstrate continuous respiratory movements and may lack muscular control of the spiracles. Others, however, utilize muscular contraction of the abdomen along with coordinated spiracle contraction and relaxation to generate cyclical gas exchange patterns and to reduce water loss into the atmosphere. The most extreme form of these patterns is termed discontinuous gas exchange cycles.


title: Respiratory system

text: Molluscs generally possess gills that allow gas exchange between the aqueous environment and their circulatory systems. These animals also possess a heart that pumps blood containing hemocyanin as its oxygen-capturing molecule. Hence, this respiratory system is similar to that of vertebrate fish. The respiratory system of gastropods can include either gills or a lung.


title: Respiratory system

text: Plants use carbon dioxide gas in the process of photosynthesis, and exhale oxygen gas as waste. The chemical equation of photosynthesis is 6 CO (carbon dioxide) and 6 HO (water), which in the presence of sunlight makes CHO (glucose) and 6 O (oxygen). Photosynthesis uses electrons on the carbon atoms as the repository for the energy obtained from sunlight. Respiration is the opposite of photosynthesis. It reclaims the energy to power chemical reactions in cells. In so doing the carbon atoms and their electrons are combined with oxygen forming CO which is easily removed from both the cells and the organism. Plants use both processes, photosynthesis to capture the energy and oxidative metabolism to use it.


title: Respiratory system

text: Plant respiration is limited by the process of diffusion. Plants take in carbon dioxide through holes, known as stomata, that can open and close on the undersides of their leaves and sometimes other parts of their anatomy. Most plants require some oxygen for catabolic processes (break-down reactions that release energy). But the quantity of O used per hour is small as they are not involved in activities that require high rates of aerobic metabolism. Their requirement for air, however, is very high as they need CO for photosynthesis, which constitutes only 0.04% of the environmental air. Thus, to make 1 g of glucose requires the removal of all the CO from "at least" 18.7 liters of air at sea level. But inefficiencies in the photosynthetic process cause considerably greater volumes of air to be used.


title: Real number

text: In mathematics, a real number is a number that can be used to measure a "continuous" one-dimensional quantity such as a distance, duration or temperature. Here, "continuous" means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion.


title: Real number

text: The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.


title: Real number

text: The adjective "real" in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality.


title: Real number

text: The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers, and include algebraic numbers (such as the square root ) and transcendental numbers (such as ).


title: Real number

text: Real numbers can be thought of as all points on an infinitely long line called the number line or real line, where the points corresponding to integers () are equally spaced.


title: Real number

text: Conversely, analytic geometry is the association of points on lines (especially axis lines) to real numbers such that geometric displacements are proportional to differences between corresponding numbers.


title: Real number

text: The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. A current axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.


title: Real number

text: More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a nonempty set of real numbers has an upper bound, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.


title: Real number

text: A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section):


title: Real number

text: A sequence ("x") of real numbers is called a "Cauchy sequence" if for any there exists an integer "N" (possibly depending on ε) such that the distance is less than ε for all "n" and "m" that are both greater than "N". This definition, originally provided by Cauchy, formalizes the fact that the "x" eventually come and remain arbitrarily close to each other.


title: Real number

text: A sequence ("x") "converges to the limit" "x" if its elements eventually come and remain arbitrarily close to "x", that is, if for any there exists an integer "N" (possibly depending on ε) such that the distance is less than ε for "n" greater than "N".


title: Real number

text: Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.


title: Real number

text: The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).


title: Real number

text: The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.


title: Real number

text: can be made arbitrarily small (independently of "M") by choosing "N" sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that formula_4 is well defined for every "x".


title: Real number

text: The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.


title: Real number

text: First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element "z", is larger).


title: Real number

text: Additionally, an order can be Dedekind-complete, see . The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.


title: Real number

text: These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that formula_1 is the "only" uniformly complete ordered field, but it is the only uniformly complete "Archimedean field", and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.


title: Real number

text: But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the "largest" Archimedean field in the sense that every other Archimedean field is a subfield of formula_1. Thus formula_1 is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.


title: Real number

text: The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers. The cardinality of the set of all real numbers is denoted by formula_8 and called the cardinality of the continuum. It is strictly greater than the cardinality of the set of all natural numbers (denoted formula_9 and called 'aleph-naught'), and equals the cardinality of the power set of the set of the natural numbers.


title: Real number

text: The statement that there is no subset of the reals with cardinality strictly greater than formula_9 and strictly smaller than formula_11 is known as the continuum hypothesis (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.


title: Real number

text: As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.


title: Real number

text: The real numbers form a metric space: the distance between "x" and "y" is defined as the absolute value . By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.


title: Real number

text: Every nonnegative real number has a square root in formula_1, although no negative number does. This shows that the order on formula_1 is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make formula_1 the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.


title: Real number

text: The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval [0;1] has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.


title: Real number

text: The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as formula_1. Ordered fields that satisfy the same first-order sentences as formula_1 are called nonstandard models of formula_1. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in formula_1), we know that the same statement must also be true of formula_1.


title: Real number

text: The field formula_1 of real numbers is an extension field of the field formula_21 of rational numbers, and formula_1 can therefore be seen as a vector space over formula_21. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set "B" of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of "B" is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.


title: Real number

text: The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on formula_1 with the property that every nonempty subset of formula_1 has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.


title: Real number

text: A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.


title: Real number

text: Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" ("The rules of chords") in include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava , who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined. Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational.


title: Real number

text: The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra). Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots and fourth roots). In Europe, such numbers, not commensurable with the numerical unit, were called "irrational" or "surd" ("deaf").


title: Real number

text: In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.


title: Real number

text: In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.


title: Real number

text: In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Lambert (1761) gave a flawed proof that cannot be rational; Legendre (1794) completed the proof and showed that is not the square root of a rational number. Liouville (1840) showed that neither nor can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Cantor (1873) extended and greatly simplified this proof. Hermite (1873) proved that is transcendental, and Lindemann (1882), showed that is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), Hilbert (1893), Hurwitz, and Gordan.


title: Real number

text: The developers of calculus used real numbers without having defined them rigorously. The first rigorous definition was published by Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Cantor's first uncountability proof was different from his famous diagonal argument published in 1891.


title: Real number

text: The real number system formula_26 can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or of Tarski), and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are isomorphic.


title: Real number

text: The last property is what differentiates the real numbers from the rational numbers (and from other more exotic ordered fields). For example, formula_33 has a rational upper bound (e.g., 1.42), but no "least" rational upper bound, because formula_34 is not rational.


title: Real number

text: These properties imply the Archimedean property (which is not implied by other definitions of completeness), which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound "N"; then, "N" – 1 would not be an upper bound, and there would be an integer "n" such that , and thus , which is a contradiction with the upper-bound property of "N".


title: Real number

text: The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields formula_35 and formula_36, there exists a unique field isomorphism from formula_35 to formula_38. This uniqueness allows us to think of them as essentially the same mathematical object.


title: Real number

text: The real numbers can be constructed as a completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...) converges to a unique real number—in this case . For details and other constructions of real numbers, see construction of the real numbers.


title: Real number

text: In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.


title: Real number

text: Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.


title: Real number

text: The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.


title: Real number

text: The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.


title: Real number

text: Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory).


title: Real number

text: The continuum hypothesis posits that the cardinality of the set of the real numbers is formula_40; i.e. the smallest infinite cardinal number after formula_9, the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.


title: Real number

text: Electronic calculators and computers cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrary definable real numbers, which are inconvenient to manipulate.


title: Real number

text: Instead, computers typically work with finite-precision approximations called floating-point numbers, a representation similar to scientific notation. The achievable precision is limited by the data storage space allocated for each number, whether as fixed-point, floating-point, or arbitrary-precision numbers, or some other representation. Most scientific computation uses binary floating-point arithmetic, often a 64-bit representation with around 16 decimal digits of precision. Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not. The field of numerical analysis studies the stability and accuracy of numerical algorithms implemented with approximate arithmetic.


title: Real number

text: Alternately, computer algebra systems can operate on irrational quantities exactly by manipulating symbolic formulas for them (such as formula_42 formula_43 or formula_44) rather than their rational or decimal approximation. But exact and symbolic arithmetic also have limitations: for instance, they are computationally more expensive; it is not in general possible to determine whether two symbolic expressions are equal (the constant problem); and arithmetic operations can cause exponential explosion in the size of representation of a single number (for instance, squaring a rational number roughly doubles the number of digits in its numerator and denominator, and squaring a polynomial roughly doubles its number of terms), overwhelming finite computer storage.


title: Real number

text: A real number is called "computable" if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.


title: Real number

text: In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".


title: Real number

text: Mathematicians use mainly the symbol R to represent the set of all real numbers. Alternatively, it may be used formula_1, the letter "R" in blackboard bold, which may be encoded in Unicode (and HTML) as . As this set is naturally endowed with the structure of a field, the expression "field of real numbers" is frequently used when its algebraic properties are under consideration.


title: Real number

text: The sets of positive real numbers and negative real numbers are often noted formula_46 and formula_47, respectively; formula_48 and formula_49 are also used. The non-negative real numbers can be noted formula_50 but one often sees this set noted formula_51 In French mathematics, the "positive real numbers" and "negative real numbers" commonly include zero, and these sets are noted respectively formula_52 and formula_53 In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted formula_54 and formula_55


title: Real number

text: The notation formula_56 refers to the set of the -tuples of elements of formula_57 (real coordinate space), which can be identified to the Cartesian product of copies of formula_58 It is an -dimensional vector space over the field of the real numbers, often called the coordinate space of dimension ; this space may be identified to the -dimensional Euclidean space as soon as a Cartesian coordinate system has been chosen in the latter. In this identification, a point of the Euclidean space is identified with the tuple of its Cartesian coordinates.


title: Real number

text: In mathematics, "real" is used as an adjective, meaning that the underlying field is the field of the real numbers (or "the real field"). For example, "real matrix", "real polynomial" and "real Lie algebra". The word is also used as a noun, meaning a real number (as in "the set of all reals").


title: Luke Skywalker

text: Luke Skywalker is a fictional character and the protagonist of the original film trilogy of the "Star Wars" franchise created by George Lucas. Portrayed by Mark Hamill, Luke first appeared in "Star Wars" (1977), and he returned in "The Empire Strikes Back" (1980) and "Return of the Jedi" (1983). Over three decades later, Hamill returned as Luke in the "Star Wars" sequel trilogy, cameoing in "" (2015) before playing a major role in "" (2017) and "" (2019). He later played a digitally de-aged version of the character in the Disney+ series "The Mandalorian", appearing in , which premiered in 2020, and "The Book of Boba Fett", in the sixth episode, released in 2022.


title: Luke Skywalker

text: Originally a farmer on Tatooine living with his uncle and aunt, Luke becomes a pivotal figure in the Rebel Alliance's struggle against the Galactic Empire. The son of fallen Jedi Knight Anakin Skywalker (turned Sith Lord Darth Vader) and Padmé Amidala, Luke is the twin brother of Rebellion leader Princess Leia and eventual brother-in-law of the smuggler Han Solo. Luke trains to be a Jedi under Jedi Masters Obi-Wan Kenobi and Yoda and rebuilds the Jedi Order. He later trains his nephew Ben Solo and mentors Rey. Though Luke dies at the end of "The Last Jedi", he returns as a Force spirit in "The Rise of Skywalker", encouraging Rey to face her grandfather, the resurrected Emperor Palpatine. At the end of the film, the spirits of Luke and Leia give Rey their blessing to adopt the Skywalker surname and continue their family's legacy.


title: Luke Skywalker

text: The character also briefly appears in the prequel film "Episode III – Revenge of the Sith" as an infant, portrayed by Aidan Barton, and in the Disney+ series "Obi-Wan Kenobi" as a child, portrayed by Grant Feely. In the de-canonized "Star Wars" Expanded Universe (renamed "Legends"), Luke is a main character in many stories set after "Return of the Jedi", which depict him as a powerful Jedi Master, the husband of Mara Jade, father of Ben Skywalker, and maternal uncle of Jaina, Jacen and Anakin Solo.


title: Luke Skywalker

text: Series creator George Lucas considered various characterizations for the protagonist of the original "Star Wars" film. This included a 60-year-old grizzled war hero, a Jedi master, a dwarf, and a woman. In an interview about his early drafts, Lucas said:


title: Luke Skywalker

text: The first [version] talked about a princess and an old general. The second version involved a father, his son, and his daughter; the daughter was the heroine of the film. Now the daughter has become Luke, Mark Hamill's character. There was also the story of two brothers where I transformed one of them into a sister. The older brother was imprisoned, and the young sister had to rescue him and bring him back to their dad.


title: Luke Skywalker

text: Though Luke's surname was "Skywalker" in Lucas's 1973 treatment of "The Star Wars", it was changed to "Starkiller" in subsequent drafts, at one point featuring in the title ("The Star Wars: From the Adventures of Luke Starkiller"). The "Starkiller" surname remained for the first few months of production; Hamill used the name "Luke Starkiller" the sole time he referred to himself during filming, in a scene during which a stormtrooper helmet obscures his mouth; this was dubbed "Skywalker" in the film. "Starkiller" was dropped due to what Lucas called "unpleasant connotations" with Charles Manson.


title: Luke Skywalker

text: In an alternative ending pitched by Lucas for the "Star Wars" trilogy's final film, "Return of the Jedi" (1983), Luke assumed his father's role as Darth Vader after the latter's death and intending to rule in his place. Though Lawrence Kasdan favored the idea, Lucas ultimately declined, since the films were made for children. Another conclusion to the film featured the character disappearing into the wilderness akin to "Clint Eastwood in the spaghetti westerns."


title: Luke Skywalker

text: Later, George Lucas would expand the character upon the prophecy revealed in the prequel trilogy. Like his father, the Chosen One, Skywalker possessed a uniquely powerful connection to the Force and became one of the most powerful Force wielders in galactic history. Indeed, in "Empire Strikes Back", Darth Sidious expresses the belief that Skywalker was strong enough in the Force to destroy the Sith. Even George Lucas himself has stated Luke Skywalker is the most powerful Jedi in the galactic history.


title: Luke Skywalker

text: According to Mark Hamill, during the filming of the original film, Lucas asked him if he would reprise the role for a cameo when he was in his 60s to pass the torch to the next generation. In 1983, Hamill stated that his return to the franchise would be "either be another plane of existence or not the same character". Hamill learned of the sequel trilogy in mid-2012 at a "Star Wars" Celebration during lunch when Lucas told him one was in development.


title: Luke Skywalker

text: Luke's lack of screen time in "The Force Awakens" was due to concerns by screenwriter Michael Arndt that his presence would mean the audience would have less interest in protagonist Rey, leading to an agreement that he be removed from the screen and instead become a plot device. Hamill attended meetings for script readings, and helped conceal Luke's role in the film; instead of dialogue, he read stage directions. According to director and co-writer J. J. Abrams, this allowed Hamill to remain involved and his reading helped make a "better experience for everyone".


title: Luke Skywalker

text: After filming "The Force Awakens", Hamill shaved his beard to portray the Trickster in "The Flash", then let it grow back to film "The Last Jedi". According to concept designer Christian Alzmann, Luke's appearance in that film was partly inspired by that of Colonel Kurtz in "Apocalypse Now" (1979).


title: Luke Skywalker

text: In the original "Star Wars" trilogy, Luke Skywalker represents the hero archetype of "the young man, called to adventure, the hero going out facing the trials and ordeals, and coming back after his victory with a boon for the community".


title: Luke Skywalker

text: In a deleted scene preceding the character's first appearance in the film (preserved in the film's radio dramatization), Luke says goodbye to his best friend Biggs Darklighter, who has just joined the Imperial Academy. His childhood friends disparagingly call him "Wormie".


title: Luke Skywalker

text: In the original 1977 film (later having the extended title, "A New Hope"), Luke lives at a moisture farm on the desert planet of Tatooine with his uncle Owen and aunt Beru. Luke takes his first steps toward his destiny when he purchases the droids and R2-D2. While examining R2-D2, he sees a message from Princess Leia Organa of Alderaan. When R2-D2 goes missing, Luke goes out to search for the droid, and is saved from a band of Tusken Raiders by Obi-Wan Kenobi, an old hermit. Luke and Obi-Wan retreat to the latter's home, and R2-D2 plays the full message for Obi-Wan from Leia, beseeching him to help her defeat the Galactic Empire. Obi-Wan says that he and Luke's father were once Jedi Knights, and that his father was murdered by a traitorous Jedi named Darth Vader. Obi-Wan presents Luke with his father's lightsaber and offers to take him to Alderaan and train him in the ways of the Force, but Luke declines his offer, feeling obligated to his family's farm.


title: Luke Skywalker

text: Luke changes his mind when he returns home to find out that Imperial stormtroopers have raided the moisture farm, killing his aunt and uncle. He and Obi-Wan then travel to Mos Eisley, where they meet smugglers Han Solo and Chewbacca at a cantina. They team up and travel on the "Millennium Falcon" to Alderaan, only to find out that it has been destroyed by the Death Star, the Empire's battle station. The "Falcon" is brought into the Death Star's hangar bay via tractor beam, where Luke and Han disguise themselves as stormtroopers and infiltrate the station. When they discover Princess Leia is there, Luke persuades a reluctant Han and Chewbacca to help rescue her. Obi-Wan deactivates the tractor beam, and he later sacrifices his life in a duel with Vader, so that Luke and his friends can board the "Falcon" and escape.


title: Luke Skywalker

text: During the Battle of Yavin, Luke joins the Rebel Alliance in attacking the Death Star. In the trench leading to the Death Star's exhaust port, Luke hears Obi-Wan's voice, telling him to "trust his feelings"; he takes Obi-Wan's advice and switches off his X-wing's missile guidance system, instead using the Force to guide the missiles and destroy the Death Star. In the film's final scene, he joins Han and Chewbacca in receiving a medal of honor for his heroism.


title: Luke Skywalker

text: Three years after the destruction of the Death Star, Luke is now the commander of the Rebel Alliance's Rogue Squadron. While on a mission on the ice planet Hoth, he is captured by a wampa, but manages to escape using his lightsaber. In the frozen wasteland, he sees Obi-Wan's Force ghost, who tells him to travel to the planet Dagobah and complete his training with the Jedi Master Yoda. Luke collapses but is later rescued by Han. When the Empire discovers the Rebel base on Hoth, Luke leads his squadron of snowspeeders to battle the enemy's , but he is forced to retreat when his wingmen are overwhelmed. Escaping in his X-wing, he travels to Dagobah and meets Yoda. He undergoes rigorous Jedi training, quickly increasing his power in the Force.


title: Luke Skywalker

text: During his training, Luke sees a vision of his friends in danger. Against both Obi-Wan and Yoda's advice to stay and complete his training, he travels to Bespin to save them, unwittingly falling into a trap set by Vader. He engages in a lightsaber duel with Vader. As his mentors warned, Luke proves to be no match for Vader; the Sith Lord easily overpowers Luke and severs his right hand. Vader then reveals that he is Luke's father, and offers him the chance to turn to the dark side of the Force and rule the galaxy at his side. Resolving that he would rather die than become an agent of evil, Luke throws himself into a deep reactor chasm. He survives, but is pulled into a garbage chute to the underside of Cloud City, and left hanging perilously onto a vane rod. Leia, flying away from Cloud City in the "Millennium Falcon", senses Luke's call out to her with the Force, and turns the ship around to save him. Aboard the ship, he hears Vader telepathically telling him that it is his destiny to join the dark side. Luke's severed hand is replaced with a bio-mechanical one.


title: Luke Skywalker

text: One year later, Luke is a Jedi Knight, and has constructed his own lightsaber. He returns to Tatooine with Leia, Chewbacca, the droids, and Lando Calrissian to save Han, frozen in carbonite, from the crime lord Jabba the Hutt. Luke offers to negotiate with Jabba, who rejects his offer and casts him into a pit to battle a rancor. When Luke kills the rancor, he, Han, and Chewbacca are sentenced to death in the sarlacc pit. Luke escapes with R2-D2's help, saving his friends and destroying Jabba's sail barge.


title: Luke Skywalker

text: Luke returns to Dagobah, and learns from a dying Yoda that Vader is indeed his father. Luke then learns from Obi-Wan's spirit that he has a twin sister, whom he immediately realizes is princess Leia. Both Jedi Masters tell Luke that he must face Vader again to finish his training and save the galaxy, but he is disturbed by the idea of killing his own father.


title: Luke Skywalker

text: Arriving on Endor as part of a Rebel commando squad, Luke surrenders to Vader in an attempt to bring his father back from the dark side of the Force. Vader brings Luke to the second Death Star orbiting around Endor, where his master, Emperor Palpatine, tries to tempt Luke to the dark side, revealing his plan to destroy the Rebel fleet. Luke lashes out at the Emperor with his lightsaber, but Vader blocks his strike, and father and son once again duel with each other. Luke regains control of his emotions until Vader senses that Luke has a sister, and threatens to turn her to the dark side if Luke will not submit. Luke snaps and overpowers Vader, severing his father's mechanical right hand. The Emperor orders Luke to kill Vader and take his place. Luke looks at his own bionic hand and realizes that he is on the verge of suffering his father's fate. He casts his lightsaber aside, declaring himself a Jedi.


title: Luke Skywalker

text: Furious, the Emperor tortures Luke with Force lightning. In agony, Luke calls out to his father for help; unwilling to let his son die, Vader throws the Emperor down a reactor shaft to his death, but is mortally wounded by the Emperor's Force lightning in the process. As Rebel fighters head toward the second Death Star's main reactor, Luke removes Vader's mask and looks upon his father's real face for the first and last time. The redeemed Anakin Skywalker reassures Luke that there was good in him after all, and to tell his sister that he was right about him, before dying peacefully. On Endor, Luke burns his father's body on a funeral pyre, giving him a proper Jedi funeral. During the Rebels' victory celebrations on Endor, Luke sees his father's spirit appear alongside those of Obi-Wan and Yoda.


title: Luke Skywalker

text: In the prequel film "Revenge of the Sith" (2005), during the waning days of the Clone Wars between the Galactic Republic and the Separatists, Senator Padmé Amidala, wife of Jedi Knight Anakin Skywalker, becomes pregnant with Luke and Leia. Anakin has a prophetic vision of Padmé dying in childbirth, and allows Chancellor Palpatine – secretly the Sith Lord Darth Sidious – to corrupt him to the dark side as Darth Vader.


title: Luke Skywalker

text: After Vader uses the dark side to choke her, Padmé is taken to Polis Massa, where she gives birth to Luke and Leia and dies, having lost the will to live after Vader's betrayal. Obi-Wan and Yoda agree to separate the twins in order to protect them from the Sith and the newly created Galactic Empire. Obi-Wan takes Luke to the desert planet Tatooine, where he is adopted by Vader's stepbrother, Owen Lars, and his wife, Beru, while Leia is adopted by Senator Bail Organa of Alderaan. The infant Luke is portrayed by Aidan Barton, the son of Roger Barton, an editor of the film.


title: Luke Skywalker

text: In the first installment of the sequel trilogy, "The Force Awakens" (2015), the opening crawl reveals that Luke Skywalker had mysteriously vanished some time in the 30 years after the destruction of the second Death Star. Luke went into hiding after his nephew and apprentice, Ben Solo, turned to the dark side and became Kylo Ren, chief enforcer of the tyrannical First Order and its leader, Snoke. When Ren killed all of his fellow apprentices and ushered in the fall of the New Republic, Luke felt responsible, and disappeared. At the end of the film, the Resistance, led by Leia, manage to reconstruct a map, which traces the location of the temple from the Empire's archives to his location, and he is subsequently found on the planet Ahch-To by the young scavenger, Rey, who presents him with the lightsaber previously wielded by both Luke and his father.


title: Luke Skywalker

text: "The Art of Star Wars: The Last Jedi" notes that Luke's exile is a reversal from his decision to help his friends in "The Empire Strikes Back".


title: Luke Skywalker

text: In "The Last Jedi" (2017), Luke throws the lightsaber away after Rey gives it to him. He then seals himself in his home, refusing to talk to Rey. However, when he sees Chewbacca without Han and that Rey came on the "Millennium Falcon", he inquires about what has gone on with his family. Rey informs Luke about Solo's death at the hands of Kylo Ren and that the First Order has risen to rule the galaxy. Rey asks Luke to train her in the ways of the Force. Luke is initially reluctant to train Rey, telling her that it is time for the Jedi Order to end.


title: Luke Skywalker

text: After some persuasion from R2-D2, however, Luke starts training Rey, but grows increasingly afraid of her power. Luke then tells Rey the history of the Jedi Order, how Darth Sidious rose to power and how the Jedi were partly responsible for his father's fall to the dark side. Luke also initially tells Rey that Ben was corrupted by Snoke, and destroyed the Jedi Temple he confronted his treasonous nephew. Ultimately, however, Luke tells Rey the truth: He had briefly considered killing the sleeping Ben after seeing a vision of the destruction he could cause, but immediately relented; Ben woke to see Luke with his lightsaber drawn and turned to the dark side because he felt betrayed. Upon learning this, Rey urges Luke to help her redeem Ben, but he refuses to participate in this nor help the Resistance. When Rey leaves, an embittered Luke tries to burn the Jedi temple, but fails. He is then visited by the spirit of Yoda, who assures Luke that he still has a purpose.


title: Luke Skywalker

text: Luke appears on the planet Crait, as the Resistance are staging a standoff against the First Order, and he apologizes to Leia for allowing Ben to fall to the dark side. Luke steps in front of the First Order walkers, and unexpectedly survives an onslaught of blaster fire ordered by Ren. Ren charges at Luke in hand-to-hand combat, seemingly bisecting him with his lightsaber, but Luke remains unscathed; still on Ahch-To, Luke has sent a projection of himself to Crait, using the Force. This distraction allows the Resistance to escape the planet. Luke tells Kylo that he will not be the last Jedi before his projection disappears. On Ahch-To, Luke collapses then looks off in the horizon to see the planet's two suns setting before he becomes one with the Force and vanishes.


title: Luke Skywalker

text: Luke appears briefly in "", the ninth and final chapter of the main series. When Rey discovers her lineage as the resurrected Palpatine's granddaughter, she exiles herself on Ahch-To, just as Luke had. Meaning to isolate herself, she throws Luke's lightsaber towards the burning wreckage of Ren's TIE whisper, in which she had traveled to her new home. Luke appears as a Force spirit, and reprimands Rey for treating the lightsaber with disrespect. Luke then admits that he was wrong not to participate in the Resistance, and thanks Rey for helping him rediscover himself. Luke convinces Rey to not view herself as a Palpatine, but as the good person Leia saw her as when training her, and not to give up on her battle against the Sith. Luke lends Rey his old T-65B X-wing "Red Five" and tells Rey to combine his father's lightsaber with Leia's in her battle on Exegol. He also restrengthened Rey when she reached out to and heard the voices of the past Jedi to help her destroyed Palpatine once and for all.


title: Luke Skywalker

text: After Rey vanquishes Palpatine once and for all, she visits the moisture farm on Tatooine where Luke was raised. She buries Luke and Leia's lightsabers, revealing that she has constructed her own. A local asks Rey who she is. As Leia, who died earlier, and Luke appear as Force spirits, Rey names herself Rey Skywalker.


title: Luke Skywalker

text: At the end of the "Star Wars Rebels" episode "Twin Suns", the silhouette of a young Luke Skywalker is seen from afar by Obi-Wan Kenobi.


title: Luke Skywalker

text: Luke appears in the animated Disney micro-series "Star Wars Forces of Destiny" (with Hamill reprising his role). The episode "The Path Ahead" details him training with Yoda on Dagobah. He also appears in the episode "Traps and Tribulations", which takes place shortly after the Battle of Endor and shows him and Leia assisting the Ewoks in stopping a rampaging monster known as a Gorax.


title: Luke Skywalker

text: Luke appears in "", the season two finale of "The Mandalorian", after Grogu contacted him via the Force in . Luke arrives on Moff Gideon's cruiser and destroys all of the Dark Troopers, saving Grogu and his guardian, Din Djarin, as well as their companions. Luke then takes Grogu with him to train him as a Jedi, with Djarin's permission.


title: Luke Skywalker

text: Luke appears in the sixth episode, "Chapter 6: From the Desert Comes a Stranger", of the spin-off series "The Book of Boba Fett". While training Grogu, he helps him remember some of his past, including his home at the Jedi Temple on Coruscant and the events of the Great Jedi Purge. Djarin comes to visit Grogu, but decides against it after speaking with Ahsoka Tano, not wanting to hinder his training; however, he gives Ahsoka a gift to deliver to Grogu: beskar chain mail forged by the Armorer. Ahsoka gives the chain mail to Luke, who confesses that he is unsure whether Grogu is fully committed to the Jedi path and that he does not know how to handle the matter. Following Ahsoka's advice to listen to his instincts, Luke decides to let Grogu choose his own destiny by asking him to choose between the chain mail and the lightsaber of his old master, Yoda. In the seventh episode, "Chapter 7: In The Name of Honour", it is revealed that Grogu chose the chain mail, leading Luke to send him back to Djarin. He places him in his X-wing starfighter, which is then flown to Tatooine by R2-D2, where Grogu and Djarin are eventually reunited.


title: Luke Skywalker

text: Luke is a playable character in "Star Wars Battlefront II", and in the online multiplayer of "Star Wars Battlefront".


title: Luke Skywalker

text: An older and wiser Luke Skywalker also appears in the "" and "" video games at the rank of Jedi Master. In "Jedi Outcast", Luke helps Kyle Katarn in his fight against Desann and Empire Reborn by driving them away from the Valley of the Jedi. Luke appears in the game "" in a downloadable alternate storyline where Starkiller duels Luke. He also appears in the Disney Lucasfilm video game "Star Wars Commander". Luke is also a playable character in "Disney Infinity 3.0", and a playable character to unlock for a limited time in "Disney Magic Kingdoms".


title: Luke Skywalker

text: Luke is the main character of the novel "Heir to the Jedi" and the junior novel "The Legends of Luke Skywalker" (the latter of which was adapted as a manga). He is also a main character of the 2015 comic "Star Wars", which takes place between the films of the original trilogy.


title: Luke Skywalker

text: Star Wars: Heir to the Jedi was announced as one of the first four canon novels to be released in 2014 and 2015. Set between "A New Hope" and "The Empire Strikes Back", "Heir to the Jedi" chronicles the adventures of Luke as he continues to battle the Empire with his Rebel friends, grows close with fellow Rebel Nakari Kelen, and begins to develop his Force abilities. The novel is written from the first-person perspective of Luke, and is only the second "Star Wars" novel to attempt this type of narrative voice.


title: Luke Skywalker

text: Star Wars: Shadow of the Sith is a forthcoming novel set between "Return of the Jedi" and "The Force Awakens". Set to be released on June 28, 2022, the book follows Luke and Lando Calrissian on a mission to locate Exegol.


title: Luke Skywalker

text: In April 2014, most of the licensed "Star Wars" novels and comics produced since the originating 1977 film "Star Wars" were rebranded by Lucasfilm as "Star Wars" Legends and declared non-canon to the franchise. The Legends-branded novels, comic books and video games detail Luke's exploits following "Return of the Jedi".


title: Luke Skywalker

text: In the novel "The Truce at Bakura", set one day after the battle of Endor, Luke and his friend Wedge Antilles recover a message droid from the titular planet, which was being invaded by the Ssi-Ruuk. Luke commands a task force, turning back the enemy army. He also meets Dev Sibwarra, a Force-sensitive human who had been captured by the Ssi-Ruuk, who is killed in the battle after turning against his captors.