Primary mathematics/Fractions - Wikiversity Primary mathematics/Fractions < Wikiversity < Wikiversity:School of Mathematics < School of Mathematics:Primary School Mathematics In the early primary grades, fractions are generally thought of as "a part" of a whole, and are used to express numbers less that one. Yet, fractions encompass much more mathematical territory than this. They can of course be larger than one. Rational numbers are by definition fractions.. in essence the same as division. Fractions are also used to express ratios. By the end of the primary grades, students should be making the connections between all of these possible meanings of for instance, the expression 5 Methodology[ edit ] Generally speaking, the progression used when teaching fractions starts with the use of manipulatives and other visual representations that make connections to real life things: pizza, money, candy, etc. As children begin to be able to articulate these understandings both verbally and with mathematical expressions, more abstract models such as area rectangles, fraction bars and grids are introduced, permitting students to expand their understandings in more abstract ways.

The Use of Manipulatives to Teach Fractions[ edit ] Candidates for manipulatives used to teach fractions are single objects or groups of the same object that can be divided equally. There are a number of commercially available "toys" that are designed just for this purpose, but these are not necessary for a resourceful teacher. Regardless, students should initially be making connections to things they "know", like pizza and candy. Origami[ edit ] Oragami used to teach fractions By simply taking a square piece of paper and folding it in half; write the fraction 1 ; fold in half again; write 1 4 {\displaystyle {\tfrac {1}{4}}} . Continue this process to create smaller fractions. Unfold the paper and trace lines where there are folds in the paper. Notice that in the case of the example, the manipulative (the folding paper) becomes a model (a more abstract visual representation) as students draw over the folds and fill in the values. Money[ edit ] As we all know, money is a great motivator (for any age). It is something all children are familiar and comfortable with, and perhaps most importantly - they want it!

This doesn't mean you have to give it to them, but you can use it to leverage their interest. The use of money as a manipulative/model to teach fractions (and many other concepts in mathematics) has many advantages. Thankfully, most all currencies use a base ten system. Using change, the US dollar can be broken up into fourths (quarters), tenths, twentieths and hundredths, which makes it a perfect model to use for teaching decimals and even percentages as well. Fraction Bars[ edit ] From Manipulatives to Models[ edit ] While manipulatives offer a tactile and visual way for students to learn about fractions, models let them take their exploration to a higher level of abstraction, while still maintaining a visual aspect to their learning. "Area rectangle" modeling the fraction 3 Area Diagrams[ edit ] "Area diagrams" or "area rectangles" are perhaps one of the best ways for children to model fractions. They have the advantage of visually showing "how much" of the whole a fraction takes up, and will become an indespensible tool students will eventually use for modeling operations with fractions. Students should be encouraged to create their own area diagrams to reinforce their learning.

"100 Grid" modeling 10 {\displaystyle {\tfrac {2}{10}}} "100 grids" are a type of area diagram commonly used to teach fractions. Because the fractions that 100 grids model at the single unit level are hundreths, they are usually introduced in the upper primary grades. Similar to the way that area rectangles can be used to model multiplication, 100 grids are a great way to model fractions and simultaneously facilitate students in making important connections to percentages and decimals. Be sure to have students demonstrate that they understand these connections by filling in the 100 grid and then writing down the equivalent expressions. For instance: = {\displaystyle {\frac {3}{5}}\times {\frac {7}{7}}=} shouldn't change it's value, but this is very difficult for many students to understand because of the level of abstraction. Models and visual representations become necessary. One very effective visual representation we can use is the Giant One. Students also need to be reminded of the fact that anything times 1 doesn't change its value (The Identity Property of Multiplication).

The Giant One can be used to set up the equation, which helps students to visualize a way to find equivalent fractions. Students should draw these themselves to reinforce their understanding of the concept. Even students with poor motor skills can draw Giant Ones; it doesn't matter how neatly they are drawn: Using Area Rectangles to Create Equivalent Fractions[ edit ] As we have seen earlier , area rectangles have the advantage of visually showing students "how much" a fraction represents. And because it is possible to show a fraction using either horizontal or vertical lines, by using both horizontal and vertical lines it is possible to simultaneously show two superimposed fractions. Note how this exercise is executed almost in the same fashion as multiplying fractions , except that the second fraction, because it is equivalent to 1 is not filled in. The super-imposition of thirds and forths creates a total of 12 equal rectangles, or 12ths, and 9 filled in rectangles. Students should become adept at creating these. Students eventually should be able to look at the image on the right and know what equivalent fractions are being modeled.

The same thing should be modeled with a Giant One problem to reinforce the connection : Basic Operations[ edit ] The more standard algorithms that most of us use to add, subtract, and multiply fractions, are to a child's perspective, a collection of rules, which at best "work", but at worst become all mixed up in students' minds. This is usually because these 'rules' have no underpinnings in the students' web of mathematical knowledge. In other words, if students don't understand why the "rules" for using these methods exist in the first place, it becomes very difficult to apply them at the right time, not to mention execute them consistently. For this reason, this "traditional" approach (just teaching them how to do things, but not why they are doing them the way they are) to teaching operations with fractions requires a "drill and kill" methodology. Thankfully, the contemporary approach to teaching these methods has students explore these concepts in many different ways at a fundamental level. Adding Fractions[ edit ] In the lowest grades, it may be best to only add fractions with a common denominator.

This can be explained as "you can't add apples and oranges together", or, since those could be added together to get the total pieces of fruit, you could use an example of adding items which are even more different, like weeks and sheets of paper. For example, "if you have 2 weeks and 10 pieces of paper to make an art project, you don't add them together to get 12". In this analogy, the denominator of the fractions describes the type of fractions they are, which determines if they can be added. In later grades, students can convert one or both fractions to have a common denominator, before adding. This should wait until after multiplication of fractions has been taught, as this is a necessary step in finding a common denominator. Subtracting Fractions[ edit ] The same logic applies to teaching subtraction of fractions as addition, with one additional element; it is now possible to get a negative result. This can be explained with real world objects and "owing". For example, "if you said you would give your brother half a cookie, but only had a quarter cookie to give him, you still owe him another quarter cookie. This is the same as saying you now have a negative quarter cookie".

Most students when first confronted with the problem 2 {\displaystyle 5\times 6=30} because five groups of six is thirty. Asking them "what 1 4 {\displaystyle {\tfrac {1}{4}}} of 100 is" helps them to better understand why multiplying with fractions changes their earlier understanding. As they begin to understand the implications of this they can then be introduced to the concept using two fractions. I.e., what is 1/2 of 1/2? At this point models should be used to reinforce their understanding of this. Area models can be very effective towards this end, where the area of the intersecting fractions represents the product. Students should be both asked to create these models from word problems as well as create word problems that fit with the model. For instance, 2/3 of the students in the class are wearing yellow shirts. 1/4 of the class goes to the library. What fraction of the class is wearing yellow shirts in the library? or 2/3 of my garden was planted with yellow flowers, and 1/4 of all of the flowers were sold. What fraction of all of the flowers sold from the garden were probably yellow?

Note that in this case the wording of the problem has created a lesson with a connection to probability . The image below models the problem 4 Dividing Fractions[ edit ] Teaching students how to divide with fractions by simply showing them how to multiply the dividend and the reciprocal of the divisor offers practically no insight to their mathematical understanding.. because no connections have been made to what they already know about division. Before students learn to divide whole numbers, they need to understand how to multiply them. The same concept applies with fractions. Students should first explore division of whole numbers by fractions at a basic level. Models should be used whenever possible. For instance, the problem "How many 1/2s are there in 4? can be modeled with any graphic representation, such as.

The next level of difficulty involves two numbers with fractions, such as can be found in the problem, "How many 2 3 {\displaystyle {\tfrac {4}{5}}\div {\tfrac {2}{3}}} Division with two fractions requires the use of the reciprical of the divisor of the problem to be inserted into the Giant One: Eventually, students who use this method recognize that the top fracton of the problem is being multiplied by the reciprical of the bottom fraction. The process of discovering this "trick" for themselves gives them ownership of the larger conceptual understanding. In other words, students who learn to divide fractions in this way understand at a fundamental level "why" the trick works Fractions that have the same or "Common" denominator are called "Like" fractions. 1/3, 2/3 To add Like fractions together such as these: 1/3 + 2/3 = ? (one-third plus two-thirds equals what?) 1. Add the numerators (the top numbers): 1 + 2 = 3 (one plus two equals three) 2.

Use the common denominator (the bottom numbers): /3 All together it looks like this: 1/3 + 2/3 = 3/3 (one-third plus two-thirds equals three-thirds) 3. Then simplify the answer as much as you can by dividing the denominator into the numerator. 3/3 = 1 (three divided by three equals one) If the numerator is now larger than the denominator it might look like this: 4/5 + 3/5 = 7/5 (four-fifths plus three-fifths equals seven-fifths) simplified: 5 goes into 7 one time, with 2 left over. Put the remainder (2) over the denominator (5). 7/5 = 1 2/5 (seven-fifths equals one and two-fifths) To Add or Subtract fractions with different denominators, you must first convert them to equivalent fractions with common denominators.

FRACTION MATHEMATICS : definition of FRACTION MATHEMATICS and synonyms of FRACTION MATHEMATICS (English) Forms of fractions Common, vulgar, or simple fractions A common fraction (also known as a vulgar fraction or simple fraction) is a rational number written as a/b or , where the integers a and b are called the numerator and the denominator, respectively. [2] The numerator represents a number of equal parts and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. In the examples 2/5 and 7/3, the slanting line is called a solidus or forward slash . In the examples and , the horizontal line is called a vinculum or, informally, a "fraction bar." Writing simple fractions In computer displays and typography , simple fractions are sometimes printed as a single character, e.g. ½ ( one half ). Scientific publishing distinguishes four ways to set fractions, together with guidelines on use: [3] case fractions: , while large and legible, these can be disruptive, particularly for simple fractions or within complex fractions. Ratios A ratio is relationship between two or more numbers that can be sometimes expressed as a fraction.

Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group n". For example, if a car lot had 12 vehicles of which 2 are white, 6 are red, and 4 are yellow, the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1. A ratio may be typically converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to the total cars in the lot is 4:12 or 1:3. We can convert these ratios to a fraction and say that 4/12 of the cars or 1/3 of the cars in the lot are yellow. Proper and improper common fractions Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise.

[4] [5] In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is, if the fraction is between -1 and 1 (but not equal to -1 or 1). [6] [7] It is said to be an improper fraction (U.S., British or Australian) or top-heavy fraction (British, occasionally North America) if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, -3/4, and 4/9; examples of improper fractions are 9/4, -4/3, and 8/3. Mixed numbers A mixed numeral (often called a mixed number, also called a mixed fraction) is the sum of a non-zero integer and a proper fraction. This sum is implied without the use of any visible operator such as "+". For example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: . This is not to be confused with the algebra rule of implied multiplication.

When two algebraic expressions are written next to each other, the operation of multiplication is said to be "understood". In algebra, for example is not a mixed number. Instead, multiplication is understood: . An improper fraction is another way to write a whole plus a part. A mixed number can be converted to an improper fraction as follows: Write the mixed number Similarly, an improper fraction can be converted to a mixed number as follows: Divide the numerator by the denominator. In the example, , divide 11 by 4. 11 ÷ 4 = 2 with remainder 3. The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part. The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4. Thus . Mixed numbers can also be negative, as in , which equals . Reciprocals and the "invisible denominator" The reciprocal of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of , for instance, is .

The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as , where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer except for zero has a reciprocal. The reciprocal of 17 is . Complex fractions In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number [8] [9] , corresponding to division of fractions. For example, and are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example: If, in a complex fraction, there is no clear way to tell which fraction line takes precedence, then the expression is improperly formed, and meaningless. Compound fractions A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of [8] [9] , corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on multiplication ). For example, of is a compound fraction, corresponding to .

The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. Decimal fractions and percentages A decimal fraction is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digits to the right of a decimal separator , the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator ). Thus for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, viz. 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, .

Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023×10−7, which represents 0.0000006023. The 10−7 represents a denominator of 107. Dividing by 107 moves the decimal point 7 places to the left. Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series . For example, 1/3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... . Another kind of fraction is the percentage (Latin per centum meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 75% means 75/100. Related concepts are the permille , with 1000 as implied denominator, and the more general parts-per notation , as in 75 parts per million, meaning that the proportion is 75/1,000,000.

Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By mental calculation , it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example 3/6 (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to the fraction 3/6. Special cases A unit fraction is a vulgar fraction with a numerator of 1, e.g. .

Unit fractions can also be expressed using negative exponents, as in 2−1 which represents 1/2, and 2−2 which represents 1/(22) or 1/4. An Egyptian fraction is the sum of distinct positive unit fractions, for example . This definition derives from the fact that the ancient Egyptians expressed all fractions except , and and . A dyadic fraction is a vulgar fraction in which the denominator is a power of two , e.g. . Arithmetic with fractions Like whole numbers, fractions obey the commutative , associative , and distributive laws, and the rule against division by zero . Equivalent fractions Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number , the fraction . Therefore, multiplying by is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction . When the numerator and denominator are both multiplied by 2, the result is , which has the same value (0.5) as .

To picture this visually, imagine cutting a cake into four pieces; two of the pieces together ( ) make up half the cake ( ). Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A simple fraction in which the numerator and denominator are coprime [that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is said to be irreducible , in lowest terms, or in simplest terms. For example, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1. Using these rules, we can show that = = . A common fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor . For example, as the greatest common divisor of 63 and 462 is 21, the fraction can be reduced to lowest terms by dividing the numerator and denominator by 21: The Euclidean algorithm gives a method for finding the greatest common divisor of any two positive integers.

Comparing fractions Comparing fractions with the same denominator only requires comparing the numerators. because 3>2. If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger. One way to compare fractions with different numerators and denominators is to find a common denominator. To compare and . Then bd is a common denominator and the numerators ad and bc can be compared. ? gives It is not necessary to determine the value of the common denominator to compare fractions. This short cut is known as "cross multiplying" – you can just compare ad and bc, without computing the denominator. ? Multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator: ? The denominators are now the same, but it is not necessary to calculate their value – only the numerators need to be compared.

Since 5×17 (= 85) is greater than 4×18 (= 72), . Also note that every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, so every negative fraction is less than any positive fraction. Addition The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows: . If of a cake is to be added to of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters. Adding unlike quantities To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction.

For adding quarters to thirds, both types of fraction are converted to (twelfths). Consider adding the following two quantities: First, convert into twelfths by multiplying both the numerator and denominator by three: . Note that is equivalent to 1, which shows that is equivalent to the resulting . into twelfths by multiplying both the numerator and denominator by four: . Note that is equivalent to 1, which shows that is equivalent to the resulting . Now it can be seen that: is equivalent to: This method can be expressed algebraically: And for expressions consisting of the addition of three fractions: This method always works, but sometimes there is a smaller denominator that can be used (a least common denominator). For example, to add and the denominator 48 can be used (the product of 4 and 12), but the smaller denominator 12 may also be used, being the least common multiple of 4 and 12. Subtraction The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator.

The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance, Multiplication Multiplying a fraction by another fraction To multiply fractions, multiply the numerators and multiply the denominators. Thus: Why does this work? First, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths. A short cut for multiplying fractions is called "cancellation". In effect, we reduce the answer to lowest terms during multiplication.

For example: A two is a common factor in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both. Multiplying a fraction by a whole number Place the whole number over one and multiply. This method works because the fraction 6/1 means six equal parts, each one of which is a whole. Mixed numbers When multiplying mixed numbers, it's best to convert the mixed number into an improper fraction. For example: In other words, is the same as , making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is , since 8 cakes, each made of quarters, is 32 quarters in total. Division To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, equals and also equals , which reduces to . To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, .

Converting between decimals and fractions To change a common fraction to a decimal, divide the denominator into the numerator. Round the answer to the desired accuracy. For example, to change 1/4 to a decimal, divide 4 into 1.00, to obtain 0.25. To change 1/3 to a decimal, divide 3 into 1.0000..., and stop when the desired accuracy is obtained. Note that 1/4 can be written exactly with two decimal digits, while 1/3 cannot be written exactly with any finite number of decimal digits. To change a decimal to a fraction, write in the denominator a 1 followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits in the original decimal, omitting the decimal point. Thus 12.3456 = 123456/10000. Converting repeating decimals to fractions See also: Repeating decimal Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite number of repeating decimals is required to convey the same kind of precision.

Thus, it is often useful to convert repeating decimals into fractions. The preferred way to indicate a repeating decimal is to place a bar over the digits that repeat, for example 0.789 = 0.789789789… For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice.

For example: 0.5 = 5/9 0.264 = 264/999 0.6291 = 6291/9999 In case leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros : 0.05 = 5/90 0.000392 = 392/999000 0.0012 = 12/9900 In case a non-repeating set of decimals precede the pattern (such as 0.1523987), we can write it as the sum of the non-repeating and repeating parts, respectively: 0.1523 + 0.0000987 Then, convert the repeating part to a fraction: 0.1523 + 987/9990000 Fractions in abstract mathematics In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable .

Mathematicians define a fraction as an ordered pair (a, b) of integers a and b ≠ 0, for which the operations addition , subtraction , multiplication , and division are defined as follows: [10] (when c ≠ 0) In addition, an equivalence relation is specified as follows: ~ if and only if . These definitions agree in every case with the definitions given above; only the notation is different. More generally, a and b may be elements of any integral domain R, in which case a fraction is an element of the field of fractions of R. For example, when a and b are polynomials in one indeterminate, the field of fractions is the field of rational fractions (also known as the field of rational functions ). When a and b are integers, the field of fractions is the field of rational numbers . Algebraic fractions Main article: Algebraic fraction An algebraic fraction is the indicated quotient of two algebraic expressions . Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions. If the numerator and the denominator are polynomials , as in , the algebraic fraction is called a rational fraction (or rational expression).

An irrational fraction is one that contains the variable under a fractional exponent, as in . The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factor common to the numerator and the denominator is 1. An algebraic fraction whose numerator or denominator, or both, contains a fraction, such as , is called a complex fraction. Rational numbers are the quotient field of integers. Rational expressions are the quotient field of the polynomials (over some integral domain ). Since a coefficient is a polynomial of degree zero, a radical expression such as √2/2 is a rational fraction. Another example (over the reals) is , the radian measure of a right angle. The term partial fraction is used when decomposing rational expressions. The goal is to write the rational expression as the sum of other rational expressions with denominators of lesser degree. For example, the rational expression can be rewritten as the sum of two fractions: and . This is useful in many areas such as integral calculus and differential equations. Radical expressions Main articles: Nth root and Rationalization (mathematics) A fraction may also contain radicals in the numerator and/or the denominator.

If the denominator contains radicals, it can be helpful to rationalize it (compare Simplified form of a radical expression ), especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is also more convenient if division is to be done manually. When the denominator is a monomial square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator: The process of rationalization of binomial denominators involves multiplying the top and the bottom of a fraction by the conjugate of the denominator so that the denominator becomes a rational number. For example: Even if this process results in the numerator being irrational, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator. Pronunciation and spelling This section may contain original research . Please improve it by verifying the claims made and adding references . Statements consisting only of original research may be removed.

(March 2012) Further information: Ordinal number (linguistics) When reading fractions, it is customary in English to pronounce the denominator using ordinal nomenclature, as in "fifths" for fractions with a 5 in the denominator. Thus, for 3/5, we would say "three-fifths" and for 5/32, we would say "five thirty-seconds". This generally applies to whole number denominators greater than 2, though large denominators that are not powers of ten are often read using the cardinal number . Therefore, 1/123 might be read "one one hundred twenty-third" but is often read "one over one hundred twenty-three". In contrast, because one million is a power of ten, 1/1,000,000 is commonly read "one-millionth" or "one one-millionth". The denominators 1, 2, and 4 are special cases. The fraction 3/1 may be read "three wholes". The fraction 3/2 is usually read "three-halves", but never "three seconds".

The fraction 3/4 may be read either "three fourths" or "three-quarters". Furthermore, since most fractions are used grammatically as adjectives of a noun, the fractional modifier is hyphenated. This is evident in standard prose in which one might write about "every two-tenths of a mile", "the quarter-mile run", or the Three-Fifths Compromise . When the fraction's numerator is one, then the word "one" may be omitted, such as "every tenth of a second" or "during the final quarter of the year". Denominator See also: History of irrational numbers The earliest fractions were reciprocals of integers : ancient symbols representing one part of two, one part of three, one part of four, and so on. [11] The Egyptians used Egyptian fractions ca. 1000 BC. About 4,000 years ago Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions . Their methods gave the same answer as modern methods. [12] The Egyptians also had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems.

The Greeks used unit fractions and later continued fractions and followers of the Greek philosopher Pythagoras , ca. 530 BC, discovered that the square root of two cannot be expressed as a fraction. In 150 BC Jain mathematicians in India wrote the " Sthananga Sutra ", which contains work on the theory of numbers, arithmetical operations, operations with fractions . The method of putting one number below the other and computing fractions first appeared in Aryabhatta 's work around 499 CE.[ citation needed ] In Sanskrit literature , fractions , or rational numbers were always expressed by an integer followed by a fraction. When the integer is written on a line, the fraction is placed below it and is itself written on two lines, the numerator called amsa part on the first line, the denominator called cheda “divisor” on the second below. If the fraction is written without any particular additional sign, one understands that it is added to the integer above it. If it is marked by a small circle or a cross (the shape of the “plus” sign in the West) placed on its right, one understands that it is subtracted from the integer.

For example, Bhaskara I writes [13] ६ १ २ 4 5 9 to denote 6+1/4, 1+1/5, and 2–1/9 Al-Hassār , a Muslim mathematician from Fez , Morocco specializing in Islamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, ." [14] This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century. [15] In discussing the origins of decimal fractions , Dirk Jan Struik states that (p.

7): [16] "The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548-1620), then settled in the Northern Netherlands . It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century). [17] " While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.

[18] [19] Pedagogical tools In primary schools , fractions have been demonstrated through Cuisenaire rods , fraction bars , fraction strips, fraction circles, paper (for folding or cutting), pattern blocks , pie-shaped pieces, plastic rectangles, grid paper, dot paper , geoboards , counters and computer software. See also Continued fraction References ^ H. Wu, The Mis-Education of Mathematics Teachers, Notices of the American Mathematical Society, Volume 58, Issue 03 (March 2011), page 374

Question: What is the reciprocal of one-and-two-thirds (in other words, what fraction when multiplied by one-and-two-thirds produces one)?
Answer: Three-fifths