Complex numbers were discovered while attempting to solve special equations that have exponents in them. These began to pose real problems for mathematicians. As a comparison, using negative numbers, it is possible to find the "x" in the equation formula_5 for all real values of "a" and "b", but if only positive numbers are allowed for "x", it is sometimes impossible to find a positive "x", as in the equation .
With exponentiation, there is a difficulty to be overcome. There is no real number that gives −1 when it is squared. In other words, −1 (or any other negative number) has no real square root. For example, there is no real number formula_6 that solves the equation formula_7. To solve this problem, mathematicians introduced a symbol "i" and called it the "imaginary unit". This is the imaginary number that will give −1 when it is squared.
The first mathematicians to have thought of this were probably Gerolamo Cardano and Raffaele Bombelli. They lived in the 16th century. It was probably Leonhard Euler who introduced writing formula_8 for that number.
All complex numbers can be written as formula_9 (or formula_10), where "a" is called the "real part" of the number, and "b" is called the "imaginary part". We write formula_11 or formula_12 for the real part of a complex number formula_13. So, if formula_14, we write formula_15. Similarly, we write formula_16 or formula_17 for the imaginary part of a complex number formula_13; formula_19, for the same . Every real number is also a complex number; it is a complex number with formula_20.
A complex number can also be written as an ordered pair , where both "a" and "b" are real numbers. Any real number can simply be written as formula_21, or as the pair .
Sometimes, formula_22 is written instead of formula_1. In electrical engineering for instance, formula_1 means electric current, so writing formula_1 can cause a lot of problems because some numbers in electrical engineering are complex numbers.
The set of all complex numbers is usually written as formula_4.
Addition, subtraction, multiplication and exponentiation (raising numbers to exponents) are all possible with complex numbers. Division is also possible with complex numbers—as long as the divisor is not zero, Some other calculations are also possible with complex numbers.
The rule for addition and subtraction of complex numbers is pretty simple:
Let formula_27, then formula_28, and formula_29.
Multiplication is a bit different:
formula_30
Another notable operation for complex numbers is "conjugation". A complex conjugate formula_31 to formula_14 is formula_33.
It is pretty simple, but is important for calculations, because formula_34 is actually a real number for "all" complex formula_13:
formula_36.
Because of that, we can use it to do division:
formula_37
formula_38
Complex numbers can be shown on a so-called complex plane. If you have a number formula_14, you can go to point "a" on the real axis and point "b" on the imaginary axis, and draw a vector from formula_40 to formula_41. The length of this vector can be calculated using the Pythagorean theorem, and the angle of this vector is simply the angle between the positive real axis and this vector—going counterclockwise. The length of a vector for a number formula_13 is called its modulus or absolute value (written as formula_43), and the angle is called its argument (formula_44).
This leads to the trigonometrical form of describing complex numbers: by the definitions of sine and cosine, it follows that for all formula_13:
formula_46
This is closely connected to De Moivre's formula.
There exists even another form, called exponential form.
With the introduction of complex numbers to math, every polynomial with complex coefficients has roots in complex numbers. This introduction also helped to open a path to the creation of another kind of numbers, which could help resolve and explain many different problems. These include the , , , and many others. For more, see .
Frederick I of Prussia
Frederick I of Prussia (), (11 July 1657–25 February 1713) of the Hohenzollern dynasty was the first King in Prussia (1701–1713).
Siege of Jerusalem (1187)
The Siege of Jerusalem took place from September 20 to October 2, 1187. It was part of the war called the Third Crusade.
Balian of Ibelin defended Jerusalem against the armies of Saladin. When he surrendered, the Muslims enslaved thousands of Christians but let many leave after they bought their freedom.
A fictionalized version of the siege (attack) is seen in the movie "Kingdom of Heaven", directed by Ridley Scott.
Beer Hall Putsch
The Beer Hall Putsch was a failed coup d'état in Munich, Germany between the evening of 8 November and the morning of 9 November 1923. The Nazi Party under Adolf Hitler tried to take over the government of Bavaria, a state in Germany. They began trying to do this by taking several hundred people hostage in a beer hall (a large building where people got together to drink beer and sometimes listen to political speeches). Several Nazis died when they got into a gun fight with police officers in another part of Munich, and Hitler then went to prison for eight months.
Hitler chose to do this for many reasons. The Weimar Republic of Germany was weak because of fighting between right and left-wing parties. Inflation had also left the Weimar Republic with very little support from the people, or money to keep working (paying bills, or even paying pensions and wages). This happened because when Germany lost World War I, other countries made Germany pay for the entire war, and Germany had to make taxes very high so it could pay for the war. This hurt the economy of Germany.
The Nazi Party was getting very strong. Hitler was a strong leader. In 1923, the year when the Weimar Republic was least popular, the Nazi Party grew from 6,000 members to 55,000 members.
Hitler also wanted to copy Mussolini, who had taken power in Italy by marching into Rome.
8 November 1923 - Hitler and his men charged into the Beer Hall and forced the people there to support him. Hitler's men, the SA took over the Army Headquarters, but forgot to take over the telegraph office. This meant that anybody could call in the police.
9 November 1923 - 16 Nazis and 4 Bavarian State Police officers were killed in a gunfight between the Nazis and the police in "Residenzstraße". Hitler was arrested two days later.
Hitler and several of his friends were taken to court for treason, because they took up arms against the government. In the end, Hitler was sent to prison for about eight months, and the Nazi Party broke up (however, it would later get back together).
More people learned about Hitler. Before this happened, he was unknown. But after this, he was a hero to many. Many Germans at that time wanted a strong leadership after the governments of the Weimar Republic.
Hitler also took the time he had to spend in prison to write a book, called Mein Kampf or "My Struggle". It sold millions and told people about his ideas. Mein Kampf also made Hitler popular. (Hitler did not actually write the words down himself. Instead, he told another Nazi, Rudolf Hess, what to write down, and Hess wrote the words down.)
Hitler also had to re-think his strategy. He realised he could not take power the same way that Mussolini had done. Instead, he had to get power through being elected.
The people who died during the Putsch are seen as the first to die for the Nazi Party and are mentioned in Mein Kampf as Martyrs.
Verona
Verona is a city in the Veneto region of Italy with 260,000 inhabitants. It is famous for being the setting for the Shakespeare play "Romeo and Juliet" and because of the Roman Arena. This city is built on the Adige river. It has a humid subtropical climate ("Cfa" in the Köppen climate classification).
Kyiv
Kyiv or Kiev () is the capital and largest city of Ukraine. Almost three million people live there.