A complex number is a number, but is different from common numbers in many ways. A complex number is made up using two numbers combined together. The first part is a real number, and the second part is an imaginary number. The most important imaginary number is called , defined as a number that will be -1 when squared ("squared" means "multiplied by itself"): . All the other imaginary numbers are  multiplied by a real number, in the same way that all real numbers can be thought of as 1 multiplied by another number. Arithmetic functions such as addition, subtraction, multiplication, and division can be used with complex numbers. They also follow commutative, associative and distributive properties, just like real numbers. The set of complex numbers is often represented using the symbol . 

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  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  that solves the equation . 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  for that number.

All complex numbers can be written as  (or ), where a is called the real part of the number, and b is called the imaginary part. We write  or  for the real part of a complex number . So, if , we write . Similarly, we write  or  for the imaginary part of a complex number ; , for the same . Every real number is also a complex number; it is a complex number  with .

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 , or as the pair .

Sometimes,  is written instead of . In electrical engineering for instance,  means electric current, so writing  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 .

Operations over complex numbers

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 , then , and .

Multiplication is a bit different: 

Another notable operation for complex numbers is conjugation. A complex conjugate  to  is .
It is pretty simple, but is important for calculations, because  is actually a real number for all complex : 

.

Because of that, we can use it to do division:

Other forms of describing complex numbers

Complex numbers can be shown on a so-called complex plane. If you have a number , you can go to point a on the real axis and point b on the imaginary axis, and draw a vector from  to . 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  is called its modulus or absolute value (written as ), and the angle is called its argument ().

This leads to the trigonometrical form of describing complex numbers: by the definitions of sine and cosine, it follows that for all :

This is closely connected to De Moivre's formula.

There exists even another form, called exponential form.

Conclusion

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 hypercomplex numbers, sedenion, hyperreal numbers, surreal numbers and many others. For more, see types of numbers.

Related pages 

 Complex plane
 Mandelbrot set
 Riemann sphere

References

Other websites 
 A search for complex number at Wikibooks

Number theory