A complex number is a number, but is different from normal numbers in many ways. A complex number is made up using two numbers combined together. The first part is a real number. The second part of a complex number is an imaginary number. The most important imaginary number is called , defined as . 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. Using arithmetic, addition, subtraction, multiplication, and division can be used. They also follow commutative properties and associative properties, just like real numbers.
Complex numbers were invented to answer 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.
With exponentiation, there is a problem. There is no real number that gives −1 when it is squared ("squared" means "multiplied by itself"). In other words, −1 (or any other negative number) has no real square root. To solve this problem, mathematicians introduced an imaginary number called i. That imaginary number 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.
A complex number can now be written as  (or ), where a is called the real part of the number, and b is called the imaginary part. Usually, the complex number is written as the set (a, b). Both a and b are real numbers.
Any real number can simply be written as or as the set (a, 0).
Addition, subtraction, multiplication, division as long as the divisor is not zero, and exponentiation (raising numbers to exponents) are all possible with complex numbers. Some other calculations are also possible with complex numbers.
The set of all complex numbers is usually written as .
Luckily, with the addition of complex numbers to maths, there is no new area of maths to be discovered, as every number you can think of (and every one you can't) can be described by complex numbers. Even really complicated things like the square root of i can be shown in some way.