To be able to calculate all algebraic operations, we need to be able to calculate roots of negative numbers. Therefore we introduce a new number , such that .
Complex numbers are numbers of form , where . Reals and are called the real and imaginary part of and we denote them by and .
Therefore any complex number can be shown on a plane as a point with coordinates . This plane will be usually called the complex plane.
Adding and multiplying
We can add and multiply complex numbers straightforward, but recall that , so: and .
It is also easy to divide a complex number by a complex number. One shall proceed similarly as when there is an irrational root in the denominator. So:
Every non-zero complex number , can be described in a yet another way — it id defined by its distance from zero (its modulus) and the angle between it and the real axis, called the argument (, ). Therefore, .
E.g.: let , then and . On the other hand, if , then .
Geometrical view on addition and multiplication
You can notice immediately, that adding two complex numbers is the same as adding two vectors on the complex plane.
Multiplication is even more interesting. Namely, multiplication of two complex numbers actually multiplies its modules and adds their arguments. This is especially useful when calculating a power of a complex number, and it implies so called de Moivre’s equality:
E.g., Let us calculate . We get:
- modulus of is and argument is , so modulus of is and argument is , in other words .
- modulus of is and argument is , so modulus is and argument is .
By the way, de Moivre’s equality implies many trigonometric equalities. E.g., because
we get that:
Notice, that for every complex number there are always numbers such that . Indeed the first one can be calculated by calculating the root of the modulus and by dividing the argument by . But if we add to this argument any multiple of , then after taking the number to the -th power we will get the same number, because this additional angle will sum up to some multiple of . All those numbers will be called -th roots of .
E.g. are numbers. Since , every root will be of modulus . Argument is , so the first root will have as its argument and we can add to it any multiple of , so we get the following roots:
- argument: , so
- argument: , so ,
- argument: , so ,
- argument: , so .
Those root of , argument of which is in , is called the main root and denoted by . So .
Therefore we can calculated roots of a quadratic equations regardless of the sign of , or even when is not real. E.g. , , has modulus and argument , so it is . And . Therefore, and .
Exponential form and calculating complex powers
We will see that for , we have: . This gives us an idea how to define complex powers. If we substitute to we get , moreover, and . Therefore, we can define to a complex power as follows:
Which implies the famous equation of the constants of mathematics . 🙂
And we can write any complex number in an exponential form, which is similar to the polar form. Namely, .