# Complex Numbers

## Understanding the Basics of Complex Numbers

In mathematics, we often work with real numbers. However, there are situations where certain problems have no solution. This usually happens when solving equations involving negative square root numbers. But with the introduction of complex numbers, we can now solve such problems.

### Definition of a Complex Number

A complex number is a type of number that consists of a real part and an imaginary part. It is denoted as a + bi in mathematical notation, where a is the real part and bi is the imaginary part.

The imaginary part, represented by the letter i, has a numerical value and follows a key identity. An equation will only have real roots when b = 0, but it will have imaginary roots when b ≠ 0.

Therefore, a complex number can be expressed as a + bi, where a and b are real numbers and i is the imaginary unit. It is usually written in the form a + bi. Here, a is the real part, and bi is the imaginary part. This can also be written as a + bi = Re(a) + Im(b)i, where Re(a) represents the real part and Im(b) represents the imaginary part.

All real numbers are also considered complex numbers, where the imaginary part is taken as 0.

### Operations with Complex Numbers

Complex numbers have similar properties to real numbers, meaning that operations such as addition and subtraction work in the same way.

For instance, when adding or subtracting complex numbers, we use them the same way as in algebra, by adding the real parts and the imaginary parts separately. This can be represented as (a + bi) + (c + di) = (a + c) + (b + d)i or (a + bi) - (c + di) = (a - c) + (b - d)i.

Multiplication of complex numbers can be tricky, so let's break it down. Suppose we have two complex numbers, (a + bi) and (c + di). To find their product, we follow these steps:

- Firstly, multiply the real parts together, a x c.
- Next, multiply the imaginary parts together, bi x di.
- Then, multiply the real and imaginary parts, a x di and c x bi.
- Finally, combine like terms to get the product (ac - bd) + (ad + bc)i.

Division of complex numbers follows a similar method to simplifying surds. We need to use the conjugate, which is the reversed sign of the imaginary part. The conjugate of (a + bi) is (a - bi).

Therefore, to simplify a fraction involving complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator. For instance, (a + bi) / (c + di) = ((a + bi) x (c - di)) / ((c + di) x (c - di)).

### Graphical Representations of Complex Numbers

Complex numbers can also be plotted on an argand diagram. This is a two-dimensional graph with an axis representing the real part and an axis representing the imaginary part. It looks like this:

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