﻿ Complex numbers: Definition. Binomial and Cartesian form

# Complex numbers

Now I’m going to explain what complex numbers are: what parts make them up and in what forms they can be expressed.

## Definition of complex numbers

I could tell you that the set of complex numbers contains the real numbers, they are represented by the symbol C and they include the roots of all the polynomials, but what does this mean? Why do we need complex numbers?

In other words, we remember that with real numbers we cannot solve the roots of pairs of negative numbers: Therefore, we need the complex numbers to be able to solve with the negative roots of even index.

### Imaginary unit

The number i, which is equal to the root of minus one, is called the imaginary unit: With this number, we can give a solution when we have negative roots: According to the properties of the roots, we can put the negative root as that same positive root, multiplied by the root of minus one.

Finally, the positive root already has a solution in the set of real numbers and the root of minus one is replaced by the number i.

## Form of complex numbers

### Binomic form

A complex number Z (not to be confused with C, which is the set to which they belong) can be represented in the form: Belonging a and b to the set of real numbers.

This way of writing complex numbers corresponds to the binomial form, which has two parts:

• a = Actual part
• b = Imaginary part

These are examples of complex numbers in binomial form:  If the real part of a complex number is 0, that number is pure imaginary, since it only has an imaginary part: The number i is a pure imaginary number.

box type=”info”]The binomial form is used in physics when operating with vectors, where the imaginary part is represented by the letter j, instead of the letter i.[/box].

On the other hand, if it has no imaginary part, we are talking about a pure real number, which is neither more nor less than a real number: ### Cartesian form

Another way to represent complex numbers is the Cartesian form: As in the binomial form, a and b belong to the set of real numbers y:

• a = Actual part
• b = Imaginary part

The above examples represented in Cartesian form are expressed as follows:     ## Conjugate of a complex number

The conjugate of a complex number is used in the division of complex numbers. It is represented by a slash above the number. To obtain the conjugate of a complex number it is necessary to change the sign of the imaginary part (the central sign): For example: ## Oposite of a complex number

To obtain the opposite of a complex number, it is necessary to change the sign of both the real part and the imaginary part, therefore the opposite of Z, will be -Z: For example: As a conclusion, the most important thing you should know is that they are composed of a real part and an imaginary part, that the number i is equal to the root of -1 and therefore, the number i squared is equal to -1 and how to obtain the conjugate of a complex number.