**protection click fraud**

To determine the opposite, the conjugate and the equality of any complex number, we need to know some fundamentals.*Opposite*

The opposite of any real number is its symmetric, the opposite of 10 is -10, the opposite of -5 is +5. The opposite of a complex number respects this same condition, as the opposite of the complex number z will be –z.

For example: Given the complex number z = 8 – 6i, its opposite will be:

- z = - 8 + 6i.*Conjugated*

To determine the conjugate of a complex number, it is enough to represent the complex number through the opposite of the imaginary part. The conjugate of z = a + bi will be:

Example:

z = 5 – 9i, its conjugate will be:

z = – 2 – 7i, its conjugate will be*Equality*

Two complex numbers will be the same if, and only if, they meet the following condition:

equal imaginary parts

Real equal parts

Given the complex numbers z1 = a + bi and z2 = d + ei, z1 and z2, they will be equal if only if a = d and bi = ei.

Comments:**The sum of opposite complex numbers will always equal zero.**

Example 1

z + (-z) = 0.

The conjugate of the conjugate of a complex number will be the complex number itself.

There is no order relationship in the set of complex numbers, so we cannot establish who is greater or less.z + (-z) = 0.

The conjugate of the conjugate of a complex number will be the complex number itself.

There is no order relationship in the set of complex numbers, so we cannot establish who is greater or less.

Example 1

Given the complex number z = - 2 + 6i, calculate its opposite, its conjugate and the opposite of the conjugate.

Opposite

- z = 2 - 6i

Conjugated

opposite of the conjugate

*Example 2*

Determine a and b so that .

-2 + 9i = a - bi

We need to establish ownership of the relationship of equality between them. Then:

a = - 2

b = - 9

by Mark Noah

Graduated in Mathematics

Brazil School Team

**Source:** Brazil School - https://brasilescola.uol.com.br/matematica/oposto-conjugado-igualdade-numeros-complexos.htm