# Further Maths - Conjugates

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/conjugates/ · Updated: 2020-09-07 · Tags: further-maths, school, a-level

# Conjugates

A pair of complex numbers with a different sign but otherwise the same values are called _conjugates_:

$$
3 + 4i
3 - 4i
$$

##### What are complex conjugates??
A pair of complex numbers with real and imaginary parts equal in magnitude but opposite in sign:
* $a+bi$ and $a-bi$

##### What is a pair of complex numbers $a+bi$ and $a-bi$ called??
A complex conjugate.

##### What is the result of multipling complex conjugates??
You get a real number.

##### What other topic links to multiplying complex conjugates??
The difference of two squares.

##### What is the result of adding complex conjugates??
* You get a real number.
* $(a+bi)+(a-bi) = 2a$

##### What is the result of subtracting complex conjugates??
* You get an imaginary number.
* $(a+bi)-(a-bi) = 2bi$

##### What is $z + z^{\ast} for $z=(a+bi)$??
$2a$.

##### What is $z - z^{\ast}$ for $z=(a+bi)$??
$2bi$.

##### What is $zz^{\ast}$ for $z=(a+bi)$??
$a^2 + b^2$.

##### Q: What is $(3 + 4i)(3 - 4i)$??
$$
25
$$

##### Q: What is $(3 + 4i) + (3 - 4i)$??
$$
6
$$

##### Q: What is $(3 + 4i) - (3 - 4i)$??
$$
8i
$$

##### What is the conjugate of a real number??
Itself.

##### What is $x^{\ast}$ where $x \in \mathbb{R}$??
$x$.

##### What does $z^{\ast}$ mean??
The complex conjugate of $z$.

##### What is the notation for the complex conjugate of $z$??
$z^{\ast}$

##### How can you divide complex numbers??
* Using complex conjugation to convert the denominator to a real number
* Multiply the top and bottom by the complex conjugate of the denominator

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