Further Maths - Conjugates

Conjugates

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

\[3 + 4i 3 - 4i\]

Defining the complex conjugate

What are complex conjugates?

complex-conjugate-definition

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?

conjugate-pair-name

A complex conjugate.

Arithmetic with conjugates

What is the result of multiplying complex conjugates?

conjugate-product-is-real

You get a real number.

What other topic links to multiplying complex conjugates?

conjugate-product-difference-of-squares

The difference of two squares.

What is the result of adding complex conjugates?

conjugate-sum-result

You get a real number.
$(a+bi)+(a-bi) = 2a$

What is the result of subtracting complex conjugates?

conjugate-difference-result

You get an imaginary number.
$(a+bi)-(a-bi) = 2bi$

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

z-plus-conjugate

$2a$.

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

z-minus-conjugate

$2bi$.

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

z-times-conjugate

$a^2 + b^2$.

Worked numerical examples

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

conjugate-product-example

\[25\]

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

conjugate-sum-example

\[6\]

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

conjugate-difference-example

\[8i\]

Conjugate notation and special cases

What is the conjugate of a real number?

real-number-conjugate

Itself.

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

real-conjugate-notation

$x$.

What does $z^{\ast}$ mean?

conjugate-star-meaning

The complex conjugate of $z$.

What is the notation for the complex conjugate of $z$?

conjugate-notation-symbol

$z^{\ast}$

Dividing complex numbers

How can you divide complex numbers?

dividing-complex-numbers-method

Using complex conjugation to convert the denominator to a real number
Multiply the top and bottom by the complex conjugate of the denominator