# What is complex conjugate of i?

Jul 3, 2018

$- i$

#### Explanation:

Conjugate of any complex number $a + b i$ is $a - b i$

Jul 3, 2018

It depends...

#### Explanation:

A conjugate of a number is a number that goes with it in the sense that multiplying the two numbers together yields a simpler kind of number.

For numbers involving radicals, we want to find a multiplier that results in a rational result. For complex numbers we want to find a multiplier that lead to a real result.

If $a + b i$ is any complex number (where $a , b \in \mathbb{R}$) then a suitable conjugate number is $a - b i$.

This is popularly known as "the" complex conjugate of $a + b i$ and we write:

$\overline{a + b i} = a - b i$

We find:

$\left(a + b i\right) \left(a - b i\right) = {a}^{2} - {b}^{2} {i}^{2} = {a}^{2} + {b}^{2} \in \mathbb{R}$

Using this convention, we find:

$\overline{i} = \overline{0 + 1 i} = 0 - 1 i = - i$

However, please note that this is not the only possible choice of conjugate.

For example:

$i \cdot i = - 1 \in \mathbb{R}$

So we could call $i$ self-conjugate.

In fact we could choose any pure imaginary number to use as a conjugate (in the multiplicative sense) for $i$.

Why do I stress this ambiguity?

Consider $\sqrt{2} + \sqrt{3}$.

What would you say is "the" radical conjugate of $\sqrt{2} + \sqrt{3}$ (i.e. the natural choice of multiplier to give a rational product) ?

We could mechanically choose $\sqrt{2} - \sqrt{3}$ and find:

$\left(\sqrt{2} - \sqrt{3}\right) \left(\sqrt{2} + \sqrt{3}\right) = 2 - 3 = - 1 \in \mathbb{Q}$

So $\sqrt{2} - \sqrt{3}$ is certainly a conjugate, but somewhat nicer is $\sqrt{3} - \sqrt{2}$ ...

$\left(\sqrt{3} - \sqrt{2}\right) \left(\sqrt{2} + \sqrt{3}\right) = 3 - 2 = 1$

Going back to complex conjugates, the standard complex conjugate $\overline{a + b i} = a - b i$ is significant for other reasons than being a multiplicative conjugate. For example, if $a + b i$ is a zero of a polynomial with real coefficients then $\overline{a + b i} = a - b i$ is also a zero.

So in general usage, it is conventional to stay with the standard complex conjugate and refer to "complex conjugate pairs" (for example).