# What is conjugate of i?

Oct 18, 2015

The conjugate of $i$ is $- i$

#### Explanation:

If $a , b \in \mathbb{R}$ then the conjugate of $a + i b$ is $a - i b$.

When you have a polynomial equation with Real coefficients, any Complex non-Real roots that it has will occur in conjugate pairs.

For example, ${x}^{2} + x + 1 = 0$ has two roots: $- \frac{1}{2} + \frac{\sqrt{3}}{2} i$ and $- \frac{1}{2} - \frac{\sqrt{3}}{2} i$.

${x}^{2} + 1 = 0$ has two roots $i$ and $- i$.

You could say that from the perspective of $\mathbb{R}$, the numbers $i$ and $- i$ are indistinguishable. When we extend $\mathbb{R}$ to make $\mathbb{C}$ we pick one of the square roots of $- 1$ and call it $i$. Then the other is $- i$, but they could just as easily be the other way around.