Is i equal to 1?

This is a part of a series on List of Common Misconceptions.

Is $i=1$?

Why?

We know that $i=\sqrt { -1 }.$ But we also have

$\begin{aligned} i &=i^{4\times \frac{1}{4}}\\ &={\left({ i }^{ 4 }\right)}^{\frac{1}{4}}\\ &= { 1 }^{ \frac { 1 }{ 4 } }\\ &=1. \end{aligned}$

So, doesn't that mean $i=1?$

Why people say its false? Because it's just $\sqrt { -1 } .$

The statement "$i$=$1$" is $\color{#D61F06}{\textbf{false}}$.

Proof:

The roots of $1$ here can easily be seen to be 4. Here the working is solely based on false roots.

Now, we will prove that $i\neq 1$ by contradiction.

Let us assume $i=1.$ Then by squaring both sides, we get

$i^2=1^2 \Rightarrow -1=1,$

which is not possible. So $i$ cannot take the value of $1.$

Therefore, our assumption that $i=1$ is wrong and hence $i\neq 1. \ _\square$

See Also

• List of Common Misconceptions

• Complex Numbers