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

\[\begin{align} i &=i^{4\times \frac{1}{4}}\\ &={\left({ i }^{ 4 }\right)}^{\frac{1}{4}}\\ &= { 1 }^{ \frac { 1 }{ 4 } }\\ &=1. \end{align}\]

So, doesn't that mean \(i=1?\)

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

The statement "\(i\)=\(1\)" is \( \color{red}{\textbf{false}}\).

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