The Imaginary Unit i | Moosa Academy

No real number squares to give a negative. Rather than accept that √(−1) simply has no answer, mathematicians defined one — and the number they invented turned out to be indispensable in engineering and physics.

Concept The problem

A square root asks: what number, multiplied by itself, gives the value under the root?

\sqrt{9} = 3

because

3 \times 3 = 9

But what is

\sqrt{-1}

Try a positive:

1 \times 1 = 1

, not −1.

Try a negative:

(-1) \times (-1) = 1

, still not −1.

Squaring any real number gives a positive result, so no real number can be the answer.

Theorem The imaginary unit

i = \sqrt{-1} \qquad i^2 = -1

The letter i stands for "imaginary". The name is unfortunate — these numbers are no less legitimate than any other, and they solve real problems.

Example Roots of negative numbers

Split off the −1, take its root as i, and deal with the rest normally.

\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i

\sqrt{-25} = \sqrt{25} \times \sqrt{-1} = 5i

\sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i

⟹ every negative root is a real multiple of i

Concept Powers of i repeat

Raising i to successive powers produces a repeating cycle. After four steps the pattern returns to where it began, so any power of i can be reduced by dividing the exponent by 4.

i^1 = i

i^2 = -1

i^3 = i^2 \times i = -i

i^4 = i^2 \times i^2 = (-1)(-1) = 1

i^5 = i^4 \times i = i

— the cycle restarts.

Concept Complex numbers

A number with both a real and an imaginary part is called complex, written in the form:

a + bi

4 + 3i

, the real part is 4 and the imaginary part is 3i.

5 + 2i

, the real part is 5 and the imaginary part is 2i.

1 - 4i

, the real part is 1 and the imaginary part is −4i.

Always write the real part first. The form $a + bi$ is the convention every later technique relies on.

Note Mistakes to avoid

Treating i as an ordinary variable — remember

i^2 = -1

, not

i^2 = i

Forgetting that the powers of i repeat every four steps.

Writing

\sqrt{-9} = -3

instead of

3i

Writing a complex number as

bi + a

rather than

a + bi

Dismissing imaginary numbers as fake; they are standard tools in engineering and physics.

Summary

No real number squares to a negative, so i is defined with i² = −1.
Any negative square root becomes a multiple of i: √(−9) = 3i.
Powers of i cycle through i, −1, −i, 1 and then repeat.
A complex number is written a + bi, real part first.
Imaginary numbers solve otherwise unsolvable equations and underpin real applications.