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?

because

But what is

Try a positive:

, not −1.

Try a negative:

, still not −1.

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

Theorem The imaginary unit

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.

⟹ 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.

— the cycle restarts.

Concept Complex numbers

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

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

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

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

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

Note Mistakes to avoid

Treating i as an ordinary variable — remember

, not

Forgetting that the powers of i repeat every four steps.

Writing

instead of

Writing a complex number as

rather than

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.