The Imaginary Unit i & Powers of i

Q: What's the difference between an imaginary number and a complex number?

A pure imaginary number is a multiple of $i$, like $6i$. A complex number is the general form $a + bi$, which has both a real part and an imaginary part. Every imaginary number is complex (with $a = 0$), and every real number is complex too (with $b = 0$).

Q: Why does $i^2 = -1$ instead of $1$?

By definition, $i = \sqrt{-1}$, and squaring a square root gives back the number inside: $i^2 = (\sqrt{-1})^2 = -1$. The most common error is treating $-9i^2$ as $-9$; since $i^2 = -1$, it's actually $-9(-1) = +9$.

For years, math class told you that you can't take the square root of a negative number. That's true — as long as you only allow real numbers. Imaginary numbers are the fix: we define a new number $i$ so that $i = \sqrt{-1}$ , which means $i^2 = -1$ . Suddenly $\sqrt{-36}$ has an answer: it's $6i$ .

A complex number is just a real number and an imaginary number glued together: $a + bi$ , like $3 + 2i$ or $-5 - 7i$ . The name sounds intimidating, but the arithmetic is nothing new — you treat $i$ like a variable, with one extra rule: whenever $i^2$ shows up, replace it with $-1$ .

Where imaginary numbers come from

Squaring any real number gives a result that is zero or positive — $4^2 = 16$ and $(-4)^2 = 16$ . So no real number squares to $-16$ , and $\sqrt{-16}$ has no real answer. Defining $i = \sqrt{-1}$ solves this: $\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i$ .

That's the whole procedure for any negative square root: pull out the $\sqrt{-1}$ as $i$ , then simplify the rest normally. So $\sqrt{-25} = 5i$ and $\sqrt{-49} = 7i$ . One warning: convert to $i$ form before doing anything else. $\sqrt{-4} \cdot \sqrt{-9}$ is $(2i)(3i) = 6i^2 = -6$ — not $\sqrt{36} = 6$ . The product rule for radicals only works when the radicands aren't both negative.

Complex numbers: the a + bi form

A complex number has the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part. In $3 + 2i$ , the real part is $3$ and the imaginary part is $2$ . Every real number is secretly complex too — $7$ is just $7 + 0i$ .

Adding and subtracting works exactly like combining like terms: real parts combine with real parts, imaginary parts with imaginary parts. The one trap is subtraction — the minus sign applies to both parts of the second number, so distribute it before combining.

Multiplying: FOIL, then replace i squared

To multiply two complex numbers, FOIL them like binomials. You'll get an $i^2$ term — replace it with $-1$ , then combine the real parts. That single substitution is where almost every mistake happens, because a term like $-8i^2$ becomes $+8$ , flipping its sign.

Powers of $i$ follow a four-step cycle: $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ , and then it repeats. To simplify a big power like $i^{30}$ , divide the exponent by $4$ and keep the remainder — the remainder tells you where you land in the cycle.

Worked examples

Example 1: the square root of a negative number

Simplify $\sqrt{-49}$ .

Split off the negative as

\sqrt{-1}

\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1}

Simplify each factor

\sqrt{49} = 7, \quad \sqrt{-1} = i

Multiply

7i

Answer: $7i$

Example 2: subtracting complex numbers

Simplify $(2 + 5i) - (7 - 3i)$ .

Distribute the minus sign to both parts of the second number

2 + 5i - 7 + 3i

Combine the real parts

2 - 7 = -5

Combine the imaginary parts

5i + 3i = 8i

Write the result in

a + bi

form

-5 + 8i

Answer: $-5 + 8i$

Example 3: multiplying complex numbers

Multiply $(3 + 2i)(1 - 4i)$ .

FOIL the two binomials

3 - 12i + 2i - 8i^2

Replace

i^2

with

-1

, so

-8i^2

becomes

+8

3 - 12i + 2i + 8

Combine the real parts and the imaginary parts

11 - 10i

Answer: $11 - 10i$

Example 4: a large power of i

Simplify $i^{30}$ .

Divide the exponent by

4

and keep the remainder

30 = 4(7) + 2

The remainder is the power that matters

i^{30} = i^2

Use the cycle:

i^2 = -1

-1

Answer: $-1$

Try one yourself

Common questions

Are imaginary numbers actually real things, or just made up?

They're as legitimate as negative numbers — which also seemed fake when first introduced. Imaginary numbers show up constantly in electrical engineering, signal processing, and physics. The name 'imaginary' is a historical insult that stuck; the math is completely solid.

What's the difference between an imaginary number and a complex number?

A pure imaginary number is a multiple of $i$ , like $6i$ . A complex number is the general form $a + bi$ , which has both a real part and an imaginary part. Every imaginary number is complex (with $a = 0$ ), and every real number is complex too (with $b = 0$ ).

Why does

i^2 = -1

instead of

1

By definition, $i = \sqrt{-1}$ , and squaring a square root gives back the number inside: $i^2 = (\sqrt{-1})^2 = -1$ . The most common error is treating $-9i^2$ as $-9$ ; since $i^2 = -1$ , it's actually $-9(-1) = +9$ .

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