Complex Number Calculator
This complex number calculator adds, subtracts, multiplies and divides numbers of the form a + bi — and keeps every answer exact, which matters more than it sounds, because dividing complex numbers produces fractions almost every time.
Try −3 ÷ (2 − i). The answer is −6/5 − 3/5 i. Most calculators will hand you −1.2 − 0.6i instead, which is a rounded version of that — and rarely what the question wanted.
What a Complex Number Actually Is
A complex number is written a + bi. The a is the real part, the b is the imaginary part, and i is defined by a single equation:
i² = −1
That is the only new fact in the whole subject. Everything else — the arithmetic, the conjugate, the division trick — follows from it and from ordinary algebra.
It helps to think of i as a symbol you can move around like x, with one extra rule waiting for you: any time you produce an i², you swap it for −1. Note also that ordinary numbers have not gone anywhere: 7 is the complex number 7 + 0i, and 3i is 0 + 3i.
Adding and Subtracting: Two Kinds That Never Mix
These are the easy ones, and the reason is worth saying: the real and imaginary parts never interact.
(1 + i) + (3 + 5i) = 4 + 6i — the 1 and 3 make 4, the 1 and 5 make 6.
(1 + i) − (3 − 5i) = −2 + 6i — and watch the second minus, which turns −5i into +5i.
It works exactly like collecting x terms and constants. Nothing about i being a square root of −1 has come into play yet, which is why these two never cause trouble beyond a sign slip.
Multiplying, and the One Rule That Matters
Multiplication starts as ordinary bracket expansion. (1 + i)(3 + 5i) gives four products:
3 + 5i + 3i + 5i²
Up to here there is nothing complex about it at all. Then the rule arrives: i² = −1, so 5i² becomes −5, leaves the imaginary side altogether and joins the real one. Collecting up: 3 − 5 = −2 for the real part, 5 + 3 = 8 for the imaginary. The answer is −2 + 8i.
That single move explains the thing beginners find strangest — that two imaginary numbers multiply to a real one. i × i = −1. And it explains the fact everything else here depends on:
(3 + 4i)(3 − 4i) = 25
A number times its conjugate — the same number with the middle sign flipped — is always an ordinary real number, because the two cross terms cancel and the i² term turns positive.
Dividing, and Why the Answer Is Nearly Always a Fraction
You cannot divide by something with an i in it directly, so the standard move is to get rid of it: multiply the top and the bottom by the conjugate of the bottom.
This is the same manoeuvre as rationalising a denominator that has a surd in it. There you multiply by the conjugate to clear a √; here you do it to clear an i. Same idea, different awkward thing being cleared.
Take (4 + 2i) ÷ (1 + i). The conjugate of the bottom is 1 − i. Multiplying both by it turns the bottom into 1² + 1² = 2, an ordinary number, and the top into 6 − 2i. Dividing through gives 3 − i.
That one came out clean. Most do not, and here is why: after the conjugate step you are always dividing by c² + d², which is seldom a factor of what sits above it. So:
−3 ÷ (2 − i) = −6/5 − 3/5 i
6i ÷ (4 + 3i) = 18/25 + 24/25 i
Those are the answers. A calculator that shows −1.2 − 0.6i and 0.72 + 0.96i has rounded them, and in a subject where the exact form is the point, that is throwing away the thing you came for. This page keeps every part as a fraction from start to finish, and it also works the division out a second time without using the conjugate at all — by asking which number multiplied by the bottom gives the top — and requires the two routes to agree before showing you anything.
How to Use This Complex Number Calculator
Four boxes and four buttons.
Type each number as two parts. The real part goes in the first box, the multiple of i in the second. An empty box counts as 0, so a lone 5 is the real number 5, and a lone 3 in the second box is 3i. Fractions like 3/4 and decimals like 0.25 both work, and the answer stays exact either way.
Choose the operation from the row between the two numbers. Switching it reworks the answer straight away.
Open the working and you get the method rather than just the result: the two kinds kept apart for adding and subtracting, all four products and the i² step for multiplying, and for division the conjugate, the denominator turning real, the division through and the independent check.
| real | The ordinary part of the number. Empty means 0 |
| imaginary (× i) | How many i s. Empty means 0 |
| + − × ÷ | Which operation to perform |
| ← → ↑ ↓ | Move between the boxes. Up and down move between the two numbers |
| ± | Flips the sign of the box you are in |
| ⁄ and . | Fraction bar and decimal point |
| ⌫ / AC | Delete one character, or clear everything |
| = | Works it out |
The page also names what the answer is, not only what it equals: whether the imaginary parts have cancelled leaving an ordinary real number, whether the real part has gone leaving something purely imaginary, or whether the whole thing has come out as zero.
Complex Numbers FAQ
What is a complex number?
How do you multiply two complex numbers?
How do you divide complex numbers?
Why is the answer to a complex division usually a fraction?
What is 1 divided by i?
When is the product of two complex numbers a real number?
One new rule, i² = −1, and ordinary algebra everywhere else. The only place it gets awkward is division, and that is exactly where an exact answer is worth having.