The Secret Life of i

We have been inventing numbers all along

Schools rarely tell you this: the numbers you know were not found in nature. People invented them, one group at a time. Every time, people first said the new numbers were nonsense. Later the new numbers became so useful that everyone forgot they had complained.

It always happens the same way. Someone finds an equation that the current numbers cannot solve. There are two choices. You can say "no solution" and stop. Or you can invent a new number that is the solution, and then check if it follows the rules. Look at the pattern:

And every new number was mocked when it first appeared:

• Negative numbers were called "absurd" and "false" in Europe for hundreds of years. What could −3 apples mean? (Then banking arrived: −3 is a debt. Suddenly it was very real.)
• Zero was not trusted — a number for nothing? Some accountants in the Middle Ages were not allowed to use it, because it made fraud too easy.
• Irrational numbers shocked the ancient Greeks. The Pythagoreans believed every number was a fraction. The story says that when one of them, Hippasus, proved √2 is not a fraction, they were so upset that they drowned him at sea. The diagonal of a simple 1×1 square had broken their view of the world.

So when you meet a number called "imaginary" and you feel it is fake, you are not alone. People said the same thing about every new number, every time. The name "imaginary" was only an insult (you will meet the person who said it soon). Your job here is not to "believe" in a magic number. Your job is to watch one more invention prove that it is useful, just like negatives and fractions did.

Numbers are tools. When our tools cannot solve a problem, we build a new tool.

The strange history of √−1

Here is a surprise: complex numbers were not invented to solve x² = −1. Nobody worried about that equation. If x² = −1 has no answer, that is fine — you just say "no answer" and move on. Complex numbers were forced into existence by a harder problem: the cubic equation. And by one of the pettiest fights in the history of mathematics.

Italy, 1500s: math was a competition

In Italy in the 1500s, mathematicians built their reputation (and won university jobs) through public math contests. Two people exchanged lists of problems. Whoever solved more problems won. If you lost badly, you could lose your job. So if you found a powerful method, you did not share it. You kept it secret.

The prize everyone wanted was a method for cubic equations — equations that contain x³, like x³ = 15x + 4. Around 1515, a professor in Bologna named Scipione del Ferro solved a whole family of them. He told almost no one, and kept the method secret until he was dying. In 1535, a self-taught mathematician named Niccolò Tartaglia found the method again on his own, and used it to beat an opponent in a contest: 30 problems to 0.

Then came Girolamo Cardano — a doctor, gambler, astrologer, and brilliant mathematician. He asked Tartaglia for the secret many times. In 1539 Tartaglia finally shared it, but only after Cardano made a serious promise never to publish it. Later, Cardano found del Ferro's earlier solution, decided his promise no longer held, and in 1545 published everything in his book Ars Magna ("The Great Art"). Tartaglia was angry, and the fight lasted the rest of their lives. But now the world had "Cardano's formula" — and with it, a very strange problem.

The formula that needed the impossible

For cubics of the form x³ = px + q, Cardano's formula says:

x = ∛( q/2 + √( (q/2)² − (p/3)³ ) ) + ∛( q/2 − √( (q/2)² − (p/3)³ ) )

Do not memorize it. Just watch what happens when we put in a normal, simple cubic:

Bombelli's bold idea

In 1572, an Italian engineer named Rafael Bombelli — a practical man who spent his career draining swamps — had a bold idea. What if we just keep going? Treat √−1 as a real quantity. Do not ask what it is. Just make it follow the normal rules of algebra, where the only new fact is (√−1)² = −1, and see where the calculation leads.

Write √−121 as 11·√−1 (check: (11√−1)² = 121 · (−1) = −121 ✓). Then Bombelli noticed something wonderful. Watch him cube (2 + √−1), using only basic algebra:

The "impossible" parts cancelled each other perfectly and gave the true, real answer. The imaginary numbers worked like temporary support: they held the calculation up in the middle, then disappeared at the end. This was the moment complex numbers proved they were worth keeping — not because people believed in them, but because they worked.

The slow road to acceptance

1. 1515Scipione del Ferro secretly solves the first cubics.
2. 1535Tartaglia finds the method again, and wins his famous contest 30–0.
3. 1545Cardano breaks his promise and publishes Ars Magna. The formula reaches the world — and the world meets √−1.
4. 1572Bombelli's bold idea: follow the algebra, and √−1 gives real answers. He writes the first rules for calculating with these numbers.
5. 1637René Descartes does not like them and calls them "imaginary" as an insult. The name stays.
6. 1748Leonhard Euler finds the amazing formula e^iθ = cos θ + i sin θ (Step 9), and in 1777 introduces the symbol i for √−1.
7. 1799Caspar Wessel, a Norwegian land surveyor, is the first to draw complex numbers as points in a plane. He publishes in Danish, and the world ignores it for a hundred years.
8. 1806Jean-Robert Argand, a bookkeeper in Paris, draws the same picture on his own — the "Argand diagram."
9. 1831Carl Friedrich Gauss, the greatest mathematician of the time, accepts the picture, gives them the name complex number, and says the mystery is over: these are just points, and nothing about them is imaginary.
10. 1830sWilliam Rowan Hamilton removes the last mystery: a complex number is officially just a pair of real numbers (a, b) with one clever multiplication rule. (This is the main point of Step 11.)

Notice who kept this idea moving forward: an engineer who drained swamps, a land surveyor, a bookkeeper. Practical people who cared whether it worked, not whether it sounded believable. Notice the time too — about 260 years passed between Bombelli's idea and Gauss making it accepted. So if complex numbers feel strange to you this week, that is normal: they felt strange to the smartest people alive for over two hundred years.

Meet i — and the four-step cycle of its powers

Now let us name the invention properly. We define one new number and one new rule. Everything else is normal algebra that you already know.

About the name. "Complex" does not mean complicated here. It means "made of several parts joined together" — the same way a building complex is several buildings joined together. A complex number is a two-part number: one real part and one imaginary part. And "imaginary," as you saw, came from Descartes making a joke in 1637. Both names are just accidents of history. The numbers themselves are as solid as −3.

Two quick checks that make everything clear:

• Every real number is already a complex number. 7 is just 7 + 0i (imaginary part zero). The old numbers are not replaced — they are a special case of the new ones, like whole numbers living inside fractions.
• Pure imaginary numbers are complex too. −5i is 0 + (−5)i (real part zero).

The powers of i: a four-step cycle

What happens when you keep multiplying by i? Use i² = −1 again and again:

i¹ = i   ·   i² = −1   ·   i³ = i²·i = −i   ·   i⁴ = i²·i² = (−1)(−1) = 1

At i⁴ we are back to 1, so the pattern repeats forever: i, −1, −i, 1, i, −1, −i, 1, … a cycle of four. Press the button below and watch the cycle. Look carefully at the shape the arrow makes, because it is quietly showing you the biggest secret in this tutorial.

Did you see it? Every multiplication by i turned the arrow a quarter-circle. Four quarter-turns = one full turn = back to 1. The algebra (a cycle of four) and the picture (four quarter-turns) are the same fact. Remember this — Step 7 will build on it in a big way.

One new number (i), one new rule (i² = −1). Its powers cycle: i, −1, −i, 1 — four quarter-turns.

The complex plane: every number gets a home

For 250 years, the hard question was "but where is i?" It is not left of zero, not right of zero, and not between any two numbers on the line. Wessel, Argand, and Gauss finally gave the answer, and it is simple: step off the line.

Think of the real number line as a street that runs east–west. Every real number has an address on that street. The new idea: i lives one step north. Not further along the street — off the street, in a direction the street never used. Real numbers measure east–west. Imaginary numbers measure north–south. A complex number a + bi is simply the point you reach by going a steps east and b steps north. Now the whole flat plane is full of numbers.

Play with it until three things feel clear:

• Drag z onto the horizontal axis → the imaginary part becomes 0 and z is just a real number. The old number line is inside the plane, running through its middle.
• Drag z onto the vertical axis → the real part becomes 0 and z is pure imaginary. The point one step up is i itself; one step down is −i.
• Left of the vertical axis, the real part is negative. Below the horizontal axis, the imaginary part is negative. Every point in the plane is a real number in this system.

This picture — drag an arrow, split it into parts — may feel familiar from the vectors tutorial. That is not a coincidence. A complex number looks exactly like a 2-D vector with an x-part and a y-part. So a fair question comes up: "then why do we need a new name for it at all?" Keep that question in mind. It is a good one, and Step 11 answers it fully.

A complex number a + bi is the point (a, b): a steps east, b steps north. The number line was just the ground floor.

Adding: walk one, then walk the other

Now some arithmetic. Adding complex numbers is very easy: add the real parts, and add the imaginary parts. Keep east–west and north–south separate, just like the independent components in the vectors tutorial.

On the plane, this is tip-to-tail: put the start of the second arrow at the end of the first arrow. The sum is the arrow from the origin straight to the final point. Do it in either order and you draw a parallelogram — the same final point both ways.

So far, complex numbers behave exactly like 2-D vectors. This widget could sit in the vectors tutorial without any change. The surprise comes in the next section: vectors cannot do what happens in Step 7.

Addition = walking arrows tip-to-tail. Real parts and imaginary parts never mix.

Every number has two addresses

Suppose you are standing at 0 and want to tell a friend where the number z is. You have two good ways to do it. Way one — the street address: "go a east, then b north." That is a + bi, the form we have used so far. Way two — the straight-line address: "face this direction, then walk this far in a straight line." Same point, different description. For multiplication (the next step), the second address is extremely useful.

The teal circle in the widget is the unit circle — all the numbers with r = 1, meaning they are at distance exactly 1 from zero. It looks like background now, but it is about to become very important: the numbers on it are the "pure rotations."

Rectangular a + bi says "east and north." Polar r∠θ says "how far and which way." Same number, two addresses.

Multiplication is rotate-and-stretch

This is the most important section in the tutorial. Adding complex numbers works just like adding vectors — nothing new. But complex numbers can do one thing vectors cannot: you can multiply two of them, and the answer is still a point in the plane. When that happens, something beautiful appears.

First, the plain algebra

To multiply (a + bi)(c + di), just expand the brackets the way you learned in algebra class, then use i² = −1 to simplify:

(a + bi)(c + di) = ac + adi + bci + bd·i² = (ac − bd) + (ad + bc)i

Now, the surprise

Change everything in that example to straight-line addresses (length and angle):

• 2 + i: length √(4+1) = √5 ≈ 2.24, angle ≈ 26.6°
• 1 + 3i: length √(1+9) = √10 ≈ 3.16, angle ≈ 71.6°
• the answer −1 + 7i: length √(1+49) = √50 ≈ 7.07, angle ≈ 98.1°

Now look carefully. √5 × √10 = √50 — the lengths multiplied. And 26.6° + 71.6° = 98.1° — the angles added. This is not just true for this example. It is always true, for every pair of complex numbers:

Use those preset buttons and see what they show:

• w = 2 (length 2, angle 0°): the product simply doubles z. Multiplying by a positive real number only stretches, with no turn. Normal multiplication is just the special case where the angle is zero.
• w = −1 (length 1, angle 180°): the product is z turned half a turn. You have always known that "multiplying by −1 changes the sign." It was secretly "rotate by 180°" the whole time.
• w = i (length 1, angle 90°): the product is z turned a quarter-turn counter-clockwise, with the same length. Multiplying by i is a 90° rotation. Nothing more, nothing less.

And now x² = −1 answers itself. It asks: "which operation, done twice, turns you all the way around?" A quarter-turn. Which quarter-turn? Either one: counter-clockwise (that is i) or clockwise (that is −i). Two solutions, both easy to see, and neither one is truly imaginary.

The mirror image, and how to divide

To multiply: lengths multiply, angles add. ×i = quarter-turn, so i² = −1 just means "two left turns = facing backwards."

Powers spiral, roots share the circle

If multiplying once means "turn by θ and stretch by r," then multiplying z by itself again and again means turning by θ every time and stretching by r every time. So the powers of a complex number walk like steps of a staircase around the origin.

That last case — |z| = 1 — is worth a moment. Numbers on the unit circle do not stretch anything; they are pure rotations. Their powers just move around the circle like the hand of a clock. This is why the unit circle is the most important part of the complex plane.

Going backwards: roots

A square root asks "what number, squared, gives this?" — that is, what angle, when doubled, reaches the target angle, and what length, when squared, gives the target length? An n-th root asks the same, but with n. Here is the beautiful surprise: there are always n different answers, and they are perfectly evenly spaced.

Powers: multiply the angle by n, raise the length to the power n — a spiral staircase. Roots: n answers, evenly spaced on a circle.

Euler's formula: the spinning number

One more improvement to the "straight-line address," and it gives us what many mathematicians call the most beautiful equation ever written. Meet the number e and its amazing trick.

Now remember Step 7: when you multiply complex numbers, angles add. And for exponentials, exponents add. Same behavior! Euler (1748) proved they are not just similar — they are literally the same operation:

Watch the machine run. The coral point is e^iθ moving around the unit circle as θ grows. Its shadow on the real axis is cos θ, its shadow on the imaginary axis is sin θ, and off to the right you can see the height draw a perfect sine wave — the same wave from the interference tutorial. Circular motion and waves are the same thing viewed from different angles, and e^iθ is the number that says so.

With Euler's formula, the polar form gets its final, clean form. Instead of writing z = r(cos θ + i sin θ), we write z = r·e^iθ — "length r, pointing at angle θ." Three symbols carry the whole geometric story, and the multiplication rule becomes one line of exponent algebra: r₁e^iα · r₂e^iβ = r₁r₂ e^i(α+β). Lengths multiply, angles add — now using the same exponent rules you learned in school.

e^(iθ) = the point at angle θ on the unit circle. Rotation became an exponent — and e^(iπ) = −1 is just "half a turn."

The "imaginary" number in your pocket

Complex numbers stopped being just a helper tool long ago. Because they turn rotation and oscillation into simple arithmetic, they are the natural language of anything that spins, waves, or vibrates — which is most of physics and almost all of electronics.

• Electricity in your walls. Mains electricity is AC — the voltage goes up and down 50–60 times per second. Engineers write that oscillation as a rotating complex number (a phasor, r·e^iωt), which turns very hard circuit calculus into simple multiplication and addition. (One detail: electrical engineers write j instead of i, because i was already used for electric current.)
• Every sound you stream. Audio compression, equalizers, and noise cancelling all use the Fourier transform, which breaks any signal into a sum of spinning e^iωt terms — pure rotations at different speeds. Your music is stored and processed as complex numbers.
• Phone signals, Wi-Fi, radar, MRI machines. All of them store information in the size and angle of waves — that is, in the length and angle of complex numbers. An MRI image is built from a plane full of complex measurements.
• Quantum mechanics. The state of an electron is described by a complex-valued wavefunction — not just for convenience, but because it is needed. Nature itself seems to keep its records in complex numbers. (The uncertainty-principle tutorial continues this idea.)
• Computer graphics and games. Rotating a 2-D image is one complex multiplication. For 3-D rotations, games use quaternions — the 4-D bigger relative of complex numbers that you will meet in Step 11.
• Fractals. The most famous image in mathematics, below, is nothing but Step 7's arithmetic repeated.

The Mandelbrot set: infinity from one line of arithmetic

Here is the whole method. Pick a point c in the complex plane. Start with z = 0 and repeat the step z → z² + c again and again (square the number — one complex multiplication — then add c). Only two things can happen: either z stays near the origin forever, or it grows toward infinity. Paint c dark if it stays near; if it grows, colour it by how quickly. That is all. No other rules. The result may be the most detailed object in mathematics — and it is made from nothing but the multiply-and-add you learned three sections ago.

Anything that rotates or oscillates — power grids, songs, signals, electrons — speaks complex numbers naturally.

"Isn't this just coordinate geometry?"

Since Step 4 you have had a good reason to be doubtful. A complex number is a point (a, b). Descartes gave us points with coordinates back in 1637. The vectors tutorial gave us arrows we can add. So why all the excitement — why is coordinate geometry, or vectors, not enough? This section is the honest answer, and it is the deepest idea on this page.

What each system can actually do

Think of it as three levels of the same plane. Each level can do more:

Level 1 — Coordinate geometry (points). A point (a, b) is an address. You can plot it, measure distances between addresses, find midpoints, and describe lines and circles. This is great for geometry. But try to calculate with the points themselves: what is Paris + London? What is (2, 3) × (1, 4)? Coordinate geometry has no answer — points are locations, not amounts. There is no arithmetic of points, only arithmetic about them (which you do by hand, on each coordinate separately).

Level 2 — Vectors (arrows). This is a real step up: arrows can be added (tip-to-tail — Step 5 used this!) and stretched by normal numbers. But now ask for multiplication: given arrows u and v, what is u × v, as an arrow in the plane? Vectors offer two products. Look carefully at what each one gives back:

• The dot product u·v gives back a plain number, not an arrow. The answer has left the plane — you cannot take the result and dot it with something else and keep going. The chain stops after one step.
• The cross product u×v gives back an arrow pointing out of the page — at a right angle to the plane, into the third dimension. The answer has left the plane, literally.
• And neither product can be reversed: there is no such thing as "dividing by a vector." Knowing u·v and u does not tell you v. Questions like "u times what equals w?" have no vector answer.

So vectors can be combined, but they have no true multiplication that stays in the plane. And without multiplication you certainly cannot square anything — so a question like x² = −1 cannot even be asked in the language of points or vectors, and definitely cannot be answered.

Level 3 — Complex numbers. Same plane, same points, plus one extra rule — and that rule changes everything.

And watch what that one rule does to the "impossible" number. In Hamilton's language, i is simply the point (0, 1) — one step north. Multiply it by itself using only the rule:

(0, 1) × (0, 1) = (0·0 − 1·1, 0·1 + 1·0) = (−1, 0)

The point one step north, squared, is the point one step west — which is the real number −1. There it is: i² = −1, computed with ordinary real numbers and one multiplication rule. No imagination was ever needed. The "mystery" was only ever a missing definition.

Try the difference yourself. Below, drag the two arrows u and v. The panel shows what each system gives back when you ask it to "multiply" them — and where each answer lives:

Why that one rule matters so much

• The plane becomes a number system. Add, subtract, multiply, divide — all four operations work, and every non-zero number can be divided by. Mathematicians call such a system a field. The real line is a field; Hamilton's rule makes the whole plane a field too. Points are no longer just background — they became numbers.
• Equations become solvable. With real multiplication you can square points, cube them, and build polynomials from them — and the Fundamental Theorem of Algebra (Step 8) promises that every polynomial equation has its full set of answers. Coordinate geometry cannot even ask the question.
• Rotation becomes arithmetic. With vectors, rotating needs extra tools (rotation matrices, trig formulas). With complex numbers, rotation IS multiplication: turn by θ = multiply by e^iθ. Geometry problems become short algebra steps. This one feature is why physics and engineering use complex numbers everywhere.
• Calculus works on it. Because you can divide, the difference quotient makes sense, and a whole subject — complex analysis — comes from doing calculus with complex numbers. It is surprisingly powerful (it can solve real integrals that normal methods cannot), but that is a topic for another tutorial.

And why the plane is special: Hamilton's 13-year search

Here is a nice ending. If one clever rule turns 2-D points into numbers, surely some rule turns 3-D points into numbers too? Hamilton thought so. He searched for the 3-D multiplication rule for thirteen years. His children would ask at breakfast, "Papa, can you multiply triplets yet?" and he had to admit he could only add them. Every rule he tried failed at division.

On 16 October 1843, while walking along the Royal Canal in Dublin, the answer came to him: it is impossible in 3-D — but it works in four dimensions, if you give up the rule that a×b = b×a. He was so happy that he carved the equations of his new "quaternions" into the stone of Broom Bridge right there (a plaque marks the spot today). Later, mathematicians proved the full result: number systems with real multiplication and division exist only in dimensions 1 (the real numbers), 2 (the complex numbers), 4 (quaternions — you lose a×b = b×a) and 8 (octonions — you lose even more). You cannot do this in 3-D, or 5-D, or anywhere else. So the plane is not just a place where geometry and arithmetic join — it is almost the place. Complex numbers are not just a convenient notation; they hold a truly special spot in mathematics.

Same plane, one extra rule: (a,b)×(c,d) = (ac−bd, ad+bc). That rule is the whole upgrade — and it only exists in dimensions 1, 2, 4, 8.

Everything on one page

Glossary — every term in one place