Visual walkthrough — Complex number a+bi — real part, imaginary part

This is the visual companion to the parent topic. There the result was stated; here we watch it grow.

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Step 1 — The ruler, and the hole in it

WHAT. Ordinary numbers — $\dots ,-2,-1,0,1,2,\dots$ and everything between — live on a single straight line. Call it the real line. A number is just a position on that line: right of $0$ is positive, left of $0$ is negative.

WHY. Before inventing anything new we must see exactly what the old system cannot do. The question "which number, multiplied by itself, gives $-1$?" has no answer here — and the picture shows why.

PICTURE. In the figure, squaring a number is "how far from $0$, then always land on the right (positive) side." A positive number squared lands right; a negative number squared also lands right (two lefts make a right, just like $(-3)\times (-3)=+9$). So no point on the ruler squares to a point on the left — nothing squares to $-1$. The ruler has a hole.

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Step 2 — Invent a new direction: the number $i$

WHAT. We refuse to accept "no solution." We declare a brand-new number and give it the name $i$, defined by the single rule $i^2=-1.$ Term by term: $i^2$ means "$i$ multiplied by itself"; $=-1$ means "the answer is one step left of $0$ on the ruler."

WHY. This is the minimal new rule that plugs the hole in Step 1. We add exactly one property and nothing else, then check (later) that it never contradicts itself.

PICTURE. Since $i$ is not anywhere on the ruler, we draw it off the ruler — a new axis pointing up, at a right angle. One step up is $i$; two steps up is $2i$; one step down is $-i$. The horizontal axis stays "real," the vertical axis becomes "imaginary." Multiplying by $i$ will turn out to be a quarter-turn, and the figure previews this: start at $1$ (pointing right), multiply by $i$, and you point up.

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Step 3 — Glue the two directions: the point $a+bi$

WHAT. A general new number is "walk $a$ steps along the ruler, then $b$ steps up the new axis." We write this as $z=a+bi.$ Here $a$ is the real part (sideways distance), $b$ is the imaginary part — a plain real number counting how many $i$-steps up, not $bi$ itself.

WHY. We need arithmetic to stay inside our new world. Any sum of a sideways piece and an upward piece has this shape, so $a+bi$ is the natural general form — a single point $(a,b)$ on a flat sheet, the Argand plane.

PICTURE. The figure plots $z=3+4i$ as an arrow from the origin to the point $3$ right and $4$ up. The horizontal leg (length $a=3$) is the real part; the vertical leg (length $b=4$) is the imaginary part. The number is the arrow.

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Step 4 — Multiply two of them, honestly, term by term

WHAT. Take two complex numbers $z=a+bi$ and $w=c+di$. We multiply them using the only tools we have: ordinary distribution (every part times every part), plus the one new rule $i^2=-1$.

$$(a+bi)(c+di)=\underset{\text{right}\times \text{right}}{\underbrace{ac}}+\underset{\text{right}\times \text{up}}{\underbrace{adi}}+\underset{\text{up}\times \text{right}}{\underbrace{bci}}+\underset{\text{up}\times \text{up}}{\underbrace{bd{i}^2}}.$$

WHY. We are not allowed to invent a new multiplication — that would risk contradictions. We reuse the rule we already trust (distributivity) and see what it forces.

PICTURE. The figure is a $2\times 2$ grid: rows labelled $a,bi$; columns labelled $c,di$. Each of the four cells is one product. Three cells are ordinary; the amber cell in the bottom-right is $bi\times di=bd{i}^2$ — the only place the special rule bites.

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Step 5 — Apply the one rule, and collect

WHAT. Replace $i^2$ by $-1$ in the amber cell, then gather the "flat" (real) pieces separately from the "$i$" (imaginary) pieces:

$$bd{i}^2=bd(-1)=-bd,$$ $$(a+bi)(c+di)=\underset{\text{new real part}}{\underbrace{(ac-bd)}}+\underset{\text{new imaginary part}}{\underbrace{(ad+bc)}}i.$$

Term by term: $ac-bd$ collects the two cells with no leftover $i$ (the $ac$ cell, and the flipped $bd$ cell). $ad+bc$ collects the two cells carrying a single $i$.

WHY. Grouping by "how many $i$'s" is exactly the equality rule: a complex number is one real amount sideways and one real amount up. Anything of the form $A+Bi$ is again a legal point on our sheet — so the world is closed under multiplication. It never spits us back out.

PICTURE. The figure re-colours the grid: the two blue cells ($ac$ and the now-negative $bd$) merge into the horizontal real part; the two cyan cells ($ad$ and $bc$) merge into the vertical imaginary part. Four cells collapse into one arrow.

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Step 6 — Degenerate check: multiply by a plain real number

WHAT. Put $d=0$, so $w=c$ is an ordinary real number. Our formula gives $(a+bi)(c+0i)=(ac-b\cdot 0)+(a\cdot 0+bc)i=ac+bci=c(a+bi).$

WHY. A brand-new rule is only trustworthy if, when the "new-ness" is switched off, it agrees with the old arithmetic we already believe. It does: multiplying by a real number $c$ just scales the arrow — longer if $c>1$, shorter if $0<c<1$, flipped through the origin if $c<0$.

PICTURE. The figure shows $z$ and $2z$ (double length, same direction) and $-z$ (same length, pointing exactly opposite). No twisting — pure stretch/flip.

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Step 7 — Degenerate check: multiply by $i$ (pure rotation)

WHAT. Put $c=0,d=1$, so $w=i$. The formula gives $(a+bi)(0+1i)=(a\cdot 0-b\cdot 1)+(a\cdot 1+b\cdot 0)i=-b+ai.$

WHY. This is the deepest sanity check. The point $(a,b)$ becomes $(-b,a)$. Ask: what geometric move sends $(a,b)\to (-b,a)$? It is exactly a $90^{\circ }$ counter-clockwise turn about the origin. So the promise from Step 2 — "$\times i$ is a quarter-turn" — is now proven from the multiplication formula, not assumed.

PICTURE. The figure shows an arrow $z=(a,b)$ in amber and its image $iz=(-b,a)$ in cyan, with the right-angle of the turn marked. Multiply by $i$ again and you reach $(-a,-b)=-z$ — the half-turn, confirming $i^2=-1$.

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The one-picture summary

WHAT the final figure shows. All four moving parts at once: the ruler with its hole (Step 1), the $i$-axis rising off it (Step 2), a sample point $a+bi$ (Step 3), the $2\times 2$ product grid with the amber $i^2$ cell (Steps 4–5), and the two special cases — real $c$ stretches, imaginary $i$ turns (Steps 6–7). Every arrow is annotated with the symbol it represents.

Recall Feynman retelling — the whole walk in plain words

We started with a plain ruler and found a hole: nothing on it, squared, gives $-1$, because every square lands on the positive side. So we invented one new number, $i$, and insisted $i\times i=-1$. Since $i$ isn't on the ruler, we drew it as a new "up" direction. Now every number is "so far right, so far up" — written $a+bi$ — a point on a flat sheet. To multiply two such points we just did ordinary "each-times-each," which makes four pieces; the only tricky piece has $i^2$ in it, and we swap that for $-1$. Collecting the flat pieces and the up pieces gives $(ac-bd)+(ad+bc)i$ — still a point on the sheet, so the world is closed. Finally we tested the extremes: multiply by an ordinary number and the arrow just stretches or flips (no turning); multiply by $i$ and the arrow rotates a clean quarter-turn. That last check proves the promise we made at the start — $i$ is literally the "quarter-turn" number.

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Connections

• Argand diagram & modulus — the flat sheet and arrow lengths used throughout.
• Addition and multiplication of complex numbers — the algebra behind these pictures.
• Complex conjugate — reflecting the arrow across the real axis.
• Polar form and Euler's formula — turns "scale + rotate" into $r{e}^{i\theta }$.
• Quadratic equations with complex roots — where the invented $i$ pays off.

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Concept Map