Foundations — Complex number a+bi — real part, imaginary part

This page assumes nothing. Before you touch the parent topic, every letter, sign, and picture it leans on is built here from the ground up, in an order where each idea rests on the one before it.

1. Numbers as points on a line

Before we can invent a new direction, we must be crystal clear on the old one.

WHAT it looks like: one horizontal arrow. Every number you have ever counted, measured, or split lives somewhere on this single line.

WHY the topic needs it: the whole point of complex numbers is that this line is too small — it has no room for a solution to $x^2=-1$. You cannot appreciate what's missing until you see how flat and one-dimensional this ruler is.

The symbol $\in$ means belongs to / "is a member of". Picture a fish $a$ swimming inside a pond $\mathbb{R}$.

2. Multiplication and squaring, and why squaring traps us on the positive side

Squaring is built on multiplication, so we pin down multiplication first — nothing skipped.

This is the exact gap the parent note means by "$x^2=-1$ has no real solution because $x^2\ge 0$ always."

The symbol $\ge$ means greater than or equal to; picture it as "sits at $0$ or further right".

3. Two directions at right angles: axis, plane, sheet

Before we can go "up" we must say precisely what up means and what surface we are drawing on.

WHY the topic needs this: the real ruler is a single axis (one dimension). To make room for $i$ we add a second axis at a right angle, and the flat sheet they span is where every complex number will live.

4. Inventing a new direction: the symbol $i$

Reals can't reach $-1$ as a square. Instead of giving up, we demand a solution and give it a name.

WHY a new direction and not a new spot on the line? Because the line is full — every point there already squares to something $\ge 0$. The only way to add $i$ without contradiction is to leave the line entirely and go along the perpendicular (up) axis from Section 3. One step up $=i$.

5. Gluing the two directions: the point $a+bi$

Now we have two directions — right (real) and up (imaginary). Combine them.

We usually abbreviate the whole point with a single letter $z$, so $z=a+bi$.

WHY the "$+$" between $a$ and $bi$? It is the same plus as in everyday arithmetic, but now it means "combine a sideways move with an upward move" — like giving directions "3 blocks east and 4 blocks north". You cannot simplify $3+4i$ into one number for the same reason "3 east + 4 north" isn't a single distance in one direction.

6. Reading off the parts — all the cases

Every point falls into one of five labelled types, sorted by whether each part is positive, negative, or zero. Cover them all so no example surprises you.

7. Two equations from one: the equality symbol

The parent note uses $⟺$. Let's earn it.

8. The prerequisite map

The diagram below is a flowchart: each box is an idea, and an arrow "$\to$" reads "is needed for / leads to". Follow the arrows top-to-bottom to see how the ruler's limitation snowballs into the complex plane. (If your reader shows raw text instead of boxes, read it as the plain outline underneath.)

Plain-outline legend (same information, in case the diagram does not render):

• Multiplication's sign rule → squaring is always $\ge 0$ → so nothing real solves $x^2=-1$.
• That gap → forces us to postulate $i$ with $i^2=-1$.
• Perpendicular axes → make a plane → give us the "up" axis to place $i$ on.
• Real axis (right) + up axis, glued with "$+$" → the complex number $a+bi$.
• From $a+bi$ we read off the real part $a$, the imaginary part $b$, the whole plane $\mathbb{C}$, and the equality rule $a=c$ and $b=d$.

9. Where each symbol reappears later

• $\mathbb{R}$, the ruler → becomes the horizontal axis in Argand diagram & modulus.
• The up-direction $i$ → becomes the vertical axis in the same diagram.
• $\mathrm{Im}(z)=b$ flips sign under Complex conjugate.
• The gluing "$+$" and rule $i^2=-1$ power Addition and multiplication of complex numbers.
• Distance from origin and turning-by-$i$ lead to Polar form and Euler's formula.
• The original gap $x^2=-1$ is exactly where Quadratic equations with complex roots begins.

Equipment checklist

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