Algebraic operations — add, subtract, multiply, divide (rectangular and polar): Foundations

Before you can operate on complex numbers, you must be able to read every mark on the page. Below, each symbol is introduced in the order that makes the next one possible — nothing is used before it is built.

1. The plane — where every arrow lives

Figure — Algebraic operations — add, subtract, multiply, divide (rectangular and polar)

Read the figure: the yellow arrow along the bottom is the ordinary number line (real, horizontal); the blue arrow going up is the new imaginary direction; the pink dot sits where two readings — "how far right" and "how far up" — cross. That single dot is what we will name. This sheet is the Complex plane (Argand diagram). Everything else on this page is a label for a point — or equivalently the arrow from the origin to that point.

2. $a$ and $b$ — the two shadows

Look at the figure: drop the arrow's tip straight down (blue dashed line) to get $a$ , straight across (pink dashed line) to get $b$ . Two numbers pin the arrow completely.

3. $i$ — the tag that means "turn 90°"

Read the figure: the yellow arrow is $1$ (pointing right). One $9 0^{\circ}$ turn (upper chalk arc) sends it to the blue arrow $i$ (pointing up). A second $9 0^{\circ}$ turn (second arc) sends it to the pink arrow $- 1$ (pointing left) — so applying $i$ twice is a half turn, which is literally $i^{2} = - 1$ . That is the whole rule, made visible.

So the two readings glue into one name: $z = a + bi .$ This is the rectangular form — "rectangular" because $a$ and $b$ are the sides of a rectangle whose diagonal is the arrow.

4. $r$ — the length of the arrow (modulus)

Why Pythagoras and not something else? Because $a$ (right) and $b$ (up) meet at a right angle — that is the one theorem that turns two perpendicular legs into a diagonal length. See the triangle in figure s02: the slanted side is $r$ .

$r$ is one half of Modulus and Argument. Multiplication will multiply these lengths, so we need a name for length before we can say that.

5. $θ$ — the tilt of the arrow (argument), and why $cos, sin, tan$ appear

Now three trig words enter. Each is earned by a question about the same right triangle (arrow = hypotenuse $r$ , base = $a$ , height = $b$ ), and we assume $r > 0$ (i.e. $z \neq = 0$ ):

Why these tools? We want to convert between (length, angle) and (right, up). Rearranging the first two gives exactly the conversion the parent uses: $a = r cos θ, b = r sin θ .$ And $tan$ is the tool for the reverse trip because it depends only on the angle, not the length — it cancels $r$ ( $\frac{b}{a} = \frac{r s i n θ}{r c o s θ}$ ), so it isolates the tilt.

$θ$ is the other half of Modulus and Argument. Multiplication will add these angles.

6. Polar form, cis, and $e^{i θ}$ — three spellings of one name

Feed $a = r cos θ$ and $b = r sin θ$ back into $z = a + bi$ : $z = r cos θ + (r sin θ) i = r (cos θ + i sin θ) .$

Recall Where does

$e^{i θ} = cos θ + i sin θ$ come from? (not needed here, but honest) $e^{x}$ is defined by an infinite sum $e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots$ . Feeding in $x = i θ$ and using $i^{2} = - 1, i^{3} = - i, i^{4} = 1$ to simplify the powers, the terms split into two groups: the ones with no $i$ collect into exactly the series for $cos θ$ , and the ones carrying an $i$ collect into $i$ times the series for $sin θ$ . That is why the shorthand is legal — it isn't a fresh definition pulled from nowhere, it's a rearranged sum. This is also why $θ$ must be in radians: those series are only equal to $cos, sin$ when $θ$ is radian-measured. Full detail lives in Euler's formula.

7. The conjugate $\overset{z}{ˉ}$ — mirror across the horizontal axis

Its magic property is $z \overset{z}{ˉ} = (a + bi) (a - bi) = a^{2} + b^{2} = r^{2},$ a plain real number (no $i$ ). This is Complex conjugate, and it is the exact trick that makes division work — multiplying top and bottom of a fraction by the denominator's conjugate turns the bottom into a real length you can divide by.

z

stand for?

The whole complex number — the arrow (point) itself;

Re (z), ∣ z ∣, ar g (z)

are all facts about that one arrow.

What does

Re (z)

read off the arrow?

The horizontal shadow

a

— how far right.

What single rule defines

i

i^{2} = - 1

; equivalently, multiplying by

i

rotates an arrow

9 0^{\circ}

anticlockwise.

z = 3 + 2 i

, what is the imaginary part

b

2

(a plain number, not

2 i

How do you get the length

r

from

a

and

b

, and why that formula?

r = a^{2} + b^{2}

, from Pythagoras because

a

and

b

are perpendicular legs.

Which ratio equals

cos θ

on the arrow's triangle?

adjacent over hypotenuse

= a / r

Which ratio equals

tan θ

, and why does it ignore the length?

opposite over adjacent

= b / a

; the

r

cancels, leaving only the tilt.

What is

ar g (z)

when

z = 0

Undefined — the zero arrow has no direction; only its modulus (

r = 0

) makes sense.

What is

θ

for a pure imaginary

z = 2 i

(where

a = 0

9 0^{\circ} = π /2

; read it from the picture since

tan

would divide by zero.

Which angle unit do

e^{i θ}

sin θ

cos θ

expect?

Radians (full turn

= 2 π

); convert degrees by

\times π /180

Why can't plain

arctan (b / a)

give the argument alone?

tan

repeats every

18 0^{\circ}

, so it can't tell a quadrant from its opposite; check signs of

a, b

/ use atan2.

Write

z

in polar form three ways.

r (cos θ + i sin θ) = r cis θ = r e^{i θ}

What does

e^{i θ}

mean geometrically, and where does the formula come from?

A unit arrow at angle

θ

; it comes from the power series of

e^{x}

with

x = i θ

splitting into

cos θ + i sin θ

What is

\overset{z}{ˉ}

geometrically and what is

z \overset{z}{ˉ}

Mirror across the horizontal axis;

z \overset{z}{ˉ} = a^{2} + b^{2} = r^{2}

, a real number.

Why is addition done in rectangular, not polar?

Independent right/up axes just add like vectors; tilts and lengths don't add for sums.

Connections

3.5.08 Algebraic operations — add, subtract, multiply, divide (rectangular and polar) (Hinglish)
Complex plane (Argand diagram)
Modulus and Argument
Complex conjugate
Euler's formula
De Moivre's Theorem
Vectors in 2D

Foundations — Algebraic operations — add, subtract, multiply, divide (rectangular and polar)

1. The plane — where every arrow lives

2. $a$ and $b$ — the two shadows

3. $i$ — the tag that means "turn 90°"

4. $r$ — the length of the arrow (modulus)

5. $θ$ — the tilt of the arrow (argument), and why $cos, sin, tan$ appear

6. Polar form, cis, and $e^{i θ}$ — three spellings of one name

7. The conjugate $\overset{z}{ˉ}$ — mirror across the horizontal axis

8. Arrows add like vectors

Prerequisite map

Equipment checklist

Connections