Argand plane — geometric representation: Foundations

This page is the toolbox. Before you can read the parent note comfortably, every squiggle it uses must first mean something to you. We build each symbol from nothing, anchor it to a picture, and say exactly why the topic needs it. Read top to bottom — each item leans on the one before it.

0. The number line you already own

Before two axes, make sure one axis is solid.

Why the topic needs this: the whole Argand plane is built by taking two copies of this line and standing one of them upright. If you're shaky on "one number = one point on a line", the plane will feel like magic instead of arithmetic.

Figure s01 (below): a horizontal number line with ticks $- 4$ to $4$ , the origin marked in orange, and arrows showing "smaller" to the left and "bigger" to the right — it drives home one number = one point.

Figure — Argand plane — geometric representation

1. The symbol $i$ — the new ingredient

Why the topic needs it: $i$ is the reason we need a second axis at all. It is the label on the vertical direction.

2. $z = x + i y$ — the complex number

Why the topic needs it: two independent numbers $(x, y)$ = two coordinates = one dot on a plane. This single line is the seed of everything.

Reveal-test:

Real part of

z = - 5 + 2 i

- 5

Imaginary part of

z = - 5 + 2 i

2

(not

2 i

)

3. Coordinates and the point $P (x, y)$

Figure s02 (below): the axes with a teal arrow going $3$ across, then a plum arrow going $4$ up, landing on the orange dot $P (3, 4)$ — the picture of "an ordered pair is a recipe for a location".

Why the topic needs it: the parent note says " $z$ is the point $P (x, y)$ ". That sentence is empty unless you know an ordered pair is a recipe for a location.

4. Vectors and the arrow $O P$

Why the topic needs it: addition of complex numbers is drawn as arrows stacking tip-to-tail. That trick only exists if you already read $z$ as an arrow. See Vectors — Position Vectors and Addition.

5. The right triangle hidden in every $z$

Drop a straight line from $P (x, y)$ down to the horizontal axis. You've just built a right triangle.

Figure s03 (below): the same $P (3, 4)$ with the orange hypotenuse $O P$ , the teal horizontal leg $x$ , the plum vertical leg $y$ , the little right-angle square, and the angle $θ$ marked at the origin — this is the master picture the whole topic reads from.

This triangle is the single most important picture on this whole page. The modulus, the argument, and the polar form all read directly off it.

6. The modulus bars $∣ \cdot ∣$ — measuring size first

We are about to measure the length of the arrow, so we first need the symbol that means "length".

Now that $∣ z ∣$ has a meaning, we can compute it.

7. Pythagoras and — why the modulus is a square root

Why the topic needs it: modulus = distance = hypotenuse. No Pythagoras, no modulus formula.

Reveal-test:

Value of

9

3

(never

- 3

)

Why do we keep only the non-negative root for

∣ z ∣

∣ z ∣

is a length; lengths cannot be negative

8. Angles, radians, and $π$

Why the topic needs it: the argument is an angle, and the principal range $(- π, π]$ is stated in radians. You must read $π$ as "half a turn" fluently.

9. The argument $ar g z$ — the direction of the arrow

Why the topic needs it: every statement about "the angle of $z$ " in the parent note is really a statement about $ar g z$ , and answers must land inside $(- π, π]$ .

Reveal-test:

What does

ar g z

measure?

the anticlockwise angle of

O P

from the positive real axis

Which of

+ π

- π

is a valid principal argument?

+ π

(the range is

(- π, π]

, so

- π

is excluded)

10. tan, and its undo-button arctan

Common mistake arctan can only see two quadrants — and here is the concrete fix

A calculator's $tan^{- 1}$ only ever returns an angle in $(- \frac{π}{2}, \frac{π}{2})$ — that's just the right half of the plane (quadrants I and IV, where $x > 0$ ). But $z$ can point anywhere. Because $tan$ repeats every half-turn ( $π$ ), the same ratio $\frac{y}{x}$ belongs to two opposite directions. The safe recipe: first find the reference angle $α = tan^{- 1} \frac{y}{x}$ (always a positive angle in $(0, \frac{π}{2})$ ), then adjust by quadrant:

Quadrant	signs of $(x, y)$	$ar g z$
I	$x > 0, y > 0$	$α$
II	$x < 0, y > 0$	$π - α$
III	$x < 0, y < 0$	$- (π - α)$
IV	$x > 0, y < 0$	$- α$

Notice quadrants II and III (both $x < 0$ ) are exactly where plain $tan^{- 1}$ gets the wrong half of the plane — you must add or subtract $π$ there. The worked cases live in Complex Numbers — Modulus and Argument.

Why the topic needs it: the argument formula $tan θ = y / x$ and its correct inversion are the reason a naive arctan gives wrong answers in half the plane.

11. sin and cos — projecting the arrow back

Reveal-test:

cos θ

in terms of triangle sides

adjacent over hypotenuse

= x / r

sin θ

in terms of triangle sides

opposite over hypotenuse

= y / r

12. Conjugate $\overset{z}{ˉ}$ and negation $- z$

Why the topic needs it: the parent's "conjugate = reflection, $- z$ = half-turn" claims are pure geometry once you can see these two moves.

Prerequisite map

How to read this diagram: each box is one foundation from this page; arrows mean "feeds into". Follow the flow and you see the number line and $i$ combine into $z$ ; $z$ becomes a point, then a triangle; the triangle plus Pythagoras plus the modulus bars build $∣ z ∣$ ; the triangle plus radians plus arctan build $ar g z$ ; and modulus + argument + the sin/cos shadows assemble the polar form that the topic runs on. If the diagram fails to render, the same chain is written out in the "feeds into" words below it.

Feeds into (text backup): number line + $i$ → $z$ → point/vector → right triangle → (with Pythagoras, modulus-bars, square-root) modulus; triangle + radians + tan + arctan → argument; triangle → sin/cos shadows; modulus + argument + shadows → polar form → the Argand-plane topic; point → conjugate/negation → topic.

Equipment checklist

Test yourself — if any line stumps you, reread that section before the parent note.

I can place any real number on the number line

yes — one number is one point, right is bigger

I can state the one rule defining

i

i^{2} = - 1

;

i

is a

9 0^{\circ}

turn, off the real line

I can name the real and imaginary parts of

z = x + i y

Re (z) = x

Im (z) = y

(a real number, no

i

)

I can turn

(x, y)

into a dot and an arrow

O P

x

across,

y

up; the arrow is

O

to that dot

I can draw the right triangle inside any

z

horizontal leg

x

, vertical leg

y

, hypotenuse

O P

I know what the modulus bars mean before computing

∣ z ∣

= distance of the dot from the origin, always

\geq 0

I can state Pythagoras and get the modulus

x^{2} + y^{2} = ∣ z ∣^{2}

, so

∣ z ∣ = x^{2} + y^{2}

(non-negative root)

I know why we keep only the non-negative root

∣ z ∣

is a length;

returns the non-negative value

I can convert

18 0^{\circ}

and a quarter turn to radians

18 0^{\circ} = π

9 0^{\circ} = \frac{π}{2}

I can define

ar g z

and its principal range

anticlockwise angle from positive real axis; range

(- π, π]

I know why

+ π

is kept but

- π

is dropped

both name the negative real axis; the convention keeps

+ π

I can write

tan θ

from the triangle

opposite over adjacent

= y / x

I know what to do when

x = 0

don't divide; read

ar g z = \pm \frac{π}{2}

(up

= +

, down

= -

);

z = 0

has no direction

I know the quadrant fix for II and III

x < 0

, so add (

π - α

) or subtract

- (π - α)

from the reference angle

α

I can recover

x, y

from

r

and

θ

x = r cos θ

y = r sin θ

I can picture conjugation and negation

\overset{z}{ˉ}

reflects across real axis;

- z

rotates

18 0^{\circ}

Connections

3.5.03 Argand plane — geometric representation (Hinglish)
Complex Numbers — Modulus and Argument
Polar Form of Complex Numbers
Multiplication as Rotation and Scaling
De Moivre's Theorem
Vectors — Position Vectors and Addition
Loci in the Complex Plane