Intuition The ONE core ideaEvery complex number is really just a point on a flat map — a left–right amount and an up–down amount. Euler's formula says that if you put a point at distance 1 θ cos θ + i sin θ e i θ
This page assumes nothing . Before you touch the parent proof , you need to already see eleven little pictures. We build them one at a time, each using only the ones before it.
Definition A number as a positionA number is a position on a straight line. The line has a centre called 0 x name for "some position on this line" — a placeholder we can fill in later.
Why do we need this? Everything that follows is "a line, then a second line stuck on at a right angle." So we start with one line.
Here is the first genuinely new idea of the whole topic.
Definition The imaginary uniti i new number that sits one step UP from 0 , off the original left–right line. It is defined by the rule
i 2 = − 1. "i − 1
i − 1 + 1 − 1 half-turn (180°) . If two multiplications by i − 1 i 2 = − 1 one multiplication by i half of a half-turn = a quarter-turn (90°) . So multiplying by i
Why the topic needs it: without a number that means "rotate 90°," there is no way to talk about spinning, and e i θ all about spinning.
The flat map in figure s02 is the complex plane : horizontal axis = real part, vertical axis = imaginary part. A complex number is just a point (or an arrow from the centre to that point ). This is the map every later picture lives on.
Worked example Reading points
3 2 i 3 + 2 i
The modulus ∣ z ∣ z = a + bi distance from the centre 0 ∣ z ∣ = a 2 + b 2 .
Why we need it: Euler's formula is specifically about points at distance exactly 1 . To say "distance 1" we first need a word for distance — that word is modulus.
Definition Argument (angle)θ argument θ how far anticlockwise you have turned from the positive-right direction to point at your number.
Definition Radians — the honest way to measure an angleA radian measures an angle by the length of arc it sweeps on the unit circle . Turn until the tip has travelled a distance equal to the radius (1), and you have turned 1 radian . A full circle has circumference 2 π
full turn = 2 π
half turn = π
quarter turn (90°) = 2 π
Common mistake Why not degrees?Degrees are an arbitrary human choice (360 because ancient calendars liked it). The series that builds cos sin arc length , i.e. in radians. Feed degrees into Euler's formula and it silently breaks. Always use π /2 90
Now the two workhorse functions. We define them on the unit circle , not on triangles, because that is how the topic uses them.
Definition Cosine and sine as coordinatesStand a clock hand of length 1 θ
horizontal position (left–right shadow) is cos θ vertical position (height) is sin θ
So the tip is literally the point ( cos θ , sin θ ) cos θ + i sin θ
Intuition Why this ratio "measures" the angleAs you turn, the shadow cos θ 1 0 − 1 sin θ 0 1 own unique pair of (shadow, height) on the circle — so knowing the pair tells you the angle, and vice versa. That two-way link is exactly why cos sin encode rotation.
Because the tip is always distance 1 from the centre, Pythagoras on figure s04 gives the identity you will lean on constantly:
cos 2 θ + sin 2 θ = 1.
x n x n multiply x n ": x 3 = x ⋅ x ⋅ x x 0 = 1
n ! n ! n multiply all whole numbers from 1 n ": 4 ! = 1 ⋅ 2 ⋅ 3 ⋅ 4 = 24 0 ! = 1
Definition The sum symbol∑ n = 0 ∑ ∞ ( expression in n ) "add up this expression for n = 0 n = 1 n = 2 The ∑ Sum .
Why the topic needs these: the parent proof writes e x cos x sin x infinite sums of powers over factorials . You cannot read a single line of the proof without knowing what ∑ x n n !
e e ≈ 2.718 e x growth rate at every instant equals its own current value — it is "continuous, self-feeding growth."
Why we need the sum form and not "2.718 to a power": you cannot raise a decimal to an imaginary power i θ i θ sum form takes any input, real or complex, and just adds terms. That is the whole trick of the proof: substitute x = i θ Taylor & Maclaurin Series for where these sums come from.)
We already have i 2 = − 1
i 0 = 1 , i 1 = i , i 2 = − 1 , i 3 = i 2 ⋅ i = − i , i 4 = i 2 ⋅ i 2 = ( − 1 ) ( − 1 ) = 1.
Intuition The four-beat cycleAfter i 4 1 repeat every four steps : 1 , i , − 1 , − i , 1 , i , … sort the exponential sum into a real pile (the 1 , − 1 i , − i cos sin
Number line and placeholder x
Complex plane: point a + bi
Modulus of z: distance from 0
Unit circle: distance 1 from 0
Multiply by i = rotate 90 degrees
cos theta = shadow, sin theta = height
Powers, factorial, sum symbol
Powers of i cycle: 1, i, -1, -i
Read it top-down: the plane and i where numbers live; modulus + angle + cos / sin unit-circle language ; powers + ∑ exponential-as-sum ; the i sorting engine . Feed all four into the substitution x = i θ
Test yourself — reveal only after answering aloud.
What does the symbol i A number one step "up" off the line with i 2 = − 1
In a + bi a b a b
What is the modulus ∣ z ∣ z = a + bi a 2 + b 2 — the distance from the centre
0 to the point.
What is the unit circle? All points at modulus exactly 1
What is 1 radian? The angle whose arc along the unit circle has length equal to the radius (1); a full turn is 2 π
On the unit circle, what are cos θ sin θ The horizontal shadow and the vertical height of the tip after turning by θ
What identity links cos sin cos 2 θ + sin 2 θ = 1
What does n = 0 ∑ ∞ Add the expression for n = 0 , 1 , 2 , …
What is 0 ! x 0 Both equal 1
Write e x ∑ n = 0 ∞ n ! x n = 1 + x + 2 ! x 2 + 3 ! x 3 + ⋯
Why must we use the sum form of e x Because raising a decimal to an imaginary power is meaningless, but the sum accepts any complex input.
List the cycle of powers of i i 0 1 , i , − 1 , − i ,
Why does the exponential series split into real and imaginary halves? Because i n 1 , i , − 1 , − i i
