Complex Plane & Argand Diagram: Visualize ℂ

From Algebra to Geometry

Every real number already has a place: a point on the number line.

Negative numbers sit to the left of

0

, positive ones to the right, and fractions and irrationals fill the gaps between. Each real number gets exactly one point, and every point is used.

The complex number 2 + 3i has nowhere to go on that line.

Try to place a few numbers. The real ones each land on a point.

2 + 3i

does not: it is not to the left or right of anything, because it carries a second number, the

3

in front of

i

, that the line has no room for.

number	its place on the line
−2	two units left of 0
0	the starting point
1.5	halfway between 1 and 2
2 + 3i	no place at all

The algebraic form

z = a + bi

contains two real numbers: the real part

a

and the imaginary part

b

. Two numbers naturally suggest two dimensions. Rather than confining complex numbers to a line, we grant them an entire plane — a flat surface where horizontal position encodes the real part and vertical position encodes the imaginary part.

The definition

The complex plane adds one new direction, at right angles to the number line. The horizontal axis carries the real part, the vertical axis carries the imaginary part, and the number

a + bi

is the point

(a, b)

.

RULE

Every complex number finds a home at exactly one point, and every point in the plane represents exactly one complex number.

So

2 + 3i

is the point

(2, 3)

: two units right, three units up — the way two numbers on a map, east and north, fix a single place.

A complex number is a point of the plane, read off by its two coordinates.

The correspondence is direct: the complex number

z = a + bi

maps to the point with coordinates

(a, b)

. The number

4 + 3i

becomes the point

(4, 3)

, located four units right of the origin and three units up. The number

-2 + i

becomes

(-2, 1)

, sitting in the second quadrant.

This visualization carries a historical name: the Argand diagram, honoring Jean-Robert Argand, who published the geometric interpretation in 1806. Other mathematicians including Caspar Wessel and Carl Friedrich Gauss developed similar ideas independently. The plane itself is called the complex plane, and its two axes receive names reflecting their contents — the real axis runs horizontally, the imaginary axis runs vertically.

z = 2 + 3i

at the point

(2, 3)

, in the first quadrant. The same state is on the complex number explorer.

Nothing was taken away: the old number line is still there, as the horizontal axis.

A real number

a

is the complex number

a + 0i

, so it lands at

(a, 0)

— on the real axis, in the same order and at the same distances as on the old line.

real number	as a complex number	point in the plane
3	3 + 0i	(3, 0)
−2	−2 + 0i	(−2, 0)
0	0 + 0i	(0, 0)

The identification of complex numbers with points transforms algebra into geometry. Questions about numbers become questions about locations. Properties of operations become properties of movements. The entire machinery of coordinate geometry — distance, angle, reflection, rotation — applies immediately to complex arithmetic, providing visual intuition that pure symbolism cannot offer.

One thing is lost: complex numbers cannot be lined up from smallest to largest.

On a line, every number is to the left or right of every other, which is what "smaller" and "larger" mean. In the plane there is no single direction to compare along —

2 + 3i

and

3 + 2i

are neither smaller nor larger than each other. The section on the lack of order explains why no ordering can be made to work.

The rest of the page names the two axes, plots complex numbers, reads them as arrows from the origin, and shows the conjugate and the arithmetic operations as movements in the plane.

The Two Axes

The complex plane organizes itself around two perpendicular lines passing through the origin. These axes partition the plane and provide reference directions for locating every complex number.

The horizontal axis carries the real numbers. Every point on this line has coordinates

(a, 0)

, corresponding to complex numbers of the form

z = a + 0i = a

. The integers

-3, -2, -1, 0, 1, 2, 3

appear at their familiar positions stretching left and right. Irrational values like

\sqrt{2}

and

\pi

occupy their proper locations. The entire real number line, in all its infinite extent, embeds as the horizontal axis of the complex plane.

The vertical axis carries the pure imaginary numbers. Every point on this line has coordinates

(0, b)

, corresponding to complex numbers

z = 0 + bi = bi

. The number

i

sits one unit above the origin;

2i

sits two units up;

-3i

sits three units down. No real numbers appear on this axis except at the origin itself.

The origin occupies a singular position in this structure. Located at

(0, 0)

, it represents the complex number

0 = 0 + 0i

. This point lies on both axes simultaneously — it belongs to the real numbers (having zero imaginary part) and to the pure imaginary numbers (having zero real part). No other point shares this dual membership. The origin serves as the reference for all measurements: distances emanate from it, angles open from the positive real axis through it, and the entire coordinate system centers upon it.

Every other complex number — those with both

a \neq 0

and

b \neq 0

— lives off both axes, somewhere in the interior of one of the four quadrants.

Plotting Complex Numbers

Placing a complex number on the plane requires reading its real and imaginary parts as horizontal and vertical coordinates. The process mirrors plotting ordered pairs in elementary coordinate geometry, with the complex notation providing a different packaging of the same information.

Consider

z = 3 + 2i

. The real part

3

dictates movement three units right from the origin. The imaginary part

2

dictates movement two units up. The point

(3, 2)

marks the location. For

z = -1 + 4i

, move one unit left and four units up to reach

(-1, 4)

. For

z = -2 - 3i

, move two units left and three units down to reach

(-2, -3)

in the third quadrant.

Pitfall: Swapping the coordinates

Plotting

-1 + 4i

at

(4, -1)

reads the imaginary part as the horizontal coordinate and lands in the fourth quadrant. The real part is horizontal:

-1 + 4i

is the point

(-1, 4)

, in the second quadrant.

The four quadrants organize complex numbers by the signs of their components. The first quadrant (upper right) contains numbers with positive real and positive imaginary parts. The second quadrant (upper left) holds numbers with negative real parts but positive imaginary parts. The third quadrant (lower left) contains both parts negative. The fourth quadrant (lower right) has positive real parts with negative imaginary parts. Knowing the quadrant immediately reveals the sign pattern of a complex number.

Special cases occupy the axes rather than the quadrants. Real numbers like

5

appear at

(5, 0)

on the horizontal axis. Pure imaginary numbers like

-2i

appear at

(0, -2)

on the vertical axis. These numbers have one zero component, placing them precisely on the boundary between quadrants rather than inside any particular region.

Complex numbers expressed differently may occupy the same point. The number

\frac{6}{2} + \frac{8}{4}i

simplifies to

3 + 2i

and plots at

(3, 2)

regardless of how the components were originally written. The geometric location depends only on the final real values of

Re(z)

and

Im(z)

— the extraction marks — not on the form of the expression.

Region	Re(z)	Im(z)	Example
Quadrant I (upper right)	> 0	> 0	3 + 2i at (3, 2)
Quadrant II (upper left)	< 0	> 0	−1 + 4i at (−1, 4)
Quadrant III (lower left)	< 0	< 0	−2 − 3i at (−2, −3)
Quadrant IV (lower right)	> 0	< 0	2 − i at (2, −1)
Real axis (excluding origin)	≠ 0	= 0	5 at (5, 0); −π at (−π, 0)
Imaginary axis (excluding origin)	= 0	≠ 0	3i at (0, 3); −2i at (0, −2)
Origin (on both axes)	= 0	= 0	0 at (0, 0) — the only such point

z = −3 + 2i plotted as (−3, 2)

The real part −3 is read along the horizontal axis and the imaginary part 2 up the vertical one, placing the number in the second quadrant; the arrow from the origin has length √13 ≈ 3.61. Plotting a complex number is plotting an ordered pair with the axes renamed. Move the point into every quadrant on the complex number explorer.

The plane makes the two-part structure of a complex number visible at a glance.

Complex Numbers as Vectors

Points in the plane sit motionless at fixed locations. Vectors capture something more dynamic — they encode magnitude and direction, representing displacement rather than mere position. Complex numbers admit both interpretations, and the vector viewpoint proves especially powerful for understanding arithmetic operations.

Fact: A surveyor drew the plane first

Caspar Wessel, a surveyor in Denmark, added such arrows tip to tail and multiplied them by adding their angles in a paper read in 1797. Written in Danish, it went unnoticed for a century, so English names the plane after Argand (1806) and German after Gauss (1831).

The complex number

z = a + bi

can be viewed as an arrow beginning at the origin and ending at the point

(a, b)

. The horizontal component

a

describes rightward displacement; the vertical component

b

describes upward displacement. Together they determine both where the vector points and how long it extends. The number

4 + 3i

becomes an arrow reaching to

(4, 3)

, angled upward and to the right with a length computable from its components.

Vector addition follows the parallelogram rule, also known as tip-to-tail addition. Given two vectors

z_1

and

z_2

, place the tail of

z_2

at the head of

z_1

. The sum

z_1 + z_2

stretches from the origin to the final endpoint. Algebraically,

(a + bi) + (c + di) = (a + c) + (b + d)i

adds components separately — and geometrically, this matches exactly how vectors combine through the parallelogram construction.

Subtraction reverses the second vector before adding. To compute

z_1 - z_2

, flip

z_2

to point in the opposite direction and then add. The result connects the two points in the plane, giving the displacement from

z_2

to

z_1

. The modulus of

z_1 - z_2

measures the distance between them.

Pitfall: Subtracting the moduli

For

z_1 = 4 + 3i

and

z_2 = 1 - i

, taking the distance as

|z_1| - |z_2| = 5 - \sqrt{2} \approx 3.59

compares two arrow lengths, not the two points. The distance is

|z_1 - z_2| = |3 + 4i| = 5

.

The vector interpretation makes the modulus geometrically obvious. The length of the arrow from the origin to

(a, b)

follows from the Pythagorean theorem:

|z| = \sqrt{a^2 + b^2}

. This is simply the distance formula applied to a vector's components, revealing modulus as nothing more exotic than length.

z₁ + z₂ as the diagonal of a parallelogram

The two arrows z₁ and z₂ are placed tail to tail and the sum is the diagonal of the parallelogram they span, exactly as vectors add: real parts add along the horizontal, imaginary parts along the vertical. Drag either summand and watch the diagonal follow on the complex addition and subtraction tool.

Treating complex numbers as vectors is what turns their addition into geometry.

Visualizing the Conjugate

The conjugate operation gains immediate clarity in the geometric setting. For

z = a + bi

, the conjugate

\bar{z} = a - bi

preserves the real part while negating the imaginary part. On the plane, this transforms the point

(a, b)

into

(a, -b)

— the same horizontal coordinate, opposite vertical coordinate.

This transformation has a name: reflection across the real axis. The real axis acts as a mirror. Points above the axis map to corresponding points below; points below map to points above. Points on the real axis itself — the real numbers — remain fixed in place, unmoved by conjugation because they have no imaginary component to negate.

Concrete examples illustrate the reflection. The number

z = 2 + 3i

sits at

(2, 3)

, two units right and three units up. Its conjugate

\bar{z} = 2 - 3i

sits at

(2, -3)

, the same distance right but now three units down. Draw both points and the horizontal axis, and the mirror symmetry becomes unmistakable. Similarly,

z = -1 + 4i

at

(-1, 4)

reflects to

\bar{z} = -1 - 4i

at

(-1, -4)

.

Pitfall: Reflecting across the imaginary axis

Placing the conjugate of

2 + 3i

at

(-2, 3)

negates the real part instead; that point is

-2 + 3i = -\bar{z}

. Only the imaginary part flips:

\bar{z} = 2 - 3i

, at

(2, -3)

.

The reflection perspective explains several algebraic facts geometrically. Taking the conjugate twice returns to the original point — reflecting across the same mirror twice restores every point to its starting position, confirming

\overline{\bar{z}} = z

. A number equals its own conjugate precisely when reflection leaves it unchanged, which happens only for points on the mirror itself — the real numbers. A number equals the negative of its conjugate when it lies on the imaginary axis.

z	Point (a, b)	z̄	Point of z̄	Geometric note
2 + 3i	(2, 3)	2 − 3i	(2, −3)	upper half-plane → lower half-plane, same x
−1 + 4i	(−1, 4)	−1 − 4i	(−1, −4)	Q2 reflects to Q3
5 (real)	(5, 0)	5	(5, 0)	fixed — points on the mirror (real axis) don't move
3i (pure imaginary)	(0, 3)	−3i	(0, −3)	reflects to its negative; z̄ = −z

z and z̄ mirrored in the real axis, |z| = 3.6

The point z and its conjugate z̄ sit at the same distance from the origin, one above the real axis and one below: conjugation keeps the real part and flips the sign of the imaginary part, so it is a reflection in the real axis. The modulus, the length of either arrow, is the same 3.6 for both. Drag z anywhere and watch its mirror image follow on the conjugate and modulus tool.

The conjugate is the mirror image, and mirror images have equal moduli.

The Lack of Order in $\mathbb{C}$

Real numbers line up in sequence. Given any two distinct reals, one is larger and one is smaller. The relation "less than" totally orders

\mathbb{R}

, allowing comparison between any pair of elements. This ordering underlies much of real analysis — limits approach from below or above, intervals have left and right endpoints, and functions increase or decrease.

Complex numbers refuse such arrangement.

THEOREM

No consistent definition of "less than" extends from

\mathbb{R}

to

\mathbb{C}

while preserving the essential properties we expect from an ordering.

The question "Is

3 + 2i

greater than

1 + 5i

?" has no meaningful answer. Neither number dominates the other in any standard sense.

Pitfall: Comparing complex numbers like reals

Writing

2i > i

because

2 > 1

leads to nonsense: subtracting

i

gives

i > 0

, and multiplying by this positive

i

gives

i^2 > 0

, that is

-1 > 0

. Only real quantities compare, such as the moduli

|2i| = 2 > |i| = 1

.

The geometry reveals why ordering fails. Real numbers occupy a one-dimensional line where direction determines comparison — rightward means greater, leftward means lesser. Complex numbers fill a two-dimensional plane where points spread in all directions. No single direction consistently represents "greater than" without privileging some numbers arbitrarily over others. Should

3 + 2i

exceed

1 + 5i

because it lies further right, or should

1 + 5i

exceed

3 + 2i

because it lies further up? Either choice can be made consistent on its own, but no such ordering agrees with multiplication.

One meaningful comparison survives: the modulus provides a measure of size. The modulus

|z| = \sqrt{a^2 + b^2}

gives the distance from the origin to the point. We can legitimately say

|3 + 4i| = 5

exceeds

|1 + i| = \sqrt{2}

because

5 > \sqrt{2}

. But modulus compares magnitudes, not the numbers themselves. Multiple complex numbers share the same modulus — every point on a circle centered at the origin has identical modulus — so comparing distances cannot substitute for a full ordering.

Moduli of complex numbers can be compared, but no ordering of the numbers themselves respects their arithmetic.

Visualizing Operations

Arithmetic operations on complex numbers translate into geometric actions on the plane. This correspondence provides visual intuition that algebraic manipulation alone cannot offer, and it foreshadows the deeper connections explored in trigonometric and exponential forms.

Addition performs vector combination. Given

z_1

and

z_2

as arrows from the origin, their sum

z_1 + z_2

completes the parallelogram formed by the two arrows. Equivalently, translate

z_2

so its tail sits at the head of

z_1

; the sum extends from the origin to the new head of

z_2

. This tip-to-tail construction matches the component-wise addition

(a + c) + (b + d)i

with perfect precision.

Fact: A treasure map without its gallows

In a puzzle from Gamow's 1947 book, the map starts at a gallows that no longer stands: walk to an oak, turn a right angle, walk as far again; repeat at a pine, turning the other way; dig midway. With the turns written as multiplying by

i

and

-i

, the gallows cancels from the sum, so the treasure is found anyway.

Subtraction measures displacement between points. The difference

z_1 - z_2

represents the vector pointing from

z_2

to

z_1

. Its modulus

|z_1 - z_2|

gives the distance between the two complex numbers. In navigation terms, subtraction answers "how far and in what direction must I travel from

z_2

to reach

z_1

?"

Multiplication introduces richer geometry involving both scaling and rotation. Multiplying by a positive real number stretches or shrinks the vector without changing its direction —

2z

points the same way as

z

but twice as far from the origin. Multiplying by

-1

rotates the vector by

180°

, flipping it to the opposite side of the plane. Multiplying by

i

rotates

90°

counterclockwise: the number

1

becomes

i

,

i

becomes

-1

,

-1

becomes

-i

, and

-i

returns to

1

.

Pitfall: Forgetting that i squared is −1

Expanding

i(3 + 2i) = 3i + 2i^2

with

2i^2

read as

2

gives

2 + 3i

, which only swaps the coordinates of

(3, 2)

. With

i^2 = -1

,

i(3 + 2i) = -2 + 3i

: the point

(3, 2)

turned

90°

counterclockwise to

(-2, 3)

.

General multiplication combines rotation with scaling. The product

z_1 \cdot z_2

has modulus

|z_1||z_2|

— the lengths multiply. Its angle from the positive real axis equals the sum of the individual angles — the rotations add. This geometric description anticipates the trigonometric form, where multiplication becomes transparently simple: multiply magnitudes, add angles.

Multiply by	Geometric action	Effect on 2 + i
k > 0 (positive real)	scale by k along the same ray; no rotation	3 · (2 + i) = 6 + 3i (3× longer, same direction)
−1	rotate 180° (reflect through origin)	−(2 + i) = −2 − i
i	rotate 90° counterclockwise	i · (2 + i) = 2i + i² = −1 + 2i
−i	rotate 90° clockwise	−i · (2 + i) = 1 − 2i
iⁿ (integer n)	rotate by n · 90° counterclockwise	cycles: 2 + i → −1 + 2i → −2 − i → 1 − 2i → 2 + i
general z = r · cis(θ)	scale by r, rotate by θ	lengths multiply, arguments add

z₁z₂: arguments add, moduli multiply

The product sits at the angle θ₁ + θ₂ marked in the picture and at distance |z₁||z₂| from the origin: multiplying by a complex number rotates by its argument and scales by its modulus. Nothing in the algebraic expansion shows this; the picture does. Turn one factor and watch the product swing round on the complex multiplication tool.

Addition translates, multiplication rotates and scales: the two operations look nothing alike on the plane.

Summary: Algebra ↔ Geometry Translation

Every complex-number concept introduced on this page has both an algebraic statement and a geometric counterpart. The table below collects them as a single algebra-to-geometry translation reference — coordinates, distances, vector sums, reflections, rotations, and scalings — so the algebraic side and its geometric meaning sit next to each other for quick lookup.

Algebraic object / operation	Geometric meaning in the complex plane
Complex number z = a + bi	point at (a, b), or vector from the origin to (a, b)
Re(z), Im(z)	horizontal and vertical coordinates
Modulus \|z\| = √(a² + b²)	distance from origin to z (Pythagorean theorem on (a, b))
\|z₁ − z₂\|	distance between the two points z₁ and z₂
z₁ + z₂	vector sum via parallelogram / tip-to-tail rule
z₁ − z₂	vector from z₂ to z₁ (displacement between points)
z̄ (conjugate)	reflection of z across the real axis
−z	reflection through the origin (rotation by 180°)
k · z (k a positive real)	scale by factor k along the same ray from the origin
i · z	rotate z by 90° counterclockwise about the origin
z₁ · z₂	scale by \|z₂\|, rotate by arg(z₂); moduli multiply, arguments add
z₁ ⁄ z₂	scale by 1⁄\|z₂\|, rotate by −arg(z₂); moduli divide, arguments subtract

Common Pitfalls

Five mistakes come up again and again when complex numbers are read as points and arrows. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Swapping the coordinates

A complex number is plotted with its two parts in the wrong order: the imaginary part is used as the horizontal coordinate and the real part as the vertical one. The point lands in the wrong place, often in the wrong quadrant.

Wrong working:

-1 + 4i

is plotted at

(4, -1)

, four units right and one unit down, in the fourth quadrant.

Right working: the real part

-1

is the horizontal coordinate and the imaginary part

4

the vertical one, so

-1 + 4i

is the point

(-1, 4)

, one unit left and four units up, in the second quadrant.

Why it happens: in

-1 + 4i

the number

4

is the one that stands out, attached to

i

, and some numbers are written with the imaginary term first, such as

4i - 1

, which is the same number.

How to check: the sign of the real part decides left or right of the imaginary axis. Here the real part

-1

is negative, so the point must lie to the left.

Subtracting the moduli

The distance between two complex numbers is found by subtracting their distances from the origin, instead of taking the modulus of their difference.

Wrong working: for

z_1 = 4 + 3i

and

z_2 = 1 - i

, the distance is taken as

|z_1| - |z_2| = 5 - \sqrt{2} \approx 3.59

.

Right working:

z_1 - z_2 = (4 - 1) + (3 - (-1))i = 3 + 4i

, so the distance is

|z_1 - z_2| = \sqrt{3^2 + 4^2} = 5

.

Why it happens: for real numbers on a line, the distance between two positive numbers is the larger minus the smaller. In the plane the two arrows usually point in different directions, so their lengths alone do not give the gap between their tips.

How to check: plot both points. From

(1, -1)

to

(4, 3)

the horizontal gap is

3

and the vertical gap is

4

, so the distance is

5

by the Pythagorean theorem.

Reflecting across the imaginary axis

The conjugate is drawn as the mirror image in the vertical axis instead of the horizontal one, so the real part changes sign and the imaginary part stays.

Wrong working: the conjugate of

z = 2 + 3i

is placed at

(-2, 3)

, which is the number

-2 + 3i

.

Right working: conjugation keeps the real part and flips the imaginary part, so

\bar{z} = 2 - 3i

, at

(2, -3)

. The point

-2 + 3i

is

-\bar{z}

, the negative of the conjugate.

Why it happens: both axes can serve as a mirror, and the word "reflection" does not say which one. The conjugate always uses the real axis.

How to check:

z

and

\bar{z}

lie on the same vertical line, so

z + \bar{z}

is real:

(2 + 3i) + (2 - 3i) = 4

. With the wrong point,

(2 + 3i) + (-2 + 3i) = 6i

, which is not real.

Comparing complex numbers like reals

An inequality sign is placed between two complex numbers, usually by comparing their coefficients, as if

\mathbb{C}

were ordered like

\mathbb{R}

.

Wrong working:

2i > i

because

2 > 1

. Subtracting

i

from both sides gives

i > 0

, and multiplying both sides by the positive number

i

gives

i^2 > 0

, that is

-1 > 0

.

Right working: complex numbers have no order, so

2i > i

has no meaning. What can be compared are real numbers attached to them, such as the moduli:

|2i| = 2 > |i| = 1

, so

2i

lies farther from the origin.

Why it happens:

2i

and

i

look like

2

and

1

with a label attached, and on the imaginary axis

2i

does sit above

i

. Being higher on one axis is not an ordering of the whole plane.

How to check: before writing an inequality sign, make sure both sides are real numbers. If either side has a nonzero imaginary part, compare moduli, real parts or imaginary parts instead, and say which.

Forgetting that i squared is −1

When a number is multiplied by

i

, the term with

i^2

is left as a positive real number, so the result is not a quarter turn.

Wrong working:

i(3 + 2i) = 3i + 2i^2 = 3i + 2 = 2 + 3i

. The point

(3, 2)

moves to

(2, 3)

, its mirror image in the diagonal line

y = x

.

Right working:

i(3 + 2i) = 3i + 2i^2 = 3i - 2 = -2 + 3i

. The point

(3, 2)

turns

90°

counterclockwise to

(-2, 3)

.

Why it happens: the expansion is done like ordinary algebra, and the step

i^2 = -1

is skipped at the end.

How to check: a quarter turn keeps the length and makes a right angle with the original arrow. Both

(-2, 3)

and

(2, 3)

have length

\sqrt{13}

, but only

(-2, 3)

is perpendicular to

(3, 2)

:

3 \cdot (-2) + 2 \cdot 3 = 0

, while

3 \cdot 2 + 2 \cdot 3 = 12

.

Complex Plane FAQ

What is the complex plane, and what is an Argand diagram?

+

They are the same thing under two names. Each complex number becomes a point, with the real part read along the horizontal axis and the imaginary part along the vertical one. The name Argand honours one of several people who described it independently, and the two terms are used interchangeably. Explore it on the complex explorer.Read more →

What is the geometric meaning of multiplying two complex numbers?

+

Multiply the distances and add the angles. The product sits at a distance equal to the product of the two moduli, turned by the sum of the two arguments. So multiplication combines a stretch with a rotation, which is why polar form makes it so much simpler than expanding brackets in rectangular form. See the multiplication tool.Read more →

Why can't complex numbers be ordered?

+

Because no ordering can be made compatible with the arithmetic. Any workable order would force every square to be non-negative, yet i squared is −1. Assuming i is positive or negative both lead to contradictions. Moduli can be compared, since those are real numbers, but the complex numbers themselves cannot.Read more →

What does multiplying by i do geometrically?

+

It rotates the point a quarter turn counterclockwise about the origin, leaving its distance unchanged. Applying it four times returns the number to where it started, which is the geometric reason the powers of i cycle with period four. This rotation is the clearest single picture of what i actually does.Read more →

What is the geometric meaning of adding two complex numbers?

+

It is vector addition. Treat each number as an arrow from the origin, place the second arrow at the tip of the first, and the sum lands at the far end. The parallelogram formed by the two arrows has the sum at its opposite corner. See it on the addition tool.Read more →

Who first drew complex numbers as points in a plane?

+

Caspar Wessel, a surveyor working in Denmark, in a paper read in 1797. He added the arrows tip to tail and multiplied them by adding their angles, the same picture this page uses. Written in Danish, the paper went unnoticed for about a century, so the plane carries other names: the Argand diagram in English, after Jean-Robert Argand (1806), and the Gaussian plane in German, after Gauss (1831).Read more →

How do complex numbers solve the treasure map puzzle with the missing gallows?

+

By making the gallows cancel. Put the oak at

a

, the pine at

b

and the gallows at

g

. One right-angle turn is multiplication by

-i

, the other by

i

, so the two marks land at

a - i(a - g)

and

b + i(b - g)

. Their midpoint is

\frac{a + b}{2} + i\,\frac{b - a}{2}

, with no

g

left in it: the treasure is fixed by the two trees alone, at the right-angled corner of the isosceles triangle built on the line between them.Read more →

Geometric Representation

Seeing Numbers in Two Dimensions

Key Terms

From Algebra to Geometry

The Two Axes

Plotting Complex Numbers

Complex Numbers as Vectors

Visualizing the Conjugate

The Lack of Order in $\mathbb{C}$

Visualizing Operations

Summary: Algebra ↔ Geometry Translation

Common Pitfalls

Complex Plane FAQ

What is the complex plane, and what is an Argand diagram?

What is the geometric meaning of multiplying two complex numbers?

Why can't complex numbers be ordered?

What does multiplying by i do geometrically?

What is the geometric meaning of adding two complex numbers?

Who first drew complex numbers as points in a plane?

How do complex numbers solve the treasure map puzzle with the missing gallows?

Geometric Representation

Seeing Numbers in Two Dimensions

Key Terms

From Algebra to Geometry

The Two Axes

Plotting Complex Numbers

Complex Numbers as Vectors

Visualizing the Conjugate

The Lack of Order in C\mathbb{C}C

Visualizing Operations

Summary: Algebra ↔ Geometry Translation

Common Pitfalls

Complex Plane FAQ

What is the complex plane, and what is an Argand diagram?

What is the geometric meaning of multiplying two complex numbers?

Why can't complex numbers be ordered?

What does multiplying by i do geometrically?

What is the geometric meaning of adding two complex numbers?

Who first drew complex numbers as points in a plane?

How do complex numbers solve the treasure map puzzle with the missing gallows?

The Lack of Order in $\mathbb{C}$