Speak Maths · University
Complex numbers: points that turn
A complex number is a point on a plane, and multiplying by it turns and stretches the plane. Seven signs let you say where a point is, how far, and which way it faces.
Draft: written by an AI helper and not yet checked by a teacher.
The alphabet
Read each sign aloud, the way it is written under it.
- Say: i squared equals minus oneMeans: i is the new number whose square is −1; multiplying by i is a quarter turn
- Say: z equals a plus b iMeans: a point a across and b up: a is the real part, b the imaginary part
- Say: the real part of z is a; the imaginary part of z is bMeans: the across and the up of the point z
- Say: z bar equals a minus b i (the conjugate of z)Means: the mirror image of z in the horizontal axis
- Say: the modulus of z equals the square root of a squared plus b squaredMeans: the distance from z to the origin
- Say: the argument of z equals thetaMeans: the angle from the right-hand axis to z, measured anticlockwise
- Say: e to the i theta equals cosine theta plus i sine thetaMeans: the point at angle θ on the circle of radius one: a turn by θ
The story: The turning wheel
Tap a line to see its words.
- A wheel of radius 2 sits on a plane. A mark on its rim starts at the right-hand end, 2 across.
- A quarter turn anticlockwise is multiplying by i. The mark is now at the top, at 2i.
- Another quarter turn is multiplying by i again. i times i is −1, so the mark lands at −2: a half turn.
- Later the mark is 1 across and √3 up.
- Its mirror image in the horizontal axis is 1 across and √3 down.
- Its distance from the centre is 2: it is still on the rim.
- Its angle from the right-hand axis is π over 3, one sixth of a full turn.
- So the mark is 2 times the point at angle π over 3 on the unit circle, which is 2 e to the i π over 3.
- Multiplying the mark by its mirror image gives 4, the square of its distance.
- A turn by π is a half turn, and it sends every point to its opposite: e to the iπ is −1.
In words this story is 788 letters long; in symbols, 109.
Read it
What does it say?
Write it
Which notation says this?
z is a complex number with real part a and imaginary part b.
The distance from z = a + bi to the origin.
The point at angle θ on the circle of radius one is cosine θ plus i sine θ, which is e to the i θ.
The mirror image of z = 3 + 4i in the horizontal axis.
The point 2 to the left and 5 up.
The real part of z is 4 and the imaginary part of z is −1.
i times i equals minus one.
The angle of z, measured anticlockwise from the right-hand axis, is π over 4.
The point at angle π over 2 on the unit circle is i.
The distance from 1 + i to the origin is the square root of 2.
z written with its distance r and its angle θ: r times e to the i θ.
A point and its mirror image are the same distance from the origin.
Build it
Tap the tiles in order. One tile does not belong.
z times z bar equals the modulus of z, squared.
e to the i π, plus one, equals zero.
e to the i θ equals cosine θ plus i sine θ.
The modulus of z equals the square root of a squared plus b squared.
The argument of i is π over 2.
The real part of z plus i times the imaginary part of z gives z back.
Challenges
Creative puzzles that use every sign of this chapter. Take your time; there is no clock.
1 Four points
Four points on the plane are listed in the rows. A tick shows when a point has the property in a column. Judge each sentence.
| ✓ | ✓ | ✓ | ||
| ✓ | ||||
| ✓ | ✓ | ✓ | ||
| ✓ | ✓ |
A tick means the point (row) has the property (column).
Mirroring in the horizontal axis flips only the sign of the imaginary part, and a point and its opposite are the same distance from the origin.
2 Sign language
Pair each line with what it says.
Tap a line, then its meaning.
Bars mirror, vertical lines measure distance, arg gives the angle, and e^{iθ} packs an angle into a point.
3 The famous identity
This should say: e to the iπ, plus one, equals zero. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
A turn by π sends 1 to −1, so e^{iπ} = −1 and e^{iπ} + 1 = 0.
4 Distance squared
This should say: the squared distance of z = a + bi from the origin is a² and b² together. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
By Pythagoras, |z|² = a² + b²: across squared plus up squared.
5 Quarter turns
Each multiplication by i is a quarter turn. What is i⁷?
Four quarter turns bring you back to 1, so i⁷ = i³ = i²·i = −i.
6 How far from the centre?
How far is the point z from the origin?
|z| = √(6² + (−8)²) = √100 = 10. The sign of b does not matter, because it is squared.
7 Half a turn
Where does the point land?
e^{iπ} is a half turn, which is −1, so z = 2·(−1) = −2.
8 The point and its mirror
A point times its mirror image is a real number. Write it using the distance of z, then compare.
One way:
(a + bi)(a − bi) = a² + b² = |z|², which is real and never negative.
9 Four points on and off the circle
Four points are listed in the rows; a tick shows that a point has the property in the column. Judge each sentence.
| ✓ | ✓ | ✓ | ||
| ✓ | ||||
| ✓ | ✓ | |||
| ✓ | ✓ |
A tick means the point (row) has the property (column).
Squaring i is a half turn, so i² = −1. Points straight up and straight down share Re z = 0 but not their angle, and e^{iπ/2} is i, not −1.
10 Four turns
Pair each turn with where it lands the point 1.
Tap a line, then its meaning.
e^{iθ} is the point at angle θ on the unit circle; a negative angle turns clockwise, and 2π is a whole turn.
11 Work it out
Each line is a number in disguise. Pair it with its value.
Tap a line, then its meaning.
√(9 + 16) = 5; the real part is just the 3; the bar flips the sign of 4i; and i(3 + 4i) = 3i + 4i² = −4 + 3i.
12 A quarter turn of a point
Multiply 2 + 3i by i: a quarter turn anticlockwise. The result should be −3 + 2i. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
i(2 + 3i) = 2i + 3i² = 2i − 3: the up becomes a left, the across becomes an up.
13 Which way does it face?
The point 1 to the left and 1 up is in the upper-left quarter, three eighths of a turn from the right-hand axis. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
The angle is measured from the right-hand axis: π/4 would be 1 across and 1 up. One to the left and one up is π − π/4 = 3π/4.
14 Find the point
A point is 13 from the origin, 5 across, and above the axis. Which point is it?
25 + b² = 169 gives b² = 144, so b = 12 (positive, since Im z > 0). Adding 5 + 8 = 13 is the trap.
15 Three turns of a sixth
z is the point one sixth of a turn round the unit circle. Where is z cubed?
Cubing triples the angle: 3 · π/3 = π, a half turn, and e^{iπ} = −1.
16 A square that turns
Square the point 1 across and 1 up. Where does it land?
(1 + i)² = 1 + 2i + i² = 1 + 2i − 1 = 2i. The angle doubles from π/4 to π/2, and the distance from √2 to 2.
17 The mirror of a turn
The mirror image of the point at angle θ on the unit circle is the point at angle −θ. Write it with the bar and with e. Then compare.
One way:
Mirroring in the horizontal axis flips the sign of the imaginary part, sin θ, which is the same as turning the other way: angle −θ.
Chapter finished. The symbols are yours.
Tell your own
Tell a two-line story in notation about a point on a turning wheel: where it starts, what multiplies it, where it lands. Check that |z| stays the same.
Kept on this device only.