Complex Numbers and the Plane Behind the Pictures, math that surprises you lesson

Squaring a number on the number line is familiar, since the square of $3$ is $9$ and the square of $-1$ is $1$ . The pictures of Julia sets and the Mandelbrot set repeat a rule of the form “square, then add a fixed number” at the points of a plane, so they rest on a stranger question. What does it mean to square a point? This page builds the complex numbers as points of a plane, shows that adding slides points while multiplying turns and stretches them, and settles the question exactly, since squaring a point squares its distance from $0$ and doubles its angle.

Key terms

Complex number

Definition A complex number is a number of the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit, a number whose square is $-1$ . The real number $a$ is the real part and the real number $b$ is the imaginary part. Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal. Complex numbers are added, subtracted and multiplied by the usual rules of algebra, treating $i$ like a letter and replacing $i^2$ by $-1$ wherever it appears. Addition therefore works part by part,

\begin{aligned} &(a + bi) + (p + qi) \\[0.4em] &\quad = (a + p) + (b + q)i, \end{aligned}

and multiplication is worked out in the card on complex multiplication below. Every real number $a$ is the complex number $a + 0i$ , so the real numbers are among the complex numbers. A number $bi$ with real part $0$ and $b \neq 0$ is called purely imaginary. Most complex numbers, such as $2 + i$ , are neither real nor purely imaginary, and it is wrong to call every complex number imaginary.

In plain words A complex number packs two real numbers, a real part and an imaginary part, into one number. Its arithmetic is ordinary algebra with one extra rule, $i^2 = -1$ .

Example Adding $2 + i$ and $1 + 2i$ part by part gives $(2 + 1) + (1 + 2)i = 3 + 3i$ . Multiplying two numbers that are not real can give a real number, as in

\begin{aligned} &(1 + i)(1 - i) \\[0.4em] &\quad = 1 - i + i - i^2 \\[0.4em] &\quad = 1 + 1 = 2. \end{aligned}

The real number $2$ is the complex number $2 + 0i$ , with imaginary part $0$ , and $2i$ is purely imaginary, with real part $0$ .

Complex plane

Definition The complex plane is a coordinate plane in which the complex number $a + bi$ is drawn as the point $(a, b)$ . The horizontal axis is the real axis, where the real numbers $a + 0i$ lie, and the vertical axis is the imaginary axis, where the numbers $bi$ lie. The number $0$ is the origin. Each complex number is exactly one point, and each point is exactly one complex number. Adding a fixed number $p + qi$ to every complex number moves every point by the same arrow, $p$ units across and $q$ units up, which means to the left when $p < 0$ and down when $q < 0$ . Such a motion is a translation, and adding two numbers is the same as placing their arrows from the origin tip to tail.

In plain words A complex number is an address in the plane. The real part says how far to go across and the imaginary part how far to go up or down. Adding a number slides the whole plane, without turning or stretching it.

Example The number $1 + i$ is the point $(1, 1)$ , the real number $2$ is the point $(2, 0)$ on the real axis, and $2i$ is the point $(0, 2)$ , two units above the origin. Adding $1 + 2i$ to $2 + i$ moves the point $(2, 1)$ one unit to the right and two units up, to $(3, 3)$ , which is the number $3 + 3i$ . Adding $-1$ to every number slides the whole plane one unit to the left, so $0$ moves to $-1$ and $i$ moves to $-1 + i$ .

Imaginary unit

Definition The imaginary unit $i$ is a number whose square is $-1$ . No real number has this property, since the square of a real number is never negative, so $i$ is a new number, placed at the point $(0, 1)$ of the complex plane. Its negative has the same square, because $(-i)^2 = i^2 = -1$ . The equation $z^2 = -1$ has exactly the two complex solutions $i$ and $-i$ , because $z^2 + 1 = (z - i)(z + i)$ and a product of complex numbers is $0$ only when one of the factors is $0$ . The powers of $i$ repeat in a cycle of four, with $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ , $i^5 = i$ and so on. Multiplying any complex number by $i$ gives

i(a + bi) = -b + ai,

which moves the point $(a, b)$ to the point $(-b, a)$ . That is a quarter turn, a rotation by $90$ degrees counterclockwise about the origin, and it leaves every distance from the origin unchanged.

In plain words The imaginary unit is the number that squares to $-1$ . In the plane, multiplying by $i$ is a quarter turn to the left, so multiplying by $i$ twice is a half turn, which sends every number to its negative. That half turn is the picture behind $i^2 = -1$ .

Example Start at $1$ and multiply by $i$ again and again. The numbers $1$ , $i$ , $-1$ , $-i$ and $1$ go round the circle of radius $1$ in quarter turns, and the fourth turn returns to the start. The point $2 + i$ goes to

\begin{aligned} i(2 + i) &= 2i + i^2 \\[0.4em] &= -1 + 2i. \end{aligned}

Both points are at distance $\sqrt{5}$ from the origin, and the arrows to $(2, 1)$ and $(-1, 2)$ have slopes $\tfrac12$ and $-2$ , whose product is $-1$ , so the arrows are perpendicular.

Modulus and argument

Definition The modulus of a complex number $z = a + bi$ is its distance from the origin,

\lvert z \rvert = \sqrt{a^2 + b^2},

by the Pythagorean theorem. For $z \neq 0$ , an argument of $z$ is an angle from the positive real axis to the arrow from $0$ to $z$ , measured counterclockwise. Angles on this page are measured in degrees. The argument is not unique, since adding or subtracting $360$ degrees, a whole turn, gives the same direction, so any two arguments of the same number differ by a whole number of turns. The number $0$ has modulus $0$ and no argument, because it has no direction. A nonzero number with modulus $r$ and argument $\theta$ can be written in polar form,

z = r(\cos\theta + i\sin\theta),

with real part $a = r\cos\theta$ and imaginary part $b = r\sin\theta$ .

In plain words The modulus says how far a number is from $0$ , and the argument says in which direction it lies. Distance and direction locate a point just as well as the two coordinates do, and they are the natural language for multiplication.

Example The number $1 + i$ has modulus $\sqrt{1 + 1} = \sqrt{2}$ and argument $45$ degrees, since it lies halfway between the positive real axis and the positive imaginary axis. The number $1 + \sqrt{3}\,i$ has modulus $\sqrt{1 + 3} = 2$ and argument $60$ degrees, because $2\cos 60^\circ = 1$ and $2\sin 60^\circ = \sqrt{3}$ . The real number $-1$ has modulus $1$ and argument $180$ degrees, and $-2i$ has modulus $2$ and argument $270$ degrees, which is the same direction as $-90$ degrees.

Complex multiplication

Definition Two complex numbers are multiplied by expanding the brackets and replacing $i^2$ by $-1$ ,

\begin{aligned} &(a + bi)(p + qi) \\[0.4em] &\quad = (ap - bq) \\[0.4em] &\qquad + (aq + bp)i. \end{aligned}

For nonzero factors this has a geometric meaning. If $z = r(\cos\alpha + i\sin\alpha)$ and $w = s(\cos\beta + i\sin\beta)$ , the same expansion together with the angle addition formulas of trigonometry gives

\begin{aligned} zw = rs\bigl(&\cos(\alpha + \beta) \\[0.4em] &+ i\sin(\alpha + \beta)\bigr), \end{aligned}

so the moduli multiply and the arguments add, up to whole turns. If either factor is $0$ , so is the product. In particular, squaring a nonzero number squares its modulus and doubles its argument, and multiplying by $i$ , which has modulus $1$ and argument $90$ degrees, is the quarter turn. Squaring is not done coordinate by coordinate. The square of $a + bi$ is

\begin{aligned} (a + bi)^2 &= (a^2 - b^2) \\[0.4em] &\quad + 2ab\,i, \end{aligned}

whose real part carries the minus sign that comes from $i^2 = -1$ and whose imaginary part is the mixed term $2ab$ , not $b^2$ .

In plain words Multiplying by a nonzero complex number stretches the plane by its modulus and turns it by its argument. Squaring a point therefore turns it to twice its angle and moves it to the square of its distance, which is farther out when the distance is more than $1$ and closer in when the distance is less than $1$ .

Example The number $1 + i$ has modulus $\sqrt{2}$ and argument $45$ degrees, so its square should have modulus $2$ and argument $90$ degrees, which is the number $2i$ . Algebra agrees,

\begin{aligned} (1 + i)^2 &= 1 + 2i + i^2 \\[0.4em] &= 2i. \end{aligned}

Squaring each coordinate separately would leave $1 + i$ at the point $(1, 1)$ , which is wrong. For two different factors, $(2 + i)(1 + 2i) = 2 + 4i + i + 2i^2 = 5i$ . Each factor has modulus $\sqrt{5}$ , and $\sqrt{5} \cdot \sqrt{5} = 5$ . Their arguments are about $26.57$ and $63.43$ degrees, because the points $(2, 1)$ and $(1, 2)$ are mirror images across the line halfway between the axes, and the two angles add to exactly $90$ degrees, the direction of $5i$ . The number $\sqrt{3} + i$ , with modulus $2$ and argument $30$ degrees, squares to $2 + 2\sqrt{3}\,i$ , with modulus $4$ and argument $60$ degrees. The number $\tfrac14(\sqrt{3} + i)$ , with modulus $\tfrac12$ and the same argument, squares to $\tfrac18(1 + \sqrt{3}\,i)$ , with modulus $\tfrac14$ and argument $60$ degrees. Both turn by the same angle, but the first moves out and the second moves in.

Squaring and adding together make the rule behind the next pictures. Fix a complex number $c$ and replace the current number $z$ by $z^2 + c$ again and again, each time squaring the modulus and doubling the argument, then sliding by the arrow of $c$ . With $c = -1$ and the start $0$ , the numbers are $0, -1, 0, -1, \dots$ forever, since $0^2 - 1 = -1$ and $(-1)^2 - 1 = 0$ . With the same rule and the start $2$ , they are $2, 3, 8, 63, \dots$ and keep growing. Which starting points stay bounded is the question of Julia Sets, One Rule and Many Starting Points. The precalculus lessons Complex Numbers and the Complex Plane and Polar Form of Complex Numbers treat this arithmetic in more detail.

Check yourself

Square $2 + 3i$ and find the modulus of the result. Compare it with the modulus of $2 + 3i$ , and with what squaring each coordinate separately would give.

Show the answer

Expanding the square and using $i^2 = -1$ gives

\begin{aligned} (2 + 3i)^2 &= 4 + 12i + 9i^2 \\[0.4em] &= -5 + 12i. \end{aligned}

Its modulus is $\sqrt{25 + 144} = \sqrt{169} = 13$ . The modulus of $2 + 3i$ is $\sqrt{4 + 9} = \sqrt{13}$ , and $13 = (\sqrt{13})^2$ , just as the rule for squaring predicts. The argument doubles too, from about $56.31$ degrees to about $112.62$ degrees. Squaring each coordinate separately would give $4 + 9i$ , with modulus $\sqrt{97} \approx 9.85$ and argument about $66.04$ degrees, so it gets both the distance and the direction wrong.

Square $-1 + i$ with the rule for modulus and argument, and then check the answer with algebra. Why is a doubled argument of more than $180$ degrees no problem?

Show the answer

The number $-1 + i$ has modulus $\sqrt{2}$ and argument $135$ degrees. Its square has modulus $(\sqrt{2})^2 = 2$ and argument $270$ degrees, which points straight down the imaginary axis, so the square is $-2i$ . Algebra gives the same number,

\begin{aligned} (-1 + i)^2 &= 1 - 2i + i^2 \\[0.4em] &= -2i. \end{aligned}

An argument only describes a direction, and directions repeat every $360$ degrees, so $270$ degrees and $-90$ degrees name the same direction. A doubled argument may pass $180$ or even $360$ degrees, and only the direction it describes matters.

Extend the pattern. Find $(1 + i)^2$ , $(1 + i)^4$ and $(1 + i)^8$ by squaring repeatedly. What is the smallest positive whole number $n$ for which $(1 + i)^n$ is a positive real number?

Show the answer

Squaring repeatedly gives $(1 + i)^2 = 2i$ , then $(1 + i)^4 = (2i)^2 = 4i^2 = -4$ , and then $(1 + i)^8 = (-4)^2 = 16$ . In polar terms, $(1 + i)^n$ has modulus $(\sqrt{2})^n$ and argument $45n$ degrees, because each factor multiplies the modulus by $\sqrt{2}$ and adds $45$ degrees to the argument. The power lies on the positive real axis exactly when $45n$ is a whole number of turns, a multiple of $360$ . The smallest such positive $n$ is $8$ , and then the power is $(\sqrt{2})^8 = 16$ , as found above. The powers $n = 1$ to $7$ are $1 + i$ , $2i$ , $-2 + 2i$ , $-4$ , $-4 - 4i$ , $-8i$ and $8 - 8i$ , one in each of the other seven directions that are multiples of $45$ degrees.

Complex Numbers and the Plane Behind the Pictures

Key terms

Complex number

Complex plane

Imaginary unit

Modulus and argument

Complex multiplication

Check yourself

How this lesson connects

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