The Complex Plane and Modulus

A number with two coordinates

A complex number is written $z = a + bi$ with $a$ and $b$ real; $a$ is its real part and $b$ its imaginary part. Notice that the imaginary part is the real number $b$, not the term $bi$. The plan is to treat the pair $(a, b)$ as coordinates. And the whole plan hinges on one fact: the pair pins the number down completely, with no ambiguity in either direction.

That perfect matching is the definition of the complex plane: draw the familiar coordinate plane, and let the point $(a, b)$ be the number $a + bi$. The horizontal axis collects the numbers with imaginary part $0$, which are exactly the real numbers. So that axis is called the real axis: your old number line, embedded unchanged inside the bigger picture. The vertical axis collects the numbers with real part $0$, which are $0$ and the pure imaginary numbers $bi$, so it is called the imaginary axis. The four quadrants are read the same way as for $(x, y)$ points: $3 + 2i$ sits in the first quadrant, $-2 + 3i$ in the second, and so on. The picture is also called an Argand diagram, after one of the people who first drew it; the history aside at the end tells that story.

Addition slides points

The previous lesson established the arithmetic rule: to add complex numbers, add the parts separately,

Read that rule with coordinate eyes. The point for the sum has horizontal coordinate $a + c$ and vertical coordinate $b + d$. So adding the fixed number $w = c + di$ sends every point $c$ units across and $d$ units up (negative values meaning the opposite direction). Adding $w$ is a rigid slide of the entire plane, the same shift applied to every point at once; no stretching, no turning.

There is a second picture of the same fact. Draw $z$ not as a dot but as an arrow from $0$ to its point. To form $z + w$, place a copy of $w$‘s arrow so its tail starts at $z$‘s tip; the copy’s tip lands on $z + w$. This is tip-to-tail addition, and it is what physics and later mathematics call adding vectors. Doing it in the other order, $w$ first and then $z$, traces the other two sides of the same figure, so both routes end at the same corner. The four points $0$, $z$, $w$, and $z + w$ form a parallelogram, and the equation $z + w = w + z$ becomes something you can see.

Subtraction has its own reading. The difference $z - w$ is, by definition of subtraction, the number you add to $w$ to reach $z$, since $w + (z - w) = z$. As an arrow, then, $z - w$ is the trip from $w$ to $z$, picked up and carried back so its tail sits at $0$. Keep that sentence; in a moment it will turn the modulus into a ruler for measuring the gap between any two complex numbers.

Conjugation reflects across the real axis

The previous lesson introduced the conjugate $\overline{z} = a - bi$ of $z = a + bi$ as the sign flip that makes multiplication produce something real. In the plane, conjugation sends the point $(a, b)$ to the point $(a, -b)$: the horizontal coordinate is untouched and the vertical coordinate changes sign. That is precisely a reflection across the real axis, the mirror image of the point in the horizontal line. A number in the upper half of the plane trades places with its twin in the lower half, directly above or below itself.

Two small facts fall out of the picture immediately, and the algebra confirms both. Reflecting twice puts every point back where it started, and indeed $\overline{\overline{z}} = \overline{a - bi} = a + bi = z$. And the points that do not move are exactly the points on the mirror: solving $z = \overline{z}$ means $a + bi = a - bi$, so $2bi = 0$, so $b = 0$. The numbers equal to their own conjugate are exactly the real numbers. This gives you a clean geometric test for realness that the practice sets will use again and again. A number $z$ is real precisely when its point sits on the real axis, which is precisely when conjugation fixes it.

The modulus is a distance

For a real number $x$, the first chapter defined the absolute value $|x|$ as the distance from $x$ to $0$ on the number line. Complex numbers now live in a plane, but the same question still makes sense: how far is the point $z$ from $0$? That distance is called the modulus of $z$ (also, still, its absolute value), and it is written with the same bars, $|z|$. The plural is moduli. The definition is geometric; what makes it usable is that the Pythagorean theorem converts it into a formula.

The axis case just checked is worth saying again, because it earns the notation. For a real number $z = a$, the new formula gives $|a + 0i| = \sqrt{a^2} = |a|$: on the real axis, the modulus is the old absolute value. The bars are not being recycled for something new; the same concept, distance from $0$, has simply grown to cover the whole plane.

Three properties come straight from the geometry, and each is one line of algebra. First, $|z| \ge 0$ always, and $|z| = 0$ exactly when $z = 0$. A sum of two real squares $a^2 + b^2$ is zero only when $a = b = 0$, matching the fact that the only point at distance $0$ from the origin is the origin itself. Second, $|\overline{z}| = |z|$: reflection across the real axis does not change how far a point is from $0$, and indeed $\sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2}$. Third, $|-z| = |z|$: the point $(-a, -b)$ sits on the exact opposite side of $0$ at the same distance. One warning while the bars are fresh: the modulus lets you compare the sizes of complex numbers, but the numbers themselves are not ordered. A statement like $2 + 3i < 3 + i$ is meaningless; only the real numbers $|2 + 3i|$ and $|3 + i|$ can be compared.

The conjugate identity

The modulus and the conjugate look like unrelated gadgets, one a distance and the other a reflection. They are joined by a single identity that powers most complex-number computations from here through the rest of the subject.

Pause on what the identity says. The left side is a product of two complex numbers; the right side is a real number that is never negative. Multiplying any number by its conjugate always lands you on the non-negative real axis, and it lands you at a geometrically meaningful spot: the squared distance from $0$. This is the secret behind the division method of the previous lesson. Multiplying a denominator $w$ by $\overline{w}$ produced the real number $w\overline{w}$, and now you know that mystery real number by name: it is $|w|^2$.

The identity also unlocks the most useful multiplication fact about the modulus.

In words: the length of a product is the product of the lengths. This is a genuine shortcut, because it lets you find the modulus of a complicated product without ever expanding it, and it is also a genuine surprise. Nothing about the multiplication rule $(ac - bd) + (ad + bc)i$ makes this obvious at a glance. The proof needed no cleverness, only the conjugate identity, which is exactly why that identity deserves its own section.

Distance between points, and circles

The subtraction picture now pays off. The arrow from $w$ to $z$ is the number $z - w$, so the distance between two complex numbers is the length of that arrow:

The coordinates confirm that this is the honest geometric distance. With $z = a + bi$ and $w = c + di$, the difference is $z - w = (a - c) + (b - d)\,i$, so

which is the Pythagorean theorem applied to the right triangle whose legs are the horizontal gap $|a - c|$ and the vertical gap $|b - d|$ between the two points. Distance from $0$ was the special case $w = 0$ all along.

Once distance is an equation, whole shapes become equations. Fix a center $c$ and a radius $r > 0$. The points $z$ with

are exactly the points at distance $r$ from $c$: the circle of radius $r$ centered at $c$. The simplest case is $|z| = r$, the circle of radius $r$ around $0$. The inequality $|z - c| < r$ describes the inside of the circle, and $|z - c| > r$ the outside. This one idea turns modulus equations into pictures. To understand the set of solutions of $|z - 1 - i| = 2$, you do not solve anything; you read off the center $1 + i$ and the radius $2$ and draw.

The triangle inequality

Walk from $0$ to $z$, then from $z$ onward to $z + w$. The first leg has length $|z|$, and the second leg is a translated copy of $w$‘s arrow, so it has length $|w|$. The straight-line distance from $0$ to the final point is $|z + w|$. A detour through $z$ can never be shorter than the straight route, so the geometry announces the triangle inequality:

The name is apt: $0$, $z$, and $z + w$ are the corners of a triangle whose side lengths are $|z|$, $|w|$, and $|z + w|$. And the inequality says one side of a triangle never exceeds the other two sides combined. Geometry makes it believable; the conjugate identity makes it a theorem.

When is the inequality an equality? The proof pinpoints the moment slack can enter: the real part of $z\overline{w}$ must equal the full modulus $|z\overline{w}|$, which happens only when $z\overline{w}$ is a non-negative real number. Geometrically that is the degenerate triangle: the arrows for $z$ and $w$ point in the same direction (or one of them is $0$). So the detour is no detour at all.

The inequality also has a useful reverse reading. Since $z = (z + w) + (-w)$, applying the triangle inequality to the right side gives $|z| \le |z + w| + |w|$, so $|z + w| \ge |z| - |w|$, and swapping the roles of $z$ and $w$ gives $|z + w| \ge |w| - |z|$ as well. A sum can never be shorter than the gap between the two lengths. Together the bounds pin $|z + w|$ into a window. For example, if $|z| = 3$ and $|w| = 4$, then $|z + w|$ can be as large as $7$, as small as $1$, and never anything outside $[1, 7]$.

Common mistakes