Give every complex number an address: a point, an arrow, a length, and a direction.
Last lesson we built a brand-new number, z = a + b i, out of a real part a and an imaginary part b. But where does it live? The real line is full — there is no room on it for i. So we do what we already did for ordered pairs in the coordinate plane: we give the real part a horizontal axis and the imaginary part a vertical one, and every complex number becomes a single point. That point is also an arrow from the origin — exactly a vector — so it carries a length and a direction. Those two numbers, the modulus and the argument, are the address that will power the rest of this stage.
Draw two perpendicular number lines, just like the coordinate plane. The horizontal one is the real axis — it carries the real part a. The vertical one is the imaginary axis — it carries the imaginary part b. Then the single rule is:
a + b i ↔ the point (a, b). The real part is how far right, the imaginary part is how far up. This picture is the complex plane (also called the Argand plane, after Jean-Robert Argand).
So 3 + 2i is the point (3, 2); −1 − 4i is the point (−1, −4); the real number 7 = 7 + 0i sits on the real axis at (7, 0); and the purely imaginary 5i = 0 + 5i sits on the imaginary axis at (0, 5). The reals are not lost — they are exactly the points on the real axis, the horizontal line through O. That is the picture of ℝ ⊂ ℂ: the old number line is one street in a whole new city.
The two axes are not "the x and y of a function." They are the two parts of one number. A complex number is a single object that happens to need two real coordinates to pin down — much as a vector needs two components.
There is a second, equally honest way to see z = a + b i: as the arrow from the origin O to the point (a, b). Point and arrow are the same data — the arrow's tip is the point. But the arrow makes two new features jump out: it has a length and it points in a direction. This is the geometric identity that quietly powers the whole stage:
A complex number is a vector on the plane. So adding two complex numbers will be the triangle rule for arrows (next lesson), and the length of the arrow is the magnitude √(a² + b²) we already know.
The length of that arrow has a name: the modulus of z, written |z| with the same vertical bars we use for the absolute value of a real number — and for a good reason, since it measures size. The legs of the right triangle are a and b, so Pythagoras gives the hypotenuse at once:
|a + b i| = √(a² + b²) ≥ 0. For z = 3 + 2i: |z| = √(3² + 2²) = √(9 + 4) = √13 ≈ 3.61.
Three facts follow straight from the formula. First, a modulus is never negative — it is a length, and √ returns the non-negative root. Second, |z| = 0 happens only for z = 0, because a² + b² = 0 forces a = 0 and b = 0. Third, the conjugate a − b i and the negative −a − b i have the same modulus as z: flipping a sign on a or b cannot change a² or b². So z, z̄ and −z all sit on one circle around O.
|z| is never negative — do not write "|z| = −5." And do not confuse |z| with either part: |3 + 2i| is √13, not 3 and not 2. The modulus is the whole arrow's length.
Length alone is not enough — many different numbers share a modulus (every point on a circle). What separates them is direction. The argument of z, written arg z or θ, is the angle from the positive real axis to the arrow, measured counter-clockwise as positive — exactly the angle convention from the unit circle.
For z = 3 + 2i the arrow rises 2 for every 3 across, so arg z = ≈ 33.7° (in quadrant I). For i = 0 + 1i the arrow points straight up: arg i = 90°. For −1 it points straight left: arg(−1) = 180°.
One subtlety: turning a full 360° (that is, ±2π) brings the arrow back to where it started, so it points the same way. The argument is therefore only defined up to whole turns of 360°; by convention we report the value in the half-open range (−180°, 180°]. The quadrant of z fixes the sign pattern of (a, b) — and so the rough size of θ: quadrant I has θ in (0°, 90°), quadrant II in (90°, 180°), quadrant III in (−180°, −90°), quadrant IV in (−90°, 0°).
The number 0 has a modulus (it is 0) but no argument — a zero-length arrow points nowhere. And remember ℂ has no order: you can compare two moduli (sizes are real numbers), but "3 + 2i < 4 + i" is meaningless — there is no "less than" between complex numbers themselves.
Put the two together and the address is complete: the modulus says how far from O, the argument says which way. A length and a direction locate z just as surely as the pair (a, b) does — and in the next lessons each description will earn its keep, the coordinates for adding, the length-and-direction for multiplying.
| Idea | Statement | For z = 3 + 2i |
|---|---|---|
| plane ↔ point | a + b i ↔ the point (a, b) | (3, 2) |
| point ↔ arrow | the arrow from O to (a, b) — a vector | O → (3, 2) |
| modulus |z| | √(a² + b²) ≥ 0; = 0 only for z = 0 | √13 ≈ 3.61 |
| argument arg z | angle from the +real axis, CCW, in (−180°, 180°] | ≈ 33.7° |
| together | modulus & argument locate z (so does (a, b)) | length + direction |
The reals ℝ live inside ℂ as the real axis (b = 0), and 0 alone has a modulus but no argument. Next: the four operations — addition is the triangle rule on these very arrows.
Plot (just describe the location, no drawing) and give the quadrant of each: 2 + 3i, −4i, −2 + i, 5.
2 + 3i → point (2, 3), quadrant I. −4i = 0 − 4i → point (0, −4), on the negative imaginary axis (no quadrant). −2 + i → point (−2, 1), quadrant II. 5 = 5 + 0i → point (5, 0), on the positive real axis (no quadrant).
Find the modulus of each: 3 + 4i, −5i, −1 − 1i, 0.
|3 + 4i| = √(9 + 16) = √25 = 5. |−5i| = √(0 + 25) = 5. |−1 − i| = √(1 + 1) = √2 ≈ 1.41. |0| = 0 — the only number with modulus 0.
Give the argument (in (−180°, 180°]) of each: 4, 2i, −3, −6i, 1 + i.
4 points east → 0°. 2i points up → 90°. −3 points left → 180°. −6i points down → −90°. 1 + i rises 1 for every 1 across, into quadrant I → 45°.
Show that z, its conjugate z̄, and its negative −z all have the same modulus, using z = 3 − 4i.
z = 3 − 4i, z̄ = 3 + 4i, −z = −3 + 4i. |z| = √(9 + 16) = 5; |z̄| = √(9 + 16) = 5; |−z| = √(9 + 16) = 5. All 5 — flipping a sign on a part squares away, so a² + b² is unchanged.
A friend writes "since 3 + 2i and 1 + 5i both look 'small', and 3 + 2i has the bigger real part, we have 3 + 2i > 1 + 5i." What is wrong, and what can you legitimately compare?
ℂ has no order — ">" between two non-real complex numbers is meaningless, so the whole inequality is undefined. What you can compare is their moduli (real numbers): |3 + 2i| = √13 ≈ 3.61 and |1 + 5i| = √26 ≈ 5.10, so 1 + 5i is farther from O. That is a statement about lengths, not about the numbers being "greater."
A number has modulus 5 and lies on the negative imaginary axis. Which number is it, and what is its argument?
On the negative imaginary axis a = 0 and b < 0, with √(0 + b²) = |b| = 5, so b = −5: the number is −5i. Its arrow points straight down, so arg = −90°.
Six questions to lock it in. Tap the answer you think is right.
This lesson is the geometric foundation of complex numbers, aligned with CCSS HSN-CN.B.4 (represent complex numbers on the complex plane in rectangular form) and HSN-CN.B.6 (distances and the modulus), building toward the polar/argument ideas of HSN-CN.B.5. It deliberately reuses earlier tools your student already owns: the coordinate plane (Stage 20), the magnitude of a vector √(x²+y²) and the triangle/vector picture (Stage 27), and the unit-circle angle convention (Stage 25). Two honesty habits are worth reinforcing at home: a modulus is a length, so it is never negative, and complex numbers cannot be ordered — you may compare their sizes (moduli) but never write one complex number as "greater than" another.