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Learn9FM0Paper 1 · Core Pure Mathematics 1Complex numbersThe Argand diagram, modulus and argument
Complex numbers · Subtopic 2 of 5

The Argand Diagram, Modulus and Argument

Plot z=x+iyz=x+iy as the point (x,y)(x,y). Its distance from the origin is the modulus ∣z∣|z|, and its angle from the positive real axis is the argument arg⁡z\arg z, with −π<arg⁡z≤π-\pi<\arg z\le\pi.

Cambridge 9709 Paper 3Edexcel YFM01 WFM01Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 35 minRevision 6 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Represent complex numbers, their sums and their conjugates on an Argand diagram.
  • Find the modulus ∣z∣=x2+y2|z|=\sqrt{x^2+y^2}.
  • Find the principal argument in the correct quadrant, exactly or to 3 significant figures.
  • Use zz∗=∣z∣2zz^*=|z|^2.
Pause and think

z=−3+4iz=-3+4i and w=3−4iw=3-4i. Both have tan⁡θ=−43\tan\theta=-\tfrac43. Do they have the same argument?

Section 2 of 14

Core idea

Start hereDistance and direction

Every complex number is a point on a plane. Instead of "3 across, 4 up" you can say "5 away, in direction 53∘53^\circ". That's the modulus–argument description, and in it multiplication becomes beautifully simple: lengths multiply, angles add.

The big idea

∣z∣=r=x2+y2|z|=r=\sqrt{x^2+y^2}; arg⁡z=θ\arg z=\theta (−π<θ≤π-\pi<\theta\le\pi) from a sketch. z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta). ∣zw∣=∣z∣∣w∣|zw|=|z||w|, arg⁡(zw)=arg⁡z+arg⁡w\arg(zw)=\arg z+\arg w.

1. Definitions

Modulus r and argument θ as the point turns
−4−224−3−2−11230ReImr = 2.5θz
The real part is rcos⁡θr\cos\theta and the imaginary part is rsin⁡θr\sin\theta.
z=r(cos⁡θ+isin⁡θ),r=x2+y2,tan⁡θ=yxz=r(\cos\theta+i\sin\theta),\quad r=\sqrt{x^2+y^2},\quad\tan\theta=\frac yx
where
rr
the modulus: distance from 0.
θ\theta
the argument: angle from the positive real axis, in (−π,π](-\pi,\pi].
Worked examplez=−1+i3z=-1+i\sqrt3 and w=−3−3iw=-3-3i

Find modulus and argument of each.

Goal
rr, θ\theta.
Strategy
Sketch; reference angle; adjust for quadrant.
  1. ∣z∣=1+3=2|z|=\sqrt{1+3}=2
    Modulus.
  2. Q2; ref tan⁡−13=π3⇒arg⁡z=2π3\text{Q2; ref }\tan^{-1}\sqrt3=\tfrac\pi3\Rightarrow\arg z=\tfrac{2\pi}3
    π−π3\pi-\frac\pi3.
  3. ∣w∣=32; Q3: arg⁡w=−π+π4=−3π4|w|=3\sqrt2;\ \text{Q3: }\arg w=-\pi+\tfrac\pi4=-\tfrac{3\pi}4
    Third quadrant: negative angle.
Answer2,2π32,\frac{2\pi}3; 32,−3π43\sqrt2,-\frac{3\pi}4.
✓ Check2(cos⁡2π3+isin⁡2π3)=−1+i32(\cos\frac{2\pi}3+i\sin\frac{2\pi}3)=-1+i\sqrt3 ✓.
Common misconceptionTrusting tan⁡−1\tan^{-1} tan⁡−1yx\tan^{-1}\frac yx only returns (−π2,π2)(-\frac\pi2,\frac\pi2). Always sketch for quadrants 2 and 3.

2. Multiplying

Moduli multiply, arguments add
  1. zw=r1r2(cos⁡α+isin⁡α)(cos⁡β+isin⁡β)zw=r_1r_2(\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta)
    Polar forms.
  2. =r1r2[(cos⁡αcos⁡β−sin⁡αsin⁡β)+i(sin⁡αcos⁡β+cos⁡αsin⁡β)]=r_1r_2[(\cos\alpha\cos\beta-\sin\alpha\sin\beta)+i(\sin\alpha\cos\beta+\cos\alpha\sin\beta)]
    Expand; i2=−1i^2=-1.
  3. =r1r2[cos⁡(α+β)+isin⁡(α+β)]=r_1r_2[\cos(\alpha+\beta)+i\sin(\alpha+\beta)]
    Compound-angle formulas.
Similarly ∣zw∣=r1r2\left|\frac zw\right|=\frac{r_1}{r_2}, arg⁡zw=α−β\arg\frac zw=\alpha-\beta. Multiplying by ii rotates by 90∘90^\circ.
Worked exampleProduct and quotient in polar form

z=2(cos⁡π3+isin⁡π3)z=2(\cos\frac\pi3+i\sin\frac\pi3), w=3(cos⁡π6+isin⁡π6)w=3(\cos\frac\pi6+i\sin\frac\pi6). Find zwzw and zw\frac zw.

Goal
Both.
Strategy
Multiply/divide moduli; add/subtract args.
  1. zw=6(cos⁡π2+isin⁡π2)=6izw=6(\cos\tfrac\pi2+i\sin\tfrac\pi2)=6i
    π3+π6=π2\frac\pi3+\frac\pi6=\frac\pi2.
  2. zw=23(cos⁡π6+isin⁡π6)\frac zw=\tfrac23(\cos\tfrac\pi6+i\sin\tfrac\pi6)
    π3−π6\frac\pi3-\frac\pi6.
Answer6i6i; 23(cos⁡π6+isin⁡π6)\frac23(\cos\frac\pi6+i\sin\frac\pi6).
✓ Checkz=1+3iz=1+\sqrt3i, w=332+32iw=\frac{3\sqrt3}2+\frac32i: zw=332+32i+92i−332=6izw=\frac{3\sqrt3}2+\frac32i+\frac{9}2i-\frac{3\sqrt3}2=6i ✓.
Pause and think

∣z∣|z|, arg⁡z\arg z for z=−5z=-5?

Summary

  • rr, θ\theta from a sketch.
  • Multiply: moduli ×, args +.
Key words
Modulus
Distance from the origin.
Argument
Angle from the positive real axis.
z = 3 + 4i on an Argand diagram
−112345−1123450real partimaginary part|z| = 53 + 4iarg z ≈ 0.927 rad
The modulus is the distance from 00: 32+42=5\sqrt{3^2+4^2}=5. The argument is the angle from the positive real axis: tan⁡−143≈0.927\tan^{-1}\tfrac43\approx0.927.
HistoryTwo inventors, nine years apart. The Norwegian–Danish surveyor Caspar Wessel presented the idea of picturing complex numbers as points in 1797, but he wrote in Danish and was ignored for about a century. In 1806 the Paris bookkeeper Jean-Robert Argand published the same idea, which is why the diagram carries his name.
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