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Worked solution

Find an exact quotient modulus without a calculator

Given z=1+iz=1+i and w=3+iw=\sqrt3+i, without the use of a calculator, find the exact value of ∣zw∣\left|\dfrac zw\right|. marks

[2009/ACJC/II/4(a)(modified)]

CJC · Review problems · Question 15

Answer

∣z/w∣=2/2|z/w|=\sqrt2/2.

Source method: rationalise before taking modulus

zw=(1+i)(3−i)(3+i)(3−i)=(3+1)+(3−1)i4\dfrac zw=\dfrac{(1+i)(\sqrt3-i)}{(\sqrt3+i)(\sqrt3-i)}=\dfrac{(\sqrt3+1)+(\sqrt3-1)i}{4}. Therefore ∣z/w∣=14(3+1)2+(3−1)2=148=2/2\left|z/w\right|=\tfrac14\sqrt{(\sqrt3+1)^2+(\sqrt3-1)^2}=\tfrac14\sqrt8=\sqrt2/2. The cross terms cancel.

How did this part go?

Independent modulus check

Also ∣z∣=2|z|=\sqrt2 and ∣w∣=2|w|=2, so ∣z/w∣=∣z∣/∣w∣=2/2|z/w|=|z|/|w|=\sqrt2/2. This shorter check confirms the full printed rationalisation method.

How did this part go?

Common mistakes

The ask is the modulus of the quotient, a nonnegative real number. Preserve the radical in √3+i and the outer modulus bars.

Solution notes

Source filename says 2026; printed header says 2025/JC2. Both identities are retained; rendered pages determine radicals and modulus bars.

Read the hint