NDA Maths · Complex Numbers

Modulus, Argument & Conjugate

The core toolkit of a complex number: its conjugate, its modulus (distance from the origin), and its argument (angle on the Argand plane) — plus the conditions that make it purely real or purely imaginary.

All Complex Numbers notes NDA Maths strategy

Why this matters

This is the largest subtopic and the foundation for the rest. Modulus + conjugate properties and the principal-argument quadrant rule answer most questions directly, and the triangle inequality cracks the max/min ones.

Concept 1 of 4

What a complex number is

Intuition

A complex number

z=a+ib

bundles a real part and an imaginary part, with

i^2=-1

. Add and multiply like binomials, replacing

i^2

-1

. Two complex numbers are equal only when both their real and imaginary parts match.

Definition

$z=a+ib$ , $a=\operatorname{Re}(z)$ , $b=\operatorname{Im}(z)$ , $i^2=-1$ . Equality: $a+ib=c+id\iff a=c$ and $b=d$ . Arithmetic: add/subtract componentwise; multiply as binomials ( $(a+ib)(c+id)=(ac-bd)+i(ad+bc)$ ). Divide by multiplying top and bottom by the denominator's conjugate.

Worked example

Express

(2+3i)(1-i)

in the form

a+ib

Expand: $2-2i+3i-3i^2$ .
$i^2=-1$ : $2+i+3=5+i$ .

Answer:

5+i

Practice this conceptself-check · 4 quick reps

Try it yourself

z=x+iy

and

z=\overline{z}

, what can you say about

z

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$i^2=$ ?
2.
$(a+ib)=(c+id)$ requires?
3.
Divide complex numbers by multiplying by?
4.
$\operatorname{Re}(3-5i)$ ?

Drill 2 more on what a complex number is

Concept 2 of 4

Conjugate; purely real / purely imaginary

Intuition

The conjugate

\bar z

flips the sign of the imaginary part — a reflection across the real axis. It's the lever for 'purely real' (

z=\bar z

) and 'purely imaginary' (

z=-\bar z

) conditions, and

z\bar z=|z|^2

turns a modulus into an algebra problem.

Definition

$\overline{a+ib}=a-ib$ . Key facts:

$z\bar z=|z|^2=a^2+b^2$ ; $z+\bar z=2\operatorname{Re}(z)$ ; $z-\bar z=2i\operatorname{Im}(z)$ .
Purely real $\iff z=\bar z$ (imaginary part 0). Purely imaginary $\iff z=-\bar z$ (real part 0).
$\overline{z_1z_2}=\bar z_1\bar z_2$ , $\overline{z_1/z_2}=\bar z_1/\bar z_2$ . A real-coefficient equation has complex roots in conjugate pairs.

Worked example

For what real

x

z=(x-2)+3i

purely imaginary?

Purely imaginary ⇒ real part $=0$ : $x-2=0$ .
$x=2$ .

Answer:

x=2

Practice this conceptself-check · 4 quick reps

Try it yourself

\dfrac{z-1}{z+1}

is purely imaginary, what is

|z|

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\overline{a+ib}=$ ?
2.
$z\bar z=$ ?
3.
Purely imaginary condition?
4.
Real-coefficient equation: complex roots come as?

From the bank · past-year question

Example 2Complex NumbersMODERATE

z

is a complex number such that

\dfrac{z-1}{z+1}

is purely imaginary, then what is

|z|

equal to?

[Q20 · Apr · 2023]

Drill 9 more on conjugate; purely real / purely imaginary

Concept 3 of 4

Modulus and the triangle inequality

Intuition

The modulus

|z|

is the distance from the origin, so it multiplies and divides cleanly across products and quotients. For sums and differences it doesn't — there you reach for the triangle inequality, which is exactly what 'maximum/minimum of

|z\pm c|

' questions want.

Definition

$|z|=\sqrt{a^2+b^2}$ . Properties: $|z_1z_2|=|z_1||z_2|$ , $\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}$ , $|z^n|=|z|^n$ , $|\bar z|=|z|$ . Triangle inequality: $\big||z_1|-|z_2|\big|\le|z_1\pm z_2|\le|z_1|+|z_2|$ — gives the max/min of $|z\pm c|$ on a disc $|z-a|\le r$ .

Worked example

Find

|(3+4i)(1-2i)|

$|z_1z_2|=|z_1||z_2|=\sqrt{9+16}\cdot\sqrt{1+4}=5\cdot\sqrt5$ .

Answer:

5\sqrt5

Practice this conceptself-check · 4 quick reps

Try it yourself

|z+4|\le 3

, find the maximum of

|z+1|

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$|z|=$ ?
2.
$|z_1 z_2|=$ ?
3.
Tool for max/min of $|z\pm c|$ ?
4.
$|3+4i|$ ?

From the bank · past-year question

Example 3Complex NumbersHARD

What is the value of

|12 + 5i| + |12 - 5i|

, where

i = \sqrt{-1}

[Q3 · Sep · 2023]

Drill 17 more on modulus and the triangle inequality

Concept 4 of 4

Argument and polar form

Intuition

Plot

z

on the Argand plane: its modulus is the distance from the origin, its argument the angle from the positive real axis. The catch is the principal argument — you must use the quadrant of

(a,b)

, not just

\tan^{-1}(b/a)

, to land it in

(-\pi,\pi]

Definition

Polar form: $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ , $r=|z|$ , $\theta=\arg z$ . The principal argument lies in $(-\pi,\pi]$ ; compute $\tan^{-1}\big|\tfrac{b}{a}\big|$ then adjust for the quadrant of $(a,b)$ . Arguments add under multiplication: $\arg(z_1z_2)=\arg z_1+\arg z_2$ , $\arg(z_1/z_2)=\arg z_1-\arg z_2$ .

Worked example

Find the modulus and principal argument of

z=1+i

$r=\sqrt{1+1}=\sqrt2$ ; $(1,1)$ is in the first quadrant.
$\theta=\tan^{-1}(1/1)=\tfrac\pi4$ .

Answer:

|z|=\sqrt2

\arg z=\tfrac\pi4

Practice this conceptself-check · 4 quick reps

Try it yourself

What is the principal argument of

-1+i

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Polar form of $z$ ?
2.
Principal argument range?
3.
$\arg(z_1 z_2)=$ ?
4.
Principal arg of $1+i$ ?

From the bank · past-year question

Example 4Complex NumbersMODERATE

The modulus and principal argument of the complex number

\dfrac{1+2i}{1-(1-i)^2}

are respectively

[Q11 · Apr · 2017]

Drill 8 more on argument and polar form

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Complex NumbersEASY

z=x+iy

, where

i=\sqrt{-1}

, then what does the equation

z\bar{z}+|z|^{2}+4(z+\bar{z})-48=0

represent?

[Q14 · Sep · 2021]

Example 2Complex NumbersMODERATE

z

is any complex number and

iz^3+z^2-z+i=0

, where

i=\sqrt{-1}

, then what is the value of

(|z|+1)^2

[Q2 · Apr · 2024]

Example 3Complex NumbersHARD

\alpha

and

\beta

are the distinct roots of equation

x^{2}-x+1=0

, then what is the value of

\left|\frac{\alpha^{100}+\beta^{100}}{\alpha^{100}-\beta^{100}}\right|

[Q5 · Apr · 2023]

Example 4Complex NumbersMODERATE

What is

\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^3

equal to?

[Q13 · Apr · 2025]

Example 5Complex NumbersEASY

Z = 1+i

, where

i = \sqrt{-1}

, then what is the modulus of

Z + \frac{2}{Z}

[Q10 · Apr · 2021]

Drill every past-year question on this subtopic

39 questions from the bank — paginated, with cart and Word-export support.

Drill the 39 questions

Drill every past-year question on this subtopic

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