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.

Fundamentals of a complex number

i^2=-1 \qquad a+ib=c+id \iff a=c,\ b=d \qquad (a+ib)(c+id)=(ac-bd)+i(ad+bc)

Worked example

Express

(2+3i)(1-i)

in the form

a+ib

Practice this conceptself-check · 4 quick reps

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.

Conjugate identities

\overline{a+ib}=a-ib \qquad z\bar z=|z|^2 \qquad \operatorname{Re}(z)=\dfrac{z+\bar z}{2} \qquad \operatorname{Im}(z)=\dfrac{z-\bar z}{2i} \qquad \overline{z_1z_2}=\bar z_1\,\bar z_2

Worked example

For what real

x

z=(x-2)+3i

purely imaginary?

Practice this conceptself-check · 4 quick reps

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]

Purely imaginary is the real-part-zero condition, not the imaginary-part-zero one

A number is purely imaginary when

z=-\bar z

(its real part is 0), and purely real when

z=\bar z

(its imaginary part is 0). Students routinely swap these. For

z=(x-2)+3i

to be purely imaginary you set the real part

x-2=0

, not the imaginary part.

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 $|z-a|\le r$ .

Modulus properties

|z|=\sqrt{a^2+b^2} \qquad |z_1z_2|=|z_1|\,|z_2| \qquad \left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|} \qquad |z|^2=z\bar z \qquad |z_1+z_2|\le|z_1|+|z_2|

Worked example

Find

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

Practice this conceptself-check · 4 quick reps

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]

$z\bar z=|z|^2$ , not $|z|$

The product

z\bar z

equals the modulus squared:

z\bar z=a^2+b^2=|z|^2

. Forgetting the square (writing

z\bar z=|z|

) is the single most common modulus slip. So for

z=3+4i

z\bar z=25

, while

|z|=5

Modulus does not distribute over a sum

|z_1+z_2|\ne|z_1|+|z_2|

in general — that's only an inequality (

|z_1+z_2|\le|z_1|+|z_2|

), with equality only when

z_1,z_2

point the same way. Modulus DOES distribute over products and quotients:

|z_1z_2|=|z_1||z_2|

. For maxima/minima of

|z\pm c|

, reach for the triangle inequality, never term-by-term addition.

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: $\arg(z_1z_2)=\arg z_1+\arg z_2$ , $\arg(z_1/z_2)=\arg z_1-\arg z_2$ .

Polar form and argument

z=r(\cos\theta+i\sin\theta)=re^{i\theta} \qquad \arg(z_1z_2)=\arg z_1+\arg z_2 \qquad \arg\!\left(\dfrac{z_1}{z_2}\right)=\arg z_1-\arg z_2

Worked example

Find the modulus and principal argument of

z=1+i

Practice this conceptself-check · 4 quick reps

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]

The principal argument depends on the quadrant, not just $\tan^{-1}(b/a)$

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

alone can't tell apart

a+ib

from

-a-ib

(same ratio, opposite quadrants). Find the reference angle

\tan^{-1}\big|\tfrac{b}{a}\big|

, then place it by the signs of

(a,b)

so the result lands in

(-\pi,\pi]

. For

-1+i

(2nd quadrant) the argument is

\tfrac{3\pi}4

, not

\tan^{-1}(-1)=-\tfrac\pi4

Drill 8 more on argument and polar form

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

What a complex number is
Fundamentals of a complex number
$i^2=-1 \qquad a+ib=c+id \iff a=c,\ b=d \qquad (a+ib)(c+id)=(ac-bd)+i(ad+bc)$
Conjugate; purely real / purely imaginary
Conjugate identities
$\overline{a+ib}=a-ib \qquad z\bar z=|z|^2 \qquad \operatorname{Re}(z)=\dfrac{z+\bar z}{2} \qquad \operatorname{Im}(z)=\dfrac{z-\bar z}{2i} \qquad \overline{z_1z_2}=\bar z_1\,\bar z_2$
Modulus and the triangle inequality
Modulus properties
$|z|=\sqrt{a^2+b^2} \qquad |z_1z_2|=|z_1|\,|z_2| \qquad \left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|} \qquad |z|^2=z\bar z \qquad |z_1+z_2|\le|z_1|+|z_2|$
Argument and polar form
Polar form and argument
$z=r(\cos\theta+i\sin\theta)=re^{i\theta} \qquad \arg(z_1z_2)=\arg z_1+\arg z_2 \qquad \arg\!\left(\dfrac{z_1}{z_2}\right)=\arg z_1-\arg z_2$

Watch out for (4)

Purely imaginary is the real-part-zero condition, not the imaginary-part-zero one
→ Conjugate; purely real / purely imaginary
$z\bar z=|z|^2$ , not $|z|$
→ Modulus and the triangle inequality
Modulus does not distribute over a sum
→ Modulus and the triangle inequality
The principal argument depends on the quadrant, not just $\tan^{-1}(b/a)$
→ Argument and polar form

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

Related notes