Introduction to Complex Numbers And Their Conjugates

Q: What is a complex number in simple terms?

A complex number is x + iy where x, y are real numbers and i = √(−1). Here x is the real part and y the imaginary part. Every real number is a complex number with zero imaginary part.

Q: What is the modulus of a complex number?

For z = x + iy: |z| = √(x² + y²). Geometrically, it's the distance from origin in the Argand plane. Always non-negative; |z| = 0 only when z = 0.

Q: How do you find the principal argument of a complex number?

Find θ from tan θ = |y/x| (Q1 angle). Determine quadrant of z. Principal argument is θ (Q1), π − θ (Q2), θ − π (Q3), or −θ (Q4). Always in (−π, π].

Q: What is the conjugate of a complex number?

z̄ = a − ib for z = a + ib — mirror image across real axis. Key: z + z̄ = 2Re(z), z − z̄ = 2i Im(z), z·z̄ = |z|².

Q: Is zero a complex number?

Yes. 0 = 0 + 0i fits x + iy. It's uniquely both purely real and purely imaginary. However, arg(0) is undefined.

Q: What is the difference between argument and principal argument?

Argument is any θ such that z = r(cos θ + i sin θ) — infinite values differing by 2π. Principal argument is the unique one in (−π, π].

Q: How to find the square root of a complex number?

For z = x + iy with y > 0: √z = ±(√((|z|+x)/2) + i√((|z|−x)/2)). If y < 0, use − inside. Example: √(8−15i) = ±(1/√2)(5−3i).

MathsComplex NumbersFor JEE aspirants

A complex number is a number of the form $z = x + i y$ , where $x$ and $y$ are real numbers and $i = - 1$ is the imaginary unit. The real number $x$ is the real part $Re (z)$ and $y$ is the imaginary part $Im (z)$ . The conjugate of $z$ is $\overset{z}{ˉ} = x - i y$ - the mirror image of $z$ across the real axis. This concept covers the definition, modulus, argument, polar form, and complete conjugate properties needed for JEE Main and Advanced.

Key Formulas - Quick Reference

$i^{2} = - 1, i^{4 n} = 1, i^{4 n + 1} = i, i^{4 n + 2} = - 1, i^{4 n + 3} = - i$
Modulus: $∣ z ∣ = x^{2} + y^{2}$
Conjugate: $\overset{z}{ˉ} = x - i y$ , $z \cdot \overset{z}{ˉ} = ∣ z ∣^{2}$
$Re (z) = \frac{z + z ˉ}{2}, Im (z) = \frac{z - z ˉ}{2 i}$
Polar form: $z = r (cos θ + i sin θ) = r e^{i θ}$
$ar g (z_{1} z_{2}) = ar g (z_{1}) + ar g (z_{2}), ar g (z^{n}) = n ar g (z)$
Triangle inequality: $∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$
$x + i y$ (for $y > 0$ ): $\pm (\frac{∣ z ∣ + x}{2} + i \frac{∣ z ∣ - x}{2})$

1. Basic Concepts of Complex Numbers

Definition A number in the form

x + i y

, where

x, y \in R

and

i = - 1

, is called a complex number.

Powers of i: $i^{2} = - 1$ , $i^{3} = - i$ , $i^{4} = 1$ . For integer $n$ : $i^{4 n} = 1, i^{4 n + 1} = i, i^{4 n + 2} = - 1, i^{4 n + 3} = - i$ .
Real / Imaginary Parts: If $z = x + i y$ : $Re (z) = x$ , $Im (z) = y$ .
Purely Real / Imaginary: $z$ is purely real if $Im (z) = 0$ ; purely imaginary if $Re (z) = 0$ . $z = 0$ is both.
Equality: $a + ib = c + i d$ iff $a = c$ and $b = d$ . No order relation exists - expressions like $z_{1} < z_{2}$ are meaningless.
Reals as Complex: Every real number $a = a + i \cdot 0$ , so $R \subset C$ .

Remark Powers of

i

cycle with period 4. For

i^{n}

, divide

n

by 4 and use the remainder.

2. Representation in the Argand Plane

A complex number

z = x + i y

is represented by the point

P (x, y)

on the axes

O X

(real) and

O Y

(imaginary). This plane is the Argand diagram or complex plane.

Figure 1: Complex number z = x + iy represented as point P(x, y) in the Argand plane

3. Modulus of a Complex Number

Definition For

z = x + i y

∣ z ∣ = x^{2} + y^{2}

. Geometrically,

∣ z ∣

is the distance of

P (x, y)

from the origin.

∣ z ∣ = x^{2} + y^{2}

4. Argument of a Complex Number

For $z = x + i y$ , the angle $θ$ satisfying $tan θ = y / x$ is the argument (or amplitude), denoted $ar g (z)$ .

Case $x = 0, y \neq = 0$ : $ar g (z) = \pm π /2$ depending on $y > 0$ or $y < 0$ . Purely imaginary.
Case $y = 0, x \neq = 0$ : $ar g (z) = 0$ or $π$ depending on $x > 0$ or $x < 0$ . Purely real.
$ar g (0)$ is undefined.

Equivalently: $cos θ = \frac{x}{x ^{2} + y ^{2}}, sin θ = \frac{y}{x ^{2} + y ^{2}}$ .

5. Principal Argument of a Complex Number

Definition The value of

θ

satisfying

- π < θ \leq π

is the principal value of the argument.

Method: How to Find the Principal Argument of $z = x + i y$

Find $tan θ = ∣ y / x ∣$ - this gives $θ$ in the first quadrant.
Determine the quadrant of $z$ from signs of $x, y$ .
Principal argument is $θ$ (Q1), $π - θ$ (Q2), $θ - π$ (Q3), or $- θ$ (Q4).

Figure 2: Principal argument formula by quadrant for complex numbers

Solved Example 1

For

z = 3 - i

, find the principal value of

ar g (z)

Solution:

Here $x = 3$ , $y = - 1$ . $tan θ = ∣ - 1/ 3 ∣ \Rightarrow θ = π /6$ .

Since $z$ is in Q4: $ar g (z) = - π /6$ .

6. Polar Form of a Complex Number

Let $O P = r$ ; then $x = r cos θ$ , $y = r sin θ$ , so:

z = r (cos θ + i sin θ)

This is the polar form (trigonometric form). Here $θ$ is the principal argument.

Figure 3: Polar form z = r(cos θ + i sin θ) with modulus r and argument θ

General Values of Argument

z = r [cos (2 nπ + θ) + i sin (2 nπ + θ)] (n \in Z)

Notation

cos θ + i sin θ

is written in short as

cis (θ)

Euler's Formula

cos θ + i sin θ = e^{i θ}

Multiplication & Division in Polar Form

Multiplied: arguments add. $(cos θ_{1} + i sin θ_{1}) (cos θ_{2} + i sin θ_{2}) = cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})$ .

Divided: arguments subtract. $\frac{cos θ _{1} + i sin θ _{1}}{cos θ _{2} + i sin θ _{2}} = cos (θ_{1} - θ_{2}) + i sin (θ_{1} - θ_{2})$ .

Solved Example 2

Find the modulus and the principal argument of:
(i)

6 (cos 31 0^{\circ} - i sin 31 0^{\circ})

(ii)

\frac{1 + i 3}{2 i ( cos \frac{π}{3} + i sin \frac{π}{3} )}

Solution (i):

$6 (cos 31 0^{\circ} - i sin 31 0^{\circ}) = 6 [cos (36 0^{\circ} - 5 0^{\circ}) - i sin (36 0^{\circ} - 5 0^{\circ})] = 6 (cos 5 0^{\circ} + i sin 5 0^{\circ})$

Modulus $= 6$ , principal argument $= 5 π /18$ .

Solution (ii):

The numerator $1 + i 3 = 2 (\frac{1}{2} + i \frac{3}{2}) = 2 (cos \frac{π}{3} + i sin \frac{π}{3})$ .

So expression $= \frac{2 ( cos \frac{π}{3} + i sin \frac{π}{3} )}{2 i ( cos \frac{π}{3} + i sin \frac{π}{3} )} = \frac{1}{i} = - i = cos (- π /2) + i sin (- π /2)$ .

Modulus $= 1$ , principal argument $= - π /2$ .

Solved Example 3

z = i^{i}

, find

∣ z ∣

Solution:

$z = i^{i} = (e^{iπ /2})^{i} = e^{- π /2}$ . Since $z$ is a positive real: $∣ z ∣ = e^{- π /2}$ .

Unimodular Complex Number

∣ z ∣ = 1 \Rightarrow z

is unimodular. It lies on the unit circle. If

∣ z ∣ = 1

z = cos θ + i sin θ

and

\frac{1}{z} = cos θ - i sin θ = \overset{z}{ˉ}

Figure 4: Unimodular complex number lies on the unit circle |z| = 1

Solved Example 4

\frac{z _{1} - i z _{2}}{z _{1} + i z _{2}}

is unimodular, show that

\frac{z _{1}}{z _{2}}

is purely real.

Solution:

\frac{z _{1} - i z _{2}}{z _{1} + i z _{2}} = 1 \Rightarrow \frac{z _{1}}{z _{2}} - i = \frac{z _{1}}{z _{2}} + i

$P (z_{1} / z_{2})$ is equidistant from $A (i)$ and $B (- i)$ , so it lies on the perpendicular bisector of $A B$ - the real axis. Hence $z_{1} / z_{2}$ is purely real.

Algebraic Operations

Operation	Formula
Addition	$(a + ib) + (c + i d) = (a + c) + i (b + d)$
Subtraction	$(a + ib) - (c + i d) = (a - c) + i (b - d)$
Multiplication	$(a + ib) (c + i d) = (a c - b d) + i (a d + b c)$
Division	$\frac{a + ib}{c + i d} = \frac{a c + b d}{c ^{2} + d ^{2}} + i \frac{b c - a d}{c ^{2} + d ^{2}}$ (if $c^{2} + d^{2} \neq = 0$ )

Square Root of a Complex Number

Let $x + i y = x_{1} + i y_{1}$ . Squaring gives $x_{1} = x^{2} - y^{2}$ , $y_{1} = 2 x y$ . Solving:

Case:

y_{1} > 0

x_{1} + i y_{1} = \pm (\frac{∣ z _{1} ∣ + x _{1}}{2} + i \frac{∣ z _{1} ∣ - x _{1}}{2})

Case:

y_{1} < 0

x_{1} + i y_{1} = \pm (\frac{∣ z _{1} ∣ + x _{1}}{2} - i \frac{∣ z _{1} ∣ - x _{1}}{2})

Solved Example 5

Find

8 - 15 i

Solution:

$y = - 15 < 0$ , $∣ z ∣ = 17$ . So $8 - 15 i = \pm (\frac{17 + 8}{2} - i \frac{17 - 8}{2}) = \pm \frac{1}{2} (5 - 3 i)$ .

7. Conjugate of a Complex Number

Definition The conjugate of

z = a + ib

\overset{z}{ˉ} = a - ib

- the mirror image of

z

in the real axis. Key identities:

$z + \overset{z}{ˉ} = 2 a$ (real), $z - \overset{z}{ˉ} = 2 ib$ (imaginary)
$z \overset{z}{ˉ} = a^{2} + b^{2} = ∣ z ∣^{2} = ∣ \overset{z}{ˉ} ∣^{2}$ (real)
$Re (z) = \frac{z + z ˉ}{2}, Im (z) = \frac{z - z ˉ}{2 i}$

Figure 5: Conjugate z̄ is the mirror image of z across the real axis

Properties of Conjugate

$\overline{(\overset{z}{ˉ})} = z$ , $∣ z ∣ = ∣ \overset{z}{ˉ} ∣$
$z + \overset{z}{ˉ} = 2 Re (z)$ , $z - \overset{z}{ˉ} = 2 i Im (z)$
$z$ purely real $\Leftrightarrow z = \overset{z}{ˉ}$ ; $z$ purely imaginary $\Leftrightarrow z + \overset{z}{ˉ} = 0$
$z \overset{z}{ˉ} = ∣ z ∣^{2} = ∣ \overset{z}{ˉ} ∣^{2}$
$\overline{z_{1} \pm z_{2}} = \overset{z}{ˉ}_{1} \pm \overset{z}{ˉ}_{2}$ , $\overline{z_{1} z_{2}} = \overset{z}{ˉ}_{1} \overset{z}{ˉ}_{2}$ , $\overline{z^{n}} = (\overset{z}{ˉ})^{n}$
$\overline{(z_{1} / z_{2})} = \overset{z}{ˉ}_{1} / \overset{z}{ˉ}_{2}$

Properties of Modulus

$∣ z ∣ = 0 \Leftrightarrow z = 0$ ; $∣ z_{1} - z_{2} ∣$ = distance between $z_{1}$ and $z_{2}$ .
$- ∣ z ∣ \leq Re (z) \leq ∣ z ∣$ , $- ∣ z ∣ \leq Im (z) \leq ∣ z ∣$
$∣ z ∣ \leq ∣ Re (z) ∣ + ∣ Im (z) ∣ \leq 2 ∣ z ∣$
$∣ z ∣^{2} = z \overset{z}{ˉ}$ , $∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$ , $∣ z^{n} ∣ = ∣ z ∣^{n}$
$∣ z_{1} / z_{2} ∣ = ∣ z_{1} ∣/∣ z_{2} ∣$
Triangle inequality: $∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$
Reverse triangle inequality: $∣ z_{1} - z_{2} ∣ \geq ∣ z_{1} ∣ - ∣ z_{2} ∣$
$∣ z_{1} + z_{2} ∣^{2} = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} + 2 Re (z_{1} \overset{z}{ˉ}_{2})$
$∣ z_{1} - z_{2} ∣^{2} = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} - 2 Re (z_{1} \overset{z}{ˉ}_{2})$

Properties of Argument

$ar g (z_{1} z_{2}) = ar g (z_{1}) + ar g (z_{2})$
$ar g (z_{1} / z_{2}) = ar g (z_{1}) - ar g (z_{2})$
$ar g (z^{n}) = n ar g (z)$

Note

θ_{1} + θ_{2}

θ_{1} - θ_{2}

are not necessarily principal values.

$ar g (z) = 0$ or $π \Rightarrow z$ is purely real $\Rightarrow z = \overset{z}{ˉ}$
$ar g (z) = \pm π /2 \Rightarrow z$ is purely imaginary $\Rightarrow z = - \overset{z}{ˉ}$

Solved Example 6

(z_{1}, z_{2})

and

(z_{3}, z_{4})

are pairs of non-zero conjugate complex numbers, find

ar g (z_{1} / z_{3}) + ar g (z_{2} / z_{4})

Solution:

$ar g (z_{1} / z_{3}) + ar g (z_{2} / z_{4}) = ar g (z_{1} z_{2} / z_{3} z_{4})$ . Since $z_{2} = \overset{z}{ˉ}_{1}, z_{4} = \overset{z}{ˉ}_{3}$ : $z_{1} z_{2} = ∣ z_{1} ∣^{2}$ , $z_{3} z_{4} = ∣ z_{3} ∣^{2}$ .

$= ar g (∣ z_{1} ∣^{2} /∣ z_{3} ∣^{2}) = 0$ (argument of positive real is zero).

Solved Example 7

∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = 1

, prove

∣ z_{1} + z_{2} + z_{3} ∣ = ∣1/ z_{1} + 1/ z_{2} + 1/ z_{3} ∣

Solution:

$∣ z_{1} + z_{2} + z_{3} ∣ = ∣ \overset{z}{ˉ}_{1} + \overset{z}{ˉ}_{2} + \overset{z}{ˉ}_{3} ∣ = ∣ \frac{∣ z _{1} ∣ ^{2}}{z _{1}} + \frac{∣ z _{2} ∣ ^{2}}{z _{2}} + \frac{∣ z _{3} ∣ ^{2}}{z _{3}} ∣ = ∣1/ z_{1} + 1/ z_{2} + 1/ z_{3} ∣$ .

Solved Example 8

Prove

∣ z_{1} ∣ + ∣ z_{2} ∣ = ∣ \frac{1}{2} (z_{1} + z_{2}) + z_{1} z_{2} ∣ + ∣ \frac{1}{2} (z_{1} + z_{2}) - z_{1} z_{2} ∣

Solution:

RHS $= \frac{1}{2} ∣ (z_{1} + z_{2})^{2} ∣ + \frac{1}{2} ∣ (z_{1} - z_{2})^{2} ∣ = \frac{1}{2} ∣ z_{1} + z_{2} ∣^{2} + \frac{1}{2} ∣ z_{1} - z_{2} ∣^{2}$ .

By parallelogram law $∣ a + b ∣^{2} + ∣ a - b ∣^{2} = 2 (∣ a ∣^{2} + ∣ b ∣^{2})$ : $= ∣ z_{1} ∣ + ∣ z_{2} ∣ =$ LHS.

Common Mistakes to Avoid

Watch out

Assuming order relations. $z_{1} < z_{2}$ is meaningless - only moduli can be compared.
Wrong quadrant for principal argument. Always check signs of $x, y$ and apply the correct formula.
Applying $a b = ab$ to negatives. $- 1 - 1 = i \cdot i = - 1$ , not $1 = 1$ .
Confusing argument with principal argument. Argument has infinite values; principal is in $(- π, π]$ .
Forgetting $ar g (0)$ is undefined. Never plug $z = 0$ into any $ar g$ formula.

Frequently Asked Questions

What is a complex number in simple terms?

A complex number is $x + i y$ where $x, y \in R$ and $i = - 1$ . Here $x$ is the real part and $y$ the imaginary part. Every real number is a complex number with zero imaginary part.

What is the modulus of a complex number?

For $z = x + i y$ : $∣ z ∣ = x^{2} + y^{2}$ . Geometrically, it's the distance from origin in the Argand plane. Always non-negative; $∣ z ∣ = 0$ only when $z = 0$ .

How do you find the principal argument of a complex number?

Find $θ$ from $tan θ = ∣ y / x ∣$ (Q1 angle). Determine quadrant of $z$ . Principal argument is $θ$ (Q1), $π - θ$ (Q2), $θ - π$ (Q3), or $- θ$ (Q4). Always in $(- π, π]$ .

What is the conjugate of a complex number?

$\overset{z}{ˉ} = a - ib$ for $z = a + ib$ - mirror image across real axis. Key: $z + \overset{z}{ˉ} = 2 Re (z)$ , $z - \overset{z}{ˉ} = 2 i Im (z)$ , $z \overset{z}{ˉ} = ∣ z ∣^{2}$ .

Is zero a complex number?

Yes. $0 = 0 + 0 i$ fits $x + i y$ . It's uniquely both purely real and purely imaginary. However, $ar g (0)$ is undefined.

What is the difference between argument and principal argument?

Argument is any $θ$ such that $z = r (cos θ + i sin θ)$ - infinite values (differing by $2 π$ ). Principal argument is the unique one in $(- π, π]$ .

How to find the square root of a complex number?

For $z = x + i y$ with $y > 0$ : $z = \pm ((∣ z ∣ + x) /2 + i (∣ z ∣ - x) /2)$ . If $y < 0$ , use $-$ inside. Example: $8 - 15 i = \pm \frac{1}{2} (5 - 3 i)$ .

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1. Basic Concepts of Complex Numbers

2. Representation in the Argand Plane

3. Modulus of a Complex Number

4. Argument of a Complex Number

5. Principal Argument of a Complex Number

Method: How to Find the Principal Argument of z=x+iy

6. Polar Form of a Complex Number

General Values of Argument

Multiplication & Division in Polar Form

Unimodular Complex Number

Algebraic Operations

Square Root of a Complex Number

7. Conjugate of a Complex Number

Properties of Conjugate

Properties of Modulus

Properties of Argument

Common Mistakes to Avoid

Frequently Asked Questions

What is a complex number in simple terms?

What is the modulus of a complex number?

How do you find the principal argument of a complex number?

What is the conjugate of a complex number?

Is zero a complex number?

What is the difference between argument and principal argument?

How to find the square root of a complex number?

Ready to master Complex Numbers?

Method: How to Find the Principal Argument of $z = x + i y$