Complex Numbers

Multiplication of complex numbers

The multiplication of complex numbers combines algebraic expansion with geometric rotation and scaling in the Argand plane, following the fundamental rule $i^2 = -1$.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

📖 Topic progress

○ Mark as complete

Definition and statement

Polar Multiplication Principle & De Moivre's Foundation

Let $z_1 = r_1(\cos \theta_1 + i \sin \theta_1) = r_1 e^{i\theta_1}$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2) = r_2 e^{i\theta_2}$ be two complex numbers. Then their product is given by $z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)] = r_1 r_2 e^{i(\theta_1 + \theta_2)}$.

📐 Geometric Interpretation

Multiplication geometrically results in scaling the length of the vector by $|z_2|$ (or $r_2$) and rotating the vector $z_1$ counterclockwise about the origin through an angle equal to the argument of $z_2$ (i.e., $\theta_2$).

⚙️ Algebraic & Structural Meaning

It establishes an isomorphism between the multiplicative group of non-zero complex numbers and the direct product of the positive reals under multiplication and the circle group under addition modulo $2\pi$. It also forms the algebraic bedrock for De Moivre's theorem and polynomial root extraction.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Cartesian to Polar Representation	`Principal argument range $(-\pi, \pi]$ or $[0, 2\pi)$`	Strictly Mandatory	Failure to use the correct principal branch or quadrant adjustment for arguments leads to extraneous angles or phase shifts of $2\pi k$.
Distributive Law over Addition	`\forall z_1, z_2, z_3 \in \mathbb{C}`	Mandatory Algebraic Axiom	Violating distributivity breaks field axioms, making algebraic factorization of complex polynomials invalid.
Modulus Multiplicativity	`\|z_1 z_2\| = \|z_1\| \|z_2\|`	Essential Property	Ignoring absolute value multiplicativity prevents rapid calculation of distances and loci in exam problems.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Modulus Squared Property

$$|z|^2 = z \bar{z}$$

Rule: Whenever you need to evaluate the magnitude or eliminate complex denominators, immediately substitute $z \bar{z}$ for $|z|^2$ instead of computing $\sqrt{x^2 + y^2}$.

Application: Directly applied in UP PGT evaluations of complex expressions and locus problems.

Argument Addition Rule

$$\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) \pmod{2\pi}$$

Rule: To find the argument of a product or power, never convert back to Cartesian form; simply add the individual arguments.

Application: Frequent 1-mark questions in DSSSB and KVS asking for $\arg(z^n)$.

Multiplication by i as Quadrant Rotation

$$i \cdot (x + iy) = -y + ix$$

Rule: Multiplying any complex number by $i$ rotates its vector representation counterclockwise by exactly $90^\circ$ ($+\pi/2$) without computing trigonometric values.

Application: Used for instant geometric identification in TGT vector-rotation problems.

Real and Imaginary Parts of Products

$$\operatorname{Re}(z_1 z_2) = x_1 x_2 - y_1 y_2, \quad \operatorname{Im}(z_1 z_2) = x_1 y_2 + x_2 y_1$$

Rule: Memorize the expansion pattern as $(ac-bd) + i(ad+bc)$ for $z_1 = a+ib$ and $z_2 = c+id$ to skip intermediate algebraic steps.

Multiplicative Inverse Shortcut

$$z^{-1} = \frac{\bar{z}}{|z|^2}$$

Rule: Never rationalize denominators manually by multiplying conjugates out the long way; use the direct inverse formula using norm squared.

Rapid Option Elimination via Modulus Check

$$|z_1 z_2 \dots z_n| = |z_1| |z_2| \dots |z_n||

Rule: Before full multiplication, compute the magnitude of the options. If moduli differ, calculate only the product of magnitudes to eliminate 2-3 options instantly.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

The Argument Addition Multi-Valued Fallacy

Common Pitfall: Assuming $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$ strictly without considering principal value boundaries $(-\pi, \pi]$.

Counterexample: Let $z_1 = -1$ and $z_2 = -1$. Here $\arg(z_1) = \pi$, $\arg(z_2) = \pi$. Their sum is $2\pi$, but $z_1 z_2 = 1$, whose principal argument is $0$.

Why it fails: Principal arguments must be adjusted modulo $2\pi$ to fall within the standard interval.

🛡️ Exam Defense: Always add $2\pi k$ ($k \in \{-1, 0, 1\}$) to bring the resulting sum back into $(-\pi, \pi]$.

⚠️ COMMON ERROR #02

Square Root of Negative Product Mistake

Common Pitfall: Applying $\sqrt{ab} = \sqrt{a}\sqrt{b}$ indiscriminately for negative real numbers.

Counterexample: \sqrt{(-1)(-1)} \neq \sqrt{-1}\sqrt{-1}

Why it fails: $\sqrt{1} = 1$, whereas $i \cdot i = i^2 = -1$. The property $\sqrt{z_1 z_2} = \sqrt{z_1}\sqrt{z_2}$ holds only under strict principal branch restrictions.

🛡️ Exam Defense: Convert negative numbers to polar form with explicit $i$ factors (e.g., $\sqrt{-a} = \sqrt{a}i$) before multiplying.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Cartesian Multiplication	Standard algebraic form $a+ib$	$(ac-bd) + i(ad+bc)$	Scaling and shearing transformation	When coefficients are small integers
Polar Multiplication	Polar form $r(\cos\theta + i\sin\theta)$	$r_1 r_2 e^{i(\theta_1 + \theta_2)}$	Scaling by $r_1 r_2$ and rotation by $\theta_1+\theta_2$	When powers, roots, or angles are given
Modulus Multiplicativity	Any $z_1, z_2 \in \mathbb{C}$	$\|z_1 z_2\| = \|z_1\|\|z_2\|$	Product of vector lengths from origin	When finding magnitudes or bounds
Conjugation Interaction	Any $z_1, z_2 \in \mathbb{C}$	$\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$	Reflection across the real axis	When dealing with real polynomials or proofs

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $z_1 = 1 + i\sqrt{3}$ and $z_2 = \sqrt{3} + i$, find the principal argument of $z_1 z_2$.

(A) $\frac{\pi}{3}$(B) $\frac{2\pi}{3}$(C) $\frac{\pi}{2}$(D) $\frac{\pi}{6}$

📚 Method A: Formal Textbook Derivation

Step 1: Find $\arg(z_1) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$. Step 2: Find $\arg(z_2) = \tan^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}$. Step 3: $\arg(z_1 z_2) = \frac{\pi}{3} + \frac{\pi}{6} = \frac{\pi}{2}$? Wait, let's recalculate: $z_1 = 2 e^{i\pi/3}$, $z_2 = 2 e^{i\pi/6}$. Sum of arguments = $\frac{\pi}{3} + \frac{\pi}{6} = \frac{\pi}{2}$. Let us check via cartesian: $z_1 z_2 = (1+i\sqrt{3})(\sqrt{3}+i) = \sqrt{3} + i + 3i - \sqrt{3} = 4i$. Argument of $4i$ is $\pi/2$. Thus option (C).

⚡ Method B: 25-Second Exam Shortcut

Use polar argument addition directly: $\arg(z_1) = 60^\circ$, $\arg(z_2) = 30^\circ$, sum = $90^\circ = \pi/2$.

💡 Tip: Always cross-verify argument addition with direct cartesian multiplication when numbers are simple.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If $|z| = 2$, what is the maximum value of $|z + 3 - 4i|$?

(A) 3(B) 5(C) 7(D) 9

📚 Method A: Formal Textbook Derivation

Step 1: Use triangle inequality $|z_1 + z_2| \le |z_1| + |z_2|$. Step 2: $|z + (3 - 4i)| \le |z| + |3 - 4i|$. Step 3: Given $|z| = 2$ and $|3-4i| = \sqrt{9+16} = 5$. Step 4: Max value = $2 + 5 = 7$.

⚡ Method B: 25-Second Exam Shortcut

Identify circle center $(-3, 4)$ radius 5, origin distance is 5, max distance = radius of circle + distance to center = $2 + 5 = 7$.

💡 Tip: Triangle inequality combined with modulus properties solves distance maximization instantly.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If $z = x + iy$ and $z(1+i) = 3+5i$, then the value of $y$ is:

A 1

B 2

C 4

D 5

Derivation & Analysis:

Multiply out $(x+iy)(1+i) = (x-y) + i(x+y) = 3+5i$. Equating real and imaginary parts: $x-y=3$ and $x+y=5$. Adding gives $2x=8 \implies x=4$, so $y=1$. Wait, $4-1=3$ and $4+1=5$, so $y=1$. Let's check options: (A) 1. Correct option index is 0.

MCQ 2: The value of $(1 + i)^4 + (1 - i)^4$ is:

A 8

B -8

C 0

D 16

Derivation & Analysis:

Convert to polar: $1+i = \sqrt{2}e^{i\pi/4}$. Then $(1+i)^4 = (\sqrt{2})^4 e^{i\pi} = 4(-1) = -4$. Similarly, $(1-i)^4 = -4$. Sum = $-4 + (-4) = -8$.

MCQ 3: If $z_1, z_2$ are complex numbers such that $|z_1| = 1$ and $|z_2| = 2$, then $|z_1 \bar{z}_2 + \bar{z}_1 z_2|$ is less than or equal to:

A 2

B 4

C 1

D 8

Derivation & Analysis:

Let $z_1 \bar{z}_2 = w$. Then the expression is $|w + \bar{w}| = 2|\operatorname{Re}(w)| \le 2|w| = 2|z_1||\bar{z}_2| = 2(1)(2) = 4$. Thus maximum value is 4.

MCQ 4: The conjugate of the complex number $\frac{(1+i)^2}{3-i}$ is:

A $\frac{-1+i}{5}$

B $\frac{1-i}{5}$

C $\frac{1+i}{5}$

D $\frac{-1-i}{5}$

Derivation & Analysis:

Numerator $(1+i)^2 = 2i$. Expression becomes $\frac{2i}{3-i}$. Multiply numerator and denominator by $3+i$: $\frac{2i(3+i)}{10} = \frac{6i - 2}{10} = \frac{-1+3i}{5}$. Its conjugate is $\frac{-1-3i}{5}$? Wait, let's re-check conjugation rules: $\overline{2i/(3-i)} = \frac{-2i}{3+i} = \frac{-2i(3-i)}{10} = \frac{-6i - 2}{10} = \frac{-1-3i}{5}$.

MCQ 5: If $z = \cos\theta + i\sin\theta$, what is the value of $z - \frac{1}{z}$?

A $2i\sin\theta$

B $2\cos\theta$

C $i\tan\theta$

D $2i\cos\theta$

Derivation & Analysis:

Since $z = e^{i\theta}$, $\frac{1}{z} = e^{-i\theta} = \cos\theta - i\sin\theta$. Thus $z - \frac{1}{z} = 2i\sin\theta$.

MCQ 6: For any two complex numbers $z_1, z_2$, the relation $|z_1 + z_2|^2 + |z_1 - z_2|^2$ equals:

A $|z_1|^2 + |z_2|^2$

B $2(|z_1|^2 + |z_2|^2)$

C $4|z_1|^2 |z_2|^2$

D $|z_1|^2 - |z_2|^2$

Derivation & Analysis:

This is the parallelogram law for complex numbers: expanding $|z_1+z_2|^2 + |z_1-z_2|^2$ gives $(z_1+z_2)(\bar{z}_1+\bar{z}_2) + (z_1-z_2)(\bar{z}_1-\bar{z}_2) = 2(|z_1|^2 + |z_2|^2)$.

MCQ 7: If $(1+x)^n = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n$, then the value of $a_0 - a_2 + a_4 - a_6 + \dots$ is:

A $2^{n/2} \cos\left(\frac{n\pi}{4}\right)$

B $2^n \cos\left(\frac{n\pi}{4}\right)$

C $2^{n/2} \sin\left(\frac{n\pi}{4}\right)$

D $0$

Derivation & Analysis:

Substitute $x = i$ into $(1+x)^n$: $(1+i)^n = (a_0 - a_2 + a_4 - \dots) + i(a_1 - a_3 + a_5 - \dots)$. Expressing $1+i = \sqrt{2}e^{i\pi/4}$, we get $(\sqrt{2})^n e^{in\pi/4} = 2^{n/2}[\cos(n\pi/4) + i\sin(n\pi/4)]$. Equating real parts yields $2^{n/2}\cos(n\pi/4)$.

MCQ 8: Assertion (A): The product of two pure imaginary numbers is a real number. Reason (R): The square of any imaginary unit $i$ is $-1$.

A Both (A) and (R) are true and (R) is the correct explanation of (A)

B Both (A) and (R) are true but (R) is not the correct explanation

C (A) is true but (R) is false

D (A) is false but (R) is true

Derivation & Analysis:

Let $z_1 = ib_1$ and $z_2 = ib_2$. Their product is $i^2 b_1 b_2 = -b_1 b_2$, which is strictly real because $i^2 = -1$. Thus both are true and R explains A.

MCQ 9: If $z$ is a complex number such that $z + |z| = 2 + 8i$, then the value of $|z|$ is:

A 10

B 15

C 17

D 20

Derivation & Analysis:

Let $z = x + iy$. Then $x + \sqrt{x^2+y^2} + iy = 2 + 8i$. Equating imaginary parts: $y = 8$. Equating real parts: $x + \sqrt{x^2+64} = 2 \implies \sqrt{x^2+64} = 2-x$. Squaring both sides: $x^2+64 = 4 - 4x + x^2 \implies 4x = -60 \implies x = -15$. Then $|z| = \sqrt{(-15)^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17$.

Q10

MCQ 10: If $\omega$ is a complex cube root of unity, then the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$ is:

A 1

B 2

C 3

D 4

Derivation & Analysis:

Since $1 + \omega + \omega^2 = 0$, we have $1 + \omega^2 = -\omega$ and $1 + \omega = --\omega^2$. The expression becomes $(-\omega - \omega)(-\omega^2 - \omega^2) = (-2\omega)(-2\omega^2) = 4\omega^3 = 4(1) = 4$.

Common questions

While Cartesian form $(a+ib)(c+id)$ is straightforward, polar form $r_1 r_2 e^{i(\theta_1 + \theta_2)}$ transforms cumbersome trigonometric expansions and high-power multiplications into simple magnitude multiplications and angle additions.

Neglecting principal argument boundaries $(-\pi, \pi]$ when adding arguments of products, and making sign errors with $i^2 = -1$ during algebraic expansion.

Always check if magnitude properties ($|z_1 z_2| = |z_1||z_2|$) or rotation shortcuts (multiplying by $i$ or $e^{i\theta}$) can eliminate options before performing full algebraic expansion.

← Previous Topic

Applications of De Moivre's theorem

Next Topic →

Separation into real and imaginary parts