Complex Numbers

Addition of complex numbers

The addition of complex numbers is defined component-wise, mapping algebraically to vector addition in the Argand plane. This topic covers this concept.

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

📖 Topic progress

○ Mark as complete

Definition and statement

Algebraic and Geometric Law of Complex Addition

Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$ be two complex numbers where $x_1, x_2, y_1, y_2 \in \mathbb{R}$ and $i = \sqrt{-1}$. Their sum is defined as $$z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)$$

📐 Geometric Interpretation

Geometrically, complex addition corresponds to the parallelogram law of vector addition in the Argand plane, where $z_1$ and $z_2$ form adjacent sides of a parallelogram whose diagonal represents the sum $z_1 + z_2$.

⚙️ Algebraic & Structural Meaning

The set of complex numbers $\mathbb{C}$ equipped with addition forms an abelian group, satisfying closure, associativity, existence of the additive identity ($0 + 0i$), and the existence of unique additive inverses ($-z$).

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Real Component Closure	`Real parts $x_1, x_2 \in \mathbb{R}$`	Strictly Mandatory / Necessary	If components do not belong to $\mathbb{R}$, standard field axioms of $\mathbb{C}$ fail to hold.
Imaginary Component Closure	`Imaginary parts $y_1, y_2 \in \mathbb{R}$`	Strictly Mandatory	Real and imaginary parts must combine independently; cross-mixing destroys the Cartesian structure of the Argand plane.
Commutativity and Associativity	`Global for all $z_1, z_2, z_3 \in \mathbb{C}$`	Essential for algebraic manipulation	Failure of group axioms leads to inconsistent algebraic evaluations in polynomial equations over $\mathbb{C}$.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Triangle Inequality Bound for Sum

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

Rule: Maximum modulus occurs when $z_1$ and $z_2$ are collinear and pointed in the same direction (arguments are equal).

Application: Frequently asked in UP PGT to find maximum/minimum values of moduli of sums.

Minimum Modulus of Sum

$$ |z_1 + z_2| \ge ||z_1| - |z_2|| $$

Rule: Minimum modulus occurs when $z_1$ and $z_2$ are collinear and pointed in opposite directions.

Application: Directly tested in DSSSB PGT and KVS for bounds of complex expressions.

Parallelogram Identity

$$ |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2) $$

Rule: Relates the lengths of diagonals of a parallelogram formed by complex numbers to the sum of the squares of its sides.

Centroid of Complex Triangle

$$ z_{centroid} = \frac{z_1 + z_2 + z_3}{3} $$

Rule: The sum of vertices divided by 3 gives the complex coordinate of the centroid instantly.

Application: Saves up to 45 seconds in coordinate-complex geometry hybrid questions.

Vector Translation via Addition

$$ z \to z + z_0 $$

Rule: Adding a constant complex number $z_0$ translates every point in the Argand plane by a vector $\vec{z_0}$ without rotation or scaling.

Application: Used in transformation and locus identification problems.

Rapid Elimination via Modulus Properties

$$ \text{If } |z_1 + z_2| = |z_1| + |z_2| \implies \arg(z_1) = \arg(z_2) $$

Rule: Instantly eliminate options where collinearity conditions are violated when maximum modulus equality holds.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

Confusing Addition with Multiplication of Arguments

Common Pitfall: Candidates mistakenly assume $\arg(z_1 + z_2) = \arg(z_1) + \arg(z_2)$, confusing addition with multiplication.

Counterexample: Let $z_1 = 1$ and $z_2 = i$. Then $\arg(z_1) = 0$, $\arg(z_2) = \pi/2$, but $z_1 + z_2 = 1 + i$ gives $\arg(z_1 + z_2) = \pi/4 \neq 0 + \pi/2$.

Why it fails: Arguments add under multiplication ($arg(z_1 z_2) = arg(z_1) + arg(z_2)$), not under addition.

🛡️ Exam Defense: Never add arguments for complex sums; always convert to Cartesian coordinates $(x_1+x_2, y_1+y_2)$ before finding the argument.

⚠️ COMMON ERROR #02

Triangle Inequality Equality Condition Misapplication

Common Pitfall: Assuming $|z_1 + z_2| = |z_1| + |z_2|$ always holds.

Counterexample: Let $z_1 = 1$ and $z_2 = i$. Then $|1+i| = \sqrt{2}$, while $|1| + |i| = 2$. Clearly $\sqrt{2} \neq 2$.

Why it fails: Equality holds if and only if $z_2 = k z_1$ for some non-negative real scalar $k$.

🛡️ Exam Defense: Check if vectors are collinear and codirectional before equating the modulus of a sum to the sum of moduli.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Complex Addition	$z_1, z_2 \in \mathbb{C}$	$z_1 + z_2 = (x_1+x_2) + i(y_1+y_2)$	Parallelogram law / Vector translation	Look for sum of coordinates or translations
Triangle Inequality	$z_1, z_2 \in \mathbb{C}$	$\|z_1 + z_2\| \le \|z_1\| + \|z_2\|$	Side length property of triangles	Look for maximum/minimum value of $\|z_1 + z_2\|$
Parallelogram Identity	$z_1, z_2 \in \mathbb{C}$	$\|z_1 + z_2\|^2 + \|z_1 - z_2\|^2 = 2(\|z_1\|^2 + \|z_2\|^2)$	Sum of squares of diagonals equals sum of squares of sides	Look for combinations of sum and difference moduli squared
Additive Inverse	$z = x + iy$	$z + (-z) = 0$	Reflection through origin in Argand plane	Look for symmetric points about origin

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $|z_1| = 3$ and $|z_2| = 4$ and $|z_1 + z_2| = 5$, find the value of $|z_1 - z_2|$.

(A) $5$(B) $7$(C) $25$(D) $12$

📚 Method A: Formal Textbook Derivation

Use the parallelogram identity: $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$. Substitute $5^2 + |z_1 - z_2|^2 = 2(3^2 + 4^2) \implies 25 + |z_1 - z_2|^2 = 2(25) = 50 \implies |z_1 - z_2|^2 = 25 \implies |z_1 - z_2| = 5$.

⚡ Method B: 25-Second Exam Shortcut

Recognize that $3^2 + 4^2 = 5^2$, meaning $z_1$ and $z_2$ are orthogonal vectors forming a right-angled triangle. Hence, the lengths of the diagonals of the rectangle are equal, so $|z_1 - z_2| = |z_1 + z_2| = 5$.

💡 Tip: Whenever side lengths form a Pythagorean triplet, the parallelogram is a rectangle, making diagonal lengths equal.

CANONICAL PROBLEM 02 Dual-Method Breakdown

What is the maximum value of $|z + 3 + 4i|$ if $|z| \le 2$?

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

📚 Method A: Formal Textbook Derivation

By triangle inequality, $|z + (3 + 4i)| \le |z| + |3 + 4i|$. Since $|z| \le 2$ and $|3 + 4i| = \sqrt{3^2 + 4^2} = 5$, the maximum value is $2 + 5 = 7$.

⚡ Method B: 25-Second Exam Shortcut

Max value = Radius of disc + Distance of center from origin = $2 + \sqrt{3^2 + 4^2} = 7$.

💡 Tip: For bounded complex regions like $|z| \le r$, maximum of $|z - z_0|$ is $r + |z_0|$ and minimum is $||z_0| - r|$.

Practice questions

Choose an answer to check it and see the solution:

If $z_1 = 2 + 3i$ and $z_2 = 1 - 4i$, then $z_1 + z_2$ is equal to:

A $3 - i$

B $3 + 7i$

C $1 + 7i$

D $3 - 7i$

Derivation & Analysis:

Add real parts and imaginary parts separately: $(2 + 1) + i(3 - 4) = 3 - i$. Correct option is (A).

The additive inverse of the complex number $z = -5 + 12i$ is:

A $5 - 12i$

B $-5 - 12i$

C $5 + 12i$

D $\frac{-5}{169} - \frac{12}{169}i$

Derivation & Analysis:

The additive inverse of $z = x + iy$ is $-z = -x - iy$. Thus, $-(-5 + 12i) = 5 - 12i$. Correct option is (A).

For any two complex numbers $z_1$ and $z_2$, the inequality $|z_1 + z_2| \le |z_1| + |z_2|$ becomes an equality if and only if:

A $z_2 = z_1$

B $z_2 = -z_1$

C $z_2 = k z_1$ where $k \ge 0$

D $z_1$ and $z_2$ are purely imaginary

Derivation & Analysis:

Equality holds in the triangle inequality when the vectors are collinear and in the same direction, which means $z_2 = k z_1$ for some real $k \ge 0$. Correct option is (C).

If $|z_1| = 2$, $|z_2| = 3$, and $|z_3| = 4$, then the maximum value of $|z_1 + z_2 + z_3|$ is:

A $9$

B $5$

C $1$

D $4$

Derivation & Analysis:

By generalized triangle inequality, $|z_1 + z_2 + z_3| \le |z_1| + |z_2| + |z_3| = 2 + 3 + 4 = 9$. Correct option is (A).

If $z$ is a complex number such that $|z - 1| + |z + 1| = 2$, then $z$ lies on:

A A circle of radius 1

B The line segment joining $-1$ and $1$

C The entire real line

D An ellipse with foci at $\pm 1$

Derivation & Analysis:

The sum of distances from $z$ to $1$ and $-1$ equals the distance between $1$ and $-1$ (which is $2$). This implies $z$ lies on the line segment connecting $-1$ and $1$. Correct option is (B).

What is the geometric representation of the addition $z + z_0$ in the Argand plane?

A Rotation about origin by $\arg(z_0)$

B Scaling by factor $|z_0|$

C Translation by vector $z_0$

D Reflection across the real axis

Derivation & Analysis:

Adding a constant complex number $z_0$ translates the point $z$ by the vector corresponding to $z_0$. Correct option is (C).

If $z_1, z_2, z_3$ are vertices of an equilateral triangle inscribed in a circle of radius $r$ centered at the origin, then $z_1 + z_2 + z_3$ equals:

A $3r$

B $0$

C $r \sqrt{3}$

D $3i r$

Derivation & Analysis:

The centroid of a regular polygon inscribed in a circle centered at the origin is at the origin. Thus $\frac{z_1 + z_2 + z_3}{3} = 0 \implies z_1 + z_2 + z_3 = 0$. Correct option is (B).

Which of the following algebraic properties does NOT hold for complex number addition?

A Closure property

B Commutative property

C Multiplicative inverse property

D Associative property

Derivation & Analysis:

Multiplicative inverse pertains to complex multiplication, not addition. Complex addition satisfies closure, commutativity, and associativity. Correct option is (C).

If $|z_1 + z_2| = |z_1 - z_2|$, what is the geometric relationship between $z_1$ and $z_2$?

A They are collinear

B They are orthogonal (perpendicular)

C Their moduli are equal

D One is zero

Derivation & Analysis:

Squaring both sides: $|z_1|^2 + |z_2|^2 + 2\text{Re}(z_1 \bar{z_2}) = |z_1|^2 + |z_2|^2 - 2\text{Re}(z_1 \bar{z_2}) \implies 4\text{Re}(z_1 \bar{z_2}) = 0$, which means $z_1$ and $z_2$ are perpendicular. Correct option is (B).

Q10

If $z_1 = 3 + 4i$ and $z_2 = 5 - 12i$, what is the real part of $z_1 + z_2$?

A $8$

B $-8$

C $2$

D $16$

Derivation & Analysis:

Real part of $z_1$ is $3$, real part of $z_2$ is $5$. Sum of real parts = $3 + 5 = 8$. Correct option is (A).

Common questions

Complex numbers are defined as ordered pairs $(x, y)$. Addition combines vectors component-wise ($(x_1+x_2, y_1+y_2)$), mimicking vector addition. Multiplication, however, arises from the distributive law and the fundamental relation $i^2 = -1$, producing cross-terms like $(x_1 x_2 - y_1 y_2) + i(x_1 y_2 + x_2 y_1)$.

Candidates often confuse inequality bounds in modulus addition, specifically mixing up the maximum condition ($|z_1| + |z_2|$) with the minimum condition ($||z_1| - |z_2||$) or misapplying the triangle equality condition when vectors are not collinear.

Always leverage geometric interpretations. Instead of expanding $|x + iy|$ algebraically, visualize expressions like $|z - z_0|$ as distances in the Argand plane and apply standard triangle inequalities or disc bounds for instant solutions.

← Previous Topic

Complex numbers

Next Topic →

Circular functions of a complex variable