MathematicsChapter 1: Complex Numbers
Equality, Addition, Subtraction
Two complex numbers are equal when real parts match and imaginary parts match: a+bi = c+di iff a=c and b=d.
Learning objectives
- Explain equality, addition, subtraction using the correct definitions, conditions, and notation.
- Apply parts equal separately, add like with like correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Two complex numbers are equal when real parts match and imaginary parts match: a+bi = c+di iff a=c and b=d. Add (or subtract) real with real, imaginary with imaginary. Additive inverse of a+bi is -a-bi.
Mathematical form
bi = c+di iff a
b=d. Add (or subtract) real with real
Where / symbols
- EQUALITY, ADDITION, SUBTRACTION
- Two complex numbers are equal when real parts match and imaginary parts match: a+bi = c+di iff a=c and b=d. Add (or subtract) real with real, imaginary with imaginary. Additive inverse of a+bi is -a-bi.
- EQUALITY
- If x + yi = 3 - 2i, then x = 3 and y = -2. You get two real equations from one complex equation. That is why equality is useful in later chapters too.
- ADD AND SUBTRACT
- (a+bi) + (c+di) = (a+c) + (b+d)i. Subtraction: (a-c) + (b-d)i. Pure imaginaries add like 3i+2i=5i.
Overview
Match parts, then combine:
Equality
If x + yi = 3 - 2i, then x = 3 and y = -2. You get two real equations from one complex equation. That is why equality is useful in later chapters too.
Examples 2+3i = 2+3i · 2+3i != 3+2i · i = 0+1i
Add and Subtract
(a+bi) + (c+di) = (a+c) + (b+d)i. Subtraction: (a-c) + (b-d)i. Pure imaginaries add like 3i+2i=5i.
Examples (3+2i)+(1+4i)=4+6i · (4+3i)-(2+i)=2+2i · -(3+2i)=-3-2i
WHAT — EQUALITY, ADDITION, SUBTRACTION
Two complex numbers are equal when real parts match and imaginary parts match: a+bi = c+di iff a=c and b=d. Add (or subtract) real with real, imaginary with imaginary. Additive inverse of a+bi is -a-bi. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Equality, Addition, Subtraction matters because it extends the real number system and makes equations with negative square roots solvable. The same rules are used later in algebra, quadratics, and graphical interpretation.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through EQUALITY, then ADD AND SUBTRACT. Worked: (5 - i) + (2 + 3i). real: 5 + 2 = 7 imaginary: -1 + 3 = 2 (5 - i) + (2 + 3i) = 7 + 2i Worked: (1 + 2i) - (3 + 5i). real: 1 - 3 = -2 imaginary: 2 - 5 = -3 (1 + 2i) - (3 + 5i) = -2 - 3i Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- (5 - i) + (2 + 3i).
- real: 5 + 2 = 7
- imaginary: -1 + 3 = 2
- (5 - i) + (2 + 3i) = 7 + 2i
- (1 + 2i) - (3 + 5i).
- real: 1 - 3 = -2
- imaginary: 2 - 5 = -3
- (1 + 2i) - (3 + 5i) = -2 - 3i
In simple words
Do not add a to b. 3+2i is not 5. The i keeps the imaginary part separate.
Key points
- parts equal separately
- add like with like
- two real conditions
- reals together, i together
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Equality, Addition, Subtraction
Sticky Notes · Quick Revision
Core Idea
- Two complex numbers are equal when real parts match and imaginary parts match: a+bi = c+di iff a=c and b=d. Add (or subtract) real with real, imaginary with imaginary. Additive inverse of a+bi is -a-bi.
EQUALITY
- If x + yi = 3 - 2i, then x = 3 and y = -2. You get two real equations from one complex equation. That is why equality is useful in later chapters too.
ADD AND SUBTRACT
- (a+bi) + (c+di) = (a+c) + (b+d)i. Subtraction: (a-c) + (b-d)i. Pure imaginaries add like 3i+2i=5i.
Worked Method
- Worked: (5 - i) + (2 + 3i).
- real: 5 + 2 = 7
- imaginary: -1 + 3 = 2
- (5 - i) + (2 + 3i) = 7 + 2i
- Worked: (1 + 2i) - (3 + 5i).
- real: 1 - 3 = -2
- imaginary: 2 - 5 = -3
- (1 + 2i) - (3 + 5i) = -2 - 3i
Common Misconception
- Do not add a to b. 3+2i is not 5. The i keeps the imaginary part separate.
FBISE Exam Check
- Do not add a to b. 3+2i is not 5. The i keeps the imaginary part separate.
60-second revision
- Two complex numbers are equal when real parts match and imaginary parts match: a+bi = c+di iff a=c and b=d.
- EQUALITY: If x + yi = 3 - 2i, then x = 3 and y = -2.
- ADD AND SUBTRACT: (a+bi) + (c+di) = (a+c) + (b+d)i.
- Worked-method reminder: (1 + 2i) - (3 + 5i) = -2 - 3i
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