MathematicsChapter 1: Complex Numbers
Division of Complex Numbers
To divide, multiply numerator and denominator by the conjugate of the denominator so the bottom becomes real (a^2+b^2).
Learning objectives
- Explain division of complex numbers using the correct definitions, conditions, and notation.
- Apply multiply by conjugate of bottom, bottom becomes real correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
To divide, multiply numerator and denominator by the conjugate of the denominator so the bottom becomes real (a^2+b^2). Then split into real and imaginary parts. 1i = -i. Never leave i in the denominator of a finished answer.
Formula
i = -i. Never leave i in the denominator of
Where / symbols
- DIVISION OF COMPLEX NUMBERS
- To divide, multiply numerator and denominator by the conjugate of the denominator so the bottom becomes real (a^2+b^2). Then split into real and imaginary parts. 1/i = -i. Never leave i in the denominator of a finished answer.
- BY i OR BY A COMPLEX
- 1/i: multiply top and bottom by i, or use 1/i = i^3/i^4 = -i. For (1+i)/(1-i), multiply by (1+i)/(1+i).
- GENERAL a+bi OVER c+di
- Multiply by c-di. Denominator c^2+d^2. Numerator (a+bi)(c-di). Then divide each part by c^2+d^2.
Overview
Clear i from the bottom:
BY i OR BY A COMPLEX
1i: multiply top and bottom by i, or use 1i = i3i4 = -i. For 1+i1-i, multiply by 1+i1+i.
Examples 1/i = -i · (1+i)/(1-i) = i · 5/(2i) = -5i/2
GENERAL a+bi OVER c+di
Multiply by c-di. Denominator c^2+d^2. Numerator (a+bi)(c-di). Then divide each part by c^2+d^2.
Examples (3+4i)/(1+2i) = (11-2i)/5 · (2+3i)/i = 3-2i · Check by multiplying back
WHAT — DIVISION OF COMPLEX NUMBERS
To divide, multiply numerator and denominator by the conjugate of the denominator so the bottom becomes real (a^2+b^2). Then split into real and imaginary parts. 1i = -i. Never leave i in the denominator of a finished answer. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Division Of Complex Numbers 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 BY i OR BY A COMPLEX, then GENERAL a+bi OVER c+di. Worked: (1 + i)/(1 - i). multiply top and bottom by (1 + i) top: (1 + i)(1 + i) = (1 + i)2 = 1 + 2i + i2 = 2i bottom: (1 - i)(1 + i) = 1 - i2 = 1 - (-1) = 2 (1 + i)/(1 - i) = 2i / 2 = i Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- (1 + i)/(1 - i).
- multiply top and bottom by (1 + i)
- top: (1 + i)(1 + i) = (1 + i)2 = 1 + 2i + i2 = 2i
- bottom: (1 - i)(1 + i) = 1 - i2 = 1 - (-1) = 2
- (1 + i)/(1 - i) = 2i / 2
- = i
In simple words
If the denominator is 0+0i, division is undefined. A non-zero complex always has a conjugate that makes the bottom a^2+b^2 > 0.
Key points
- multiply by conjugate of bottom
- bottom becomes real
- conjugate of denominator
- (c^2+d^2) on the bottom
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Division of Complex Numbers
Sticky Notes · Quick Revision
Core Idea
- To divide, multiply numerator and denominator by the conjugate of the denominator so the bottom becomes real (a^2+b^2). Then split into real and imaginary parts. 1i = -i. Never leave i in the denominator of a finished answer.
BY i OR BY A COMPLEX
- 1i: multiply top and bottom by i, or use 1i = i3i4 = -i. For 1+i1-i, multiply by 1+i1+i.
GENERAL a+bi OVER c+di
- Multiply by c-di. Denominator c^2+d^2. Numerator (a+bi)(c-di). Then divide each part by c^2+d^2.
Worked Method
- Worked: (1 + i)/(1 - i).
- multiply top and bottom by (1 + i)
- top: (1 + i)(1 + i) = (1 + i)2 = 1 + 2i + i2 = 2i
- bottom: (1 - i)(1 + i) = 1 - i2 = 1 - (-1) = 2
- (1 + i)/(1 - i) = 2i / 2
- = i
Common Misconception
- If the denominator is 0+0i, division is undefined. A non-zero complex always has a conjugate that makes the bottom a^2+b^2 > 0.
FBISE Exam Check
- If the denominator is 0+0i, division is undefined. A non-zero complex always has a conjugate that makes the bottom a^2+b^2 > 0.
60-second revision
- To divide, multiply numerator and denominator by the conjugate of the denominator so the bottom becomes real (a^2+b^2).
- BY i OR BY A COMPLEX: 1i: multiply top and bottom by i, or use 1i = i3i4 = -i.
- GENERAL a+bi OVER c+di: Multiply by c-di.
- Worked-method reminder: = i
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