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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.
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Division of Complex Numbers
What is it?
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.
In simple words
- 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
Important terms
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.
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.
What you should learn
- 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.
Examples
- 1i = -i
- 1+i1-i = i
- 52i = -5i/2
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.
Recommended for Class 10
Full school-level explanation
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.
Clear i from the bottom:
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 + i1 - i. multiply top and bottom by (1 + i) top: (1 + i)(1 + i) = (1 + i)^2 = 1 + 2i + i^2 = 2i bottom: (1 - i)(1 + i) = 1 - i^2 = 1 - (-1) = 2 1 + i1 - i = 2i / 2 = i Finish by checking signs, restrictions, units, and whether the answer matches the question.
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.
Remember
- conjugate of denominator
Examples
- 1i = -i
- 1+i1-i = i
- 52i = -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.
Remember
- (c^2+d^2) on the bottom
Examples
- 3+4i1+2i = 11-2i5
- 2+3ii = 3-2i
- Check by multiplying back
Remember
- multiply by conjugate of bottom
- bottom becomes real
- i
- denominator not zero
- conjugate of denominator
- (c^2+d^2) on the bottom
- multiply by conjugate of bottom
- bottom becomes real
How the ideas connect
Worked: 1 + i1 - i. multiply top and bottom by (1 + i) top: (1 + i)(1 + i) = (1 + i)^2 = 1 + 2i + i^2 = 2i bottom: (1 - i)(1 + i) = 1 - i^2 = 1 - (-1) = 2 1 + i1 - i = 2i / 2 = i
Deeper understanding
Go beyond memorising Division of Complex Numbers. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: 1 + i1 - i. multiply top and bottom by (1 + i) top: (1 + i)(1 + i) = (1 + i)^2 = 1 + 2i + i^2 = 2i bottom: (1 - i)(1 + i) = 1 - i^2 = 1 - (-1) = 2 1 + i1 - i = 2i / 2 = i
Common mistakes
Exam focus
Core Idea
BY i OR BY A COMPLEX
GENERAL a+bi OVER c+di
Worked Method
Exam focus
Quick revision
- multiply by conjugate of bottom
- bottom becomes real
- i
- denominator not zero
- DIVISION OF COMPLEX NUMBERS: To divide, multiply numerator and denominator by the conjugate of the denominator so the bottom beco
- 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+i)/(1+
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.
For long answers on Division of Complex Numbers: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- multiply by conjugate of bottom
- bottom becomes real
- i
- denominator not zero
Important formulas
- multiply by conjugate of bottom
- bottom becomes real
- i
Common exam traps
- 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.
- Do not treat BY i OR BY A COMPLEX and GENERAL a+bi OVER c+di as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
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.
- Mistake
Do not treat BY i OR BY A COMPLEX and GENERAL a+bi OVER c+di as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
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.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
multiply by conjugate of bottom
bottom becomes real
i
denominator not zero
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
- 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
- Example
- 1i = -i
- Critical warning
- If the denominator is 0+0i, division is undefined. A non-zero complex always has a conjugate that makes the bottom a^2+b
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Exam Practice
Define
Q1. Define Division of Complex Numbers.
Explain
Q2. Explain: BY i OR BY A COMPLEX.
Example
Q3. Give one clear example of Division of Complex Numbers and say why it fits.
Mistake
Q4. State one common mistake students make about Division of Complex Numbers.
Exam note
Q5. Write a short exam note on Division of Complex Numbers: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.