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Multiplication of Complex Numbers
Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A number times its conjugate is a real square sum: (a+bi)(a-bi) = a^2 + b^2.
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Multiplication of Complex Numbers
What is it?
Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A number times its conjugate is a real square sum: (a+bi)(a-bi) = a^2 + b^2.
In simple words
- Use FOIL (or the distributive law) and replace i^2 by -1.
- TWO COMPLEX FACTORS: (2+3i)(1+i) = 2*1 + 2*i + 3i*1 + 3i*i = 2 + 2i + 3i + 3i2 = 2 + 5i - 3 = -1 + 5i.
- SQUARES AND CONJUGATES: (1+i)2 = 1 + 2i + i2 = 2i.
- Worked-method reminder: = -1 + 5i
Important terms
MULTIPLICATION OF COMPLEX NUMBERS: Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A number times its conjugate is a real square sum: (a+bi)(a-bi) = a^2 + b^2.
TWO COMPLEX FACTORS: (2+3i)(1+i) = 21 + 2i + 3i1 + 3ii = 2 + 2i + 3i + 3i^2 = 2 + 5i - 3 = -1 + 5i. Collect real leftover after i^2, then i terms.
SQUARES AND CONJUGATES: (1+i)^2 = 1 + 2i + i^2 = 2i. Product with conjugate: (3+2i)(3-2i) = 9 - (2i)^2 = 9 - 4(-1) = 13.
What you should learn
- Explain multiplication of complex numbers using the correct definitions, conditions, and notation.
- Apply i^2 -> -1, ac - bd correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- (2i)(3i)=6i2=-6
- (1+i)(1-i)=2
- i(4+2i)=-2+4i
TWO COMPLEX FACTORS
(2+3i)(1+i) = 21 + 2i + 3i1 + 3ii = 2 + 2i + 3i + 3i^2 = 2 + 5i - 3 = -1 + 5i. Collect real leftover after i^2, then i terms.
SQUARES AND CONJUGATES
(1+i)^2 = 1 + 2i + i^2 = 2i. Product with conjugate: (3+2i)(3-2i) = 9 - (2i)^2 = 9 - 4(-1) = 13.
Recommended for Class 10
Full school-level explanation
Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A number times its conjugate is a real square sum: (a+bi)(a-bi) = a^2 + b^2.
Expand, then tidy i^2:
WHAT — MULTIPLICATION OF COMPLEX NUMBERS
Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A number times its conjugate is a real square sum: (a+bi)(a-bi) = a^2 + b^2. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Multiplication 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 TWO COMPLEX FACTORS, then SQUARES AND CONJUGATES. Worked: (2 + 3i)(1 + i). (2 + 3i)(1 + i) = 2(1) + 2(i) + 3i(1) + 3i(i) = 2 + 2i + 3i + 3i^2 i^2 = -1 so 3i^2 = -3 = (2 - 3) + (2 + 3)i = -1 + 5i Finish by checking signs, restrictions, units, and whether the answer matches the question.
TWO COMPLEX FACTORS
(2+3i)(1+i) = 21 + 2i + 3i1 + 3ii = 2 + 2i + 3i + 3i^2 = 2 + 5i - 3 = -1 + 5i. Collect real leftover after i^2, then i terms.
Remember
- FOIL then i^2
Examples
- (2i)(3i)=6i2=-6
- (1+i)(1-i)=2
- i(4+2i)=-2+4i
SQUARES AND CONJUGATES
(1+i)^2 = 1 + 2i + i^2 = 2i. Product with conjugate: (3+2i)(3-2i) = 9 - (2i)^2 = 9 - 4(-1) = 13.
Remember
- conjugate product is real
Examples
- (1+i)2 = 2i
- (3+2i)(3-2i)=13
- Not 9+4i
Remember
- i^2 -> -1
- ac - bd
- -1 + 5i
- ac - bd
- FOIL then i^2
- conjugate product is real
- i^2 -> -1
- ac - bd
How the ideas connect
Worked: (2 + 3i)(1 + i). (2 + 3i)(1 + i) = 2(1) + 2(i) + 3i(1) + 3i(i) = 2 + 2i + 3i + 3i2 i2 = -1 so 3i2 = -3 = (2 - 3) + (2 + 3)i = -1 + 5i
Deeper understanding
Go beyond memorising Multiplication of Complex Numbers. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: (2 + 3i)(1 + i). (2 + 3i)(1 + i) = 2(1) + 2(i) + 3i(1) + 3i(i) = 2 + 2i + 3i + 3i2 i2 = -1 so 3i2 = -3 = (2 - 3) + (2 + 3)i = -1 + 5i
Common mistakes
Exam focus
Core Idea
TWO COMPLEX FACTORS
SQUARES AND CONJUGATES
Worked Method
Exam focus
Quick revision
- i^2 -> -1
- ac - bd
- -1 + 5i
- MULTIPLICATION OF COMPLEX NUMBERS: Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A n
- TWO COMPLEX FACTORS: (2+3i)(1+i) = 21 + 2i + 3i1 + 3ii = 2 + 2i + 3i + 3i^2 = 2 + 5i - 3 = -1 + 5i. Collect real left
- SQUARES AND CONJUGATES: (1+i)^2 = 1 + 2i + i^2 = 2i. Product with conjugate: (3+2i)(3-2i) = 9 - (2i)^2 = 9 - 4(-1) = 13.
The real part is ac - bd, not ac + bd. The minus comes from i^2 = -1.
For long answers on Multiplication of Complex Numbers: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- i^2 -> -1
- ac - bd
- -1 + 5i
- MULTIPLICATION OF COMPLEX NUMBERS: Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A n
Important formulas
- i^2 -> -1
- ac - bd
- -1 + 5i
Common exam traps
- The real part is ac - bd, not ac + bd. The minus comes from i^2 = -1.
- Not 9+4i
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The real part is ac - bd, not ac + bd. The minus comes from i^2 = -1.
- Mistake
Not 9+4i
- Mistake
Do not treat TWO COMPLEX FACTORS and SQUARES AND CONJUGATES as the same step; each has a different purpose.
- Mistake
The real part is ac - bd, not ac + bd. The minus comes from i^2 = -1.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
i^2 -> -1
ac - bd
-1 + 5i
MULTIPLICATION OF COMPLEX NUMBERS: Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A n
Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A number times its conjugate is a real square sum: (a+bi)(a-bi)
- Definition
- Use FOIL (or the distributive law) and replace i^2 by -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i. A number times its conjugate is a real square sum: (a+bi)(a-bi)
- Example
- (2i)(3i)=6i2=-6
- Critical warning
- The real part is ac - bd, not ac + bd. The minus comes from i^2 = -1.
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Exam Practice
Define
Q1. Define Multiplication of Complex Numbers.
Explain
Q2. Explain: TWO COMPLEX FACTORS.
Example
Q3. Give one clear example of Multiplication of Complex Numbers and say why it fits.
Mistake
Q4. State one common mistake students make about Multiplication of Complex Numbers.
Exam note
Q5. Write a short exam note on Multiplication of Complex Numbers: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.