MathematicsChapter 1: Complex Numbers
Complex Numbers Recalled
A complex number is z = x + i y, with real x (real part) and real y (imaginary part).
Learning objectives
- State the main ideas and notation used in complex numbers recalled.
- Apply the rules of complex numbers recalled in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
A complex number is z = x + i y, with real x (real part) and real y (imaginary part). i is the unit with i^2 = -1. The set is C. Two complex numbers are equal when real parts match and imaginary parts match. Add and subtract like with like. Meaning: one complex equation is two real equations. Grade 11 goes on to modulus, polar form, and polar arithmetic.
Formula
z = x + i y
Where / symbols
- COMPLEX NUMBERS RECALLED
- A complex number is z = x + i y, with real x (real part) and real y (imaginary part). i is the unit with i^2 = -1. The set is C. Two complex numbers are equal when real parts match and imaginary parts match. Add and subtract like with like. Meaning: one complex equation is two real equations. Grade 11 goes on to modulus, polar form, and polar arithmetic.
- 1. NAME THE PARTS
- Re(z) = x. Im(z) = y. Pure imaginary: x = 0, y != 0. Real: y = 0. 0 = 0 + 0i is still in C. Powers of i cycle every 4 (i, -1, -i, 1).
- 2. ADD, SUBTRACT, MULTIPLY
- Add: (a+bi)+(c+di)=(a+c)+(b+d)i. Multiply: (a+bi)(c+di) = ac + adi + bci + bdi^2, then i^2 = -1. Foil, then tidy.
Relationship
- z = x + i y
Overview
Parts, then the four operations start:
Name the Parts
Re(z) = x. Im(z) = y. Pure imaginary: x = 0, y != 0. Real: y = 0. 0 = 0 + 0i is still in C. Powers of i cycle every 4 (i, -1, -i, 1).
Examples 3 + 2i · 7i pure imaginary · i^2 = -1
Add, Subtract, Multiply
Add: (a+bi)+(c+di)=(a+c)+(b+d)i. Multiply: (a+bi)(c+di) = ac + adi + bci + bdi^2, then i^2 = -1. Foil, then tidy.
Examples (1+2i)+(3-i)=4+i · (2+i)(3-i)=7+i · i * i = -1
WHAT — COMPLEX NUMBERS RECALLED
A complex number is z = x + i y, with real x (real part) and real y (imaginary part). i is the unit with i^2 = -1. The set is C. Two complex numbers are equal when real parts match and imaginary parts match. Add and subtract like with like. Meaning: one complex equation is two real equations. Grade 11 goes on to modulus, polar form, and polar arithmetic.
RULE — 1. NAME THE PARTS
Re(z) = x. Im(z) = y. Pure imaginary: x = 0, y != 0. Real: y = 0. 0 = 0 + 0i is still in C. Powers of i cycle every 4 (i, -1, -i, 1).
METHOD — 2. ADD, SUBTRACT, MULTIPLY
Add: (a+bi)+(c+di)=(a+c)+(b+d)i. Multiply: (a+bi)(c+di) = ac + adi + bci + bdi^2, then i^2 = -1. Foil, then tidy.
WORKED EXAMPLE
Worked: (2 + 3i)(1 - i). 2*1 + 2*(-i) + 3i*1 + 3i*(-i) = 2 - 2i + 3i - 3 i2 i2 = -1 so -3 i2 = 3 = 5 + i
How to answer a complex numbers recalled question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- (2 + 3i)(1 - i).
- 21 + 2(-i) + 3i1 + 3i(-i)
- = 2 - 2i + 3i - 3 i2
- i2 = -1 so -3 i2 = 3
- = 5 + i
In simple words
Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
Key points
- z = x + i y
- equal <-> both parts equal
- Re and Im
- like with like, then foil
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Complex Numbers Recalled
Sticky Notes · Quick Revision
Core Idea
- A complex number is z = x + i y, with real x (real part) and real y (imaginary part).
- i is the unit with i2 = -1.
- The set is C.
1. NAME THE PARTS
- Re(z) = x.
- Im(z) = y.
- Pure imaginary: x = 0, y != 0.
- Real: y = 0.
2. ADD, SUBTRACT, MULTIPLY
- Add: (a+bi)+(c+di)=(a+c)+(b+d)i.
- Multiply: (a+bi)(c+di) = ac + adi + bci + bdi2, then i2 = -1.
- Foil, then tidy.
Worked Example
- Worked: (2 + 3i)(1 - i).
- 21 + 2(-i) + 3i1 + 3i(-i)
- = 2 - 2i + 3i - 3 i2
- i2 = -1 so -3 i2 = 3
- = 5 + i
Common Misconception
- Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
FBISE Exam Check
- State the rule for complex numbers recalled before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- A complex number is z = x + i y, with real x (real part) and real y (imaginary part).
- i is the unit with i2 = -1.
- The set is C.
- Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
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