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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.
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Complex Numbers Recalled
What is it?
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.
In simple words
- 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.
Important terms
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.
What you should learn
- 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.
Examples
- 3 + 2i
- 7i pure imaginary
- i2 = -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).
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.
Recommended for Class 11
Full school-level explanation
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.
Parts, then the four operations start:
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
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).
Remember
- Re and Im
Examples
- 3 + 2i
- 7i pure imaginary
- i2 = -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.
Remember
- like with like, then foil
Examples
- (1+2i)+(3-i)=4+i
- (2+i)(3-i)=7+i
- i * i = -1
How to answer a complex numbers recalled question
How it works
- 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.
Remember
- z = x + i y
- equal <-> both parts equal
- 5 + i
- R inside C
- Re and Im
- like with like, then foil
- z = x + i y
- equal <-> both parts equal
How the ideas connect
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
Deeper understanding
Go beyond memorising Complex Numbers Recalled. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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
Common mistakes
Exam focus
Core Idea
1. NAME THE PARTS
2. ADD, SUBTRACT, MULTIPLY
Worked Example
Exam focus
Quick revision
- z = x + i y
- equal <-> both parts equal
- 5 + i
- R inside C
- 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
- 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 o
Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
For long answers on Complex Numbers Recalled: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- z = x + i y
- equal <-> both parts equal
- 5 + i
- R inside C
Important formulas
- z = x + i y
- equal <-> both parts equal
- 5 + i
Common exam traps
- Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
z = x + i y
equal <-> both parts equal
5 + i
R inside C
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 w
z = x + i y
- 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 w
- Formula
- z = x + i y
- Example
- 3 + 2i
- Critical warning
- Reals are complex numbers with Im = 0. Do not say 'not complex' for 5.
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Exam Practice
Define
Q1. Define Complex Numbers Recalled.
Explain
Q2. Explain: NAME THE PARTS.
Example
Q3. Give one clear example of Complex Numbers Recalled and say why it fits.
Mistake
Q4. State one common mistake students make about Complex Numbers Recalled.
Exam note
Q5. Write a short exam note on Complex Numbers Recalled: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.