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COMPLEX NUMBERS a + bi
A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Every real is complex: a = a + 0i, so R is inside C. Pure imaginary: real part 0 and b not 0, e.g. 7i. sqrt(-9) = 3i in this course.
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COMPLEX NUMBERS a + bi
What is it?
A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Every real is complex: a = a + 0i, so R is inside C. Pure imaginary: real part 0 and b not 0, e.g. 7i. sqrt(-9) = 3i in this course.
In simple words
- A complex number is z = a + bi with real a (real part) and real b (imaginary part).
- REAL, IMAGINARY, BOTH: 5 + 0i is real.
- SQUARE ROOTS OF NEGATIVES: sqrt(-4) = 2i.
- Worked-method reminder: so 5 + 0i is real
Important terms
COMPLEX NUMBERS a + bi: A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Every real is complex: a = a + 0i, so R is inside C. Pure imaginary: real part 0 and b not 0, e.g. 7i. sqrt(-9) = 3i in this course.
REAL, IMAGINARY, BOTH: 5 + 0i is real. 0 + 7i is pure imaginary. 3 + 2i is non-real complex. 0 = 0 + 0i is both real and (trivially) a complex number.
SQUARE ROOTS OF NEGATIVES: sqrt(-4) = 2i. sqrt(-25) = 5i. 2 sqrt(-3) = 2 sqrt(3) i. Write the i at the end so you do not mix it with the radicand.
What you should learn
- Explain complex numbers a + bi using the correct definitions, conditions, and notation.
- Apply z = a + bi, R inside C correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- -4 is real
- 0 - 6i is pure imaginary
- 4 + 3i is not real
REAL, IMAGINARY, BOTH
5 + 0i is real. 0 + 7i is pure imaginary. 3 + 2i is non-real complex. 0 = 0 + 0i is both real and (trivially) a complex number.
SQUARE ROOTS OF NEGATIVES
sqrt(-4) = 2i. sqrt(-25) = 5i. 2 sqrt(-3) = 2 sqrt(3) i. Write the i at the end so you do not mix it with the radicand.
Recommended for Class 10
Full school-level explanation
A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Every real is complex: a = a + 0i, so R is inside C. Pure imaginary: real part 0 and b not 0, e.g. 7i. sqrt(-9) = 3i in this course.
Name the parts:
WHAT — COMPLEX NUMBERS a + bi
A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Every real is complex: a = a + 0i, so R is inside C. Pure imaginary: real part 0 and b not 0, e.g. 7i. sqrt(-9) = 3i in this course. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Complex Numbers A + Bi 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 REAL, IMAGINARY, BOTH, then SQUARE ROOTS OF NEGATIVES. Worked: classify 0 - 6i and 5 + 0i. 0 - 6i has real part 0, imaginary part -6 so 0 - 6i is pure imaginary 5 + 0i has imaginary part 0 so 5 + 0i is real Finish by checking signs, restrictions, units, and whether the answer matches the question.
REAL, IMAGINARY, BOTH
5 + 0i is real. 0 + 7i is pure imaginary. 3 + 2i is non-real complex. 0 = 0 + 0i is both real and (trivially) a complex number.
Remember
- a and b
Examples
- -4 is real
- 0 - 6i is pure imaginary
- 4 + 3i is not real
SQUARE ROOTS OF NEGATIVES
sqrt(-4) = 2i. sqrt(-25) = 5i. 2 sqrt(-3) = 2 sqrt(3) i. Write the i at the end so you do not mix it with the radicand.
Remember
- sqrt(-k) = sqrt(k) i
Examples
- sqrt(-9) = 3i
- sqrt(-1) = i
- Not -3 for sqrt(-9)
Remember
- z = a + bi
- R inside C
- pure imaginary vs real
- reals are complex
- a and b
- sqrt(-k) = sqrt(k) i
- z = a + bi
- R inside C
How the ideas connect
Worked: classify 0 - 6i and 5 + 0i. 0 - 6i has real part 0, imaginary part -6 so 0 - 6i is pure imaginary 5 + 0i has imaginary part 0 so 5 + 0i is real
Deeper understanding
Go beyond memorising COMPLEX NUMBERS a + bi. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: classify 0 - 6i and 5 + 0i. 0 - 6i has real part 0, imaginary part -6 so 0 - 6i is pure imaginary 5 + 0i has imaginary part 0 so 5 + 0i is real
Common mistakes
Exam focus
Core Idea
REAL, IMAGINARY, BOTH
SQUARE ROOTS OF NEGATIVES
Worked Method
Exam focus
Quick revision
- z = a + bi
- R inside C
- pure imaginary vs real
- reals are complex
- COMPLEX NUMBERS a + bi: A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Ev
- REAL, IMAGINARY, BOTH: 5 + 0i is real. 0 + 7i is pure imaginary. 3 + 2i is non-real complex. 0 = 0 + 0i is both real and (t
Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
For long answers on COMPLEX NUMBERS a + bi: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- z = a + bi
- R inside C
- pure imaginary vs real
- reals are complex
Important formulas
- z = a + bi
- R inside C
- pure imaginary vs real
Common exam traps
- Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
- Not -3 for sqrt(-9)
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
- Mistake
Not -3 for sqrt(-9)
- Mistake
Do not treat REAL, IMAGINARY, BOTH and SQUARE ROOTS OF NEGATIVES as the same step; each has a different purpose.
- Mistake
Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
z = a + bi
R inside C
pure imaginary vs real
reals are complex
A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Every real is complex: a = a + 0i, so R is inside C. Pure imag
z = a + bi
- Definition
- A complex number is z = a + bi with real a (real part) and real b (imaginary part). The set is C. Every real is complex: a = a + 0i, so R is inside C. Pure imag
- Formula
- z = a + bi
- Example
- -4 is real
- Critical warning
- Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
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Exam Practice
Define
Q1. Define COMPLEX NUMBERS a + bi.
Explain
Q2. Explain: REAL, IMAGINARY, BOTH.
Example
Q3. Give one clear example of COMPLEX NUMBERS a + bi and say why it fits.
Mistake
Q4. State one common mistake students make about COMPLEX NUMBERS a + bi.
Exam note
Q5. Write a short exam note on COMPLEX NUMBERS a + bi: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.