MathematicsChapter 1: Complex Numbers
COMPLEX NUMBERS a + bi
A complex number is z = a + bi with real a (real part) and real b (imaginary part).
Learning objectives
- 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.
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 imaginary: real part 0 and b not 0, e.g. 7i. sqrt(-9) = 3i in this course.
Mathematical form
z = a + bi with real a (real part) and real
a = a + 0i
Where / symbols
- 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.
Relationship
- z = a + bi
Overview
Name the parts:
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.
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.
Examples sqrt(-9) = 3i · sqrt(-1) = i · Not -3 for sqrt(-9)
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.
Worked example
- 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
In simple words
Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
Key points
- z = a + bi
- R inside C
- a and b
- sqrt(-k) = sqrt(k) i
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.COMPLEX NUMBERS a + bi
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- 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 Misconception
- Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
FBISE Exam Check
- Do not say a complex number is 'not real' if b = 0. Reals are complex numbers with b = 0.
60-second revision
- 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
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