Chapter 4.1: Introduction to Complex Numbers
- Imaginary Unit (i): Defined as √(-1).
- Complex Number: A number of the form a + bi, where a and b are real numbers and i is the imaginary unit.
- Real and Imaginary Parts: The real part of a + bi is a, and the imaginary part is b.
- Equality of Complex Numbers: Two complex numbers are equal if their real and imaginary parts are equal.
Chapter 4.2: Algebraic Operations with Complex Numbers
- Addition and Subtraction: Add or subtract the corresponding real and imaginary parts.
- Multiplication
- Division
Chapter 4.3: Polar Representation of a Complex Number
- Modulus of a Complex Number: The distance of the complex number from the origin in the complex plane.
- Argument of a Complex Number
- Polar Form: A complex number can be represented as r(cos θ + i sin θ), where r is the modulus and θ is the argument.
Chapter 4.4: Quadratic Equations
- Quadratic Equation: An equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
- Roots of a Quadratic Equation: The values of x that satisfy the equation.
- Discriminant (Δ): Δ = b² – 4ac
- If Δ > 0, the equation has two distinct real roots.
- If Δ = 0, the equation has one real root (repeated).
- If Δ < 0, the equation has no real roots (but two complex conjugate roots).
- Quadratic Formula: x = (-b ± √Δ) / (2a)
Key Concepts:
- Complex numbers and their operations
- Polar representation of complex numbers
- Quadratic equations, their roots, and the discriminant
- The relationship between quadratic equations and complex numbers
Exercise 4.1
Express each of the complex number given in the Exercises 1 to 10 in the form
a + ib.
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2.
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3. i-39
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4. 3(7 + i7) + i (7 + i7)
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5. (1 – i) – ( –1 + i6)
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(1 – i) – ( –1 + i6) = 1 – i + 1 – 6i
= 2 – 7i
6.
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7.
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8. (1-i)4
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9.
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10.
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Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.
11. 4 – 3i
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12.
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13. – i
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14. Express the following expression in the form of a + ib :
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