Complex_Numbers_Chapter1 for Grade 11.pptx

complex numbers

1. Complex Numbers Chapter 1 – Mathematics Grade 11 Teaching Version with Notes

2. Introduction Complex numbers extend real numbers. Needed to solve equations like x² + 1 = 0. Imaginary unit introduced: i = √-1.

3. Definition A complex number is written as z = a + bi. a: real part, b: imaginary part. i = √-1.

4. Powers of i i² = -1 i³ = -i i⁴ = 1 Pattern repeats every 4 powers

5. Algebra of Complex Numbers Addition: (a+bi) + (c+di) = (a+c) + (b+d)i Subtraction: (a+bi) - (c+di) = (a-c) + (b-d)i Multiplication: (a+bi)(c+di) = (ac-bd) + (ad+bc)i Division: Multiply numerator & denominator by conjugate

6. Conjugate & Modulus Conjugate of z = a + bi is z̄ = a - bi Modulus: |z| = √(a² + b²) Useful for division and polar form

7. Argand Diagram Visual representation of complex numbers. x-axis: real part, y-axis: imaginary part. z = a + bi → point (a, b).

8. Polar Form z = r(cosθ + i sinθ) r = |z| = √(a² + b²) θ = argument = tan⁻¹(b/a)

9. Euler’s Form z = re^(iθ) e^(iθ) = cosθ + i sinθ Powerful in simplifying calculations

10. Solved Example Example: Simplify (2 + 3i)(1 - 4i). Solution: (2)(1) + (2)(-4i) + (3i)(1) + (3i)(-4i) = 2 - 8i + 3i -12i² = 2 - 5i + 12 = 14 - 5i

11. Practice Problems 1. Simplify (3 + 2i)(3 - 2i). 2. Find modulus and conjugate of z = -1 + i√3. 3. Express 1 + i√3 in polar form. 4. Compute (cos30° + i sin30°)^4.

12. Summary & Key Formulas z = a + bi, Re(z) = a, Im(z) = b i² = -1, i³ = -i, i⁴ = 1 Conjugate: z̄ = a - bi, |z| = √(a² + b²) Polar form: z = r(cosθ + i sinθ) Euler’s form: z = re^(iθ)