Complex Numbers Tutorial Notes

Finding Roots of Equations

  • Q1: Find the roots of the following equations:-

    • (a) For the equation x² + 2x + 1 = 0:

      • Using the quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = 2, c = 1:

      • Discriminant: (2)² - 4(1)(1) = 0

      • Roots: −1 ± i, which simplifies to −1 + i, −1 - i.

    • (b) For the equation x² + 2x + 3 = 0:

      • Again using the quadratic formula:

      • Here, a = 1, b = 2, c = 3:

      • Discriminant: (2)² - 4(1)(3) = -8, indicating complex roots.

      • Roots: −2, 1 ± i√3.

Expressions in Form x + yi

  • Q2: Express the following expressions:-

    • (a) 1/(2 - 3 - 4i) \ 5/(−8i):

      • Multiply by the conjugate to rationalize the denominator:

      • Result: −5/17 - (4/17)i.

    • (b) 1/(5 - 3i) - 1/(5 + 3i):

      • Combine fractions over a common denominator:

      • Result: 3/17 i.

Simultaneous Equations

  • Q3: Solve for complex numbers w and z:- Equations:

    1. 4z + 3w = 23

    2. z + iw = 6 + 8i

    • To solve, substitute one equation into the other and isolate variables:

      • Result:

      • w = 5 - 4i and z = 2 + 3i.

Finding z!

  • Q4: Find z! in the form x + yi:- Given: [ \frac{1}{z!} = \frac{1}{z"} + \frac{1}{(z"z#)} ] with z" = 3 - 4i and z# = 5 + 2i

    • Find a common denominator:

      • Result: 11/4 - (13/4)i.

Argand Diagram

  • Q5: Plot points and express in polar form:-

    • (a) 4 + 3i:

      • Modulus: √(4² + 3²) = 5, Argument: tan⁻¹(3/4).

    • (b) −3 + 5i:

      • Modulus: √34, Argument: π - tan⁻¹(5/3).

    • (c) −3 – 2i:

      • Modulus: √13, Argument: tan⁻¹(-2/-3) - π.

    • (d) 7 – 5i:

      • Modulus: √74, Argument: -tan⁻¹(5/7).

Evaluating Complex Products and Powers

  • Q6: Evaluate products and express in polar form:-

    • Given z" = 2(8 cos(3π/4) + isin(3π/4)) and z# = 8(cos(π/6) + isin(π/6):

      • Use polar multiplication:

      • Evaluate z"z# = r"r# (cos(θ" + θ#) + isin(θ" + θ#)).

General Polar Forms

  • Q7: Find general polar forms:-

    • (a) For (8 + 8i)²:

      • Use polar conversion, then square:

      • Result: 2 ∠ θ + kπ, k = 0, 1, 2.

    • (b) For (2 + 2i)^(3/2):

      • Apply polar exponents:

      • Result: 4, ∠ θ + kπ, k = 0, 1, 2.

General Exponential Form

  • Q8: Find (5 - 3i)^(3/5) in general exponential form:-

    • Convert to polar form then apply exponential rule:

      • Result: 34^(1/5)e^(iθ).

Solving Quadratic Equations

  • Q9: Solve quadratic equation z² - (3 + 5i)z + (8i - 5) = 0:-

    • Use the quadratic formula:

      • Solutions:

      • z = (3 + 5i) ± √(1 - 4(8i - 5)) / 2 .

      • Result includes complex solutions.

Finding Roots of Equations

When we want to find the values of x that make an equation true, especially quadratic equations of the form ax² + bx + c = 0, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a.
1. Example: For x² + 2x + 1 = 0:

  • Here, a = 1, b = 2, c = 1.

  • Calculate the discriminant: (2)² - 4(1)(1) = 0.

  • This tells us we have one unique root, leading to x = -1 + i and x = -1 - i (complex numbers).

2. Example: For x² + 2x + 3 = 0:

  • Here, a = 1, b = 2, c = 3.

  • The discriminant is (2)² - 4(1)(3) = -8, indicating we have complex roots since it is negative.

  • The roots are expressed as -2, 1 ± i√3 (which means there are two answers).

Expressions in Form x + yi

This refers to complex numbers which can be written as x + yi, where x is the real part and yi is the imaginary part.
1. Example: To simplify 1/(2 - 3 - 4i):

  • We multiply by the conjugate (2 + 3 + 4i) to eliminate i from the denominator.

  • The result simplifies to −5/17 - (4/17)i.

2. Example: For 1/(5 - 3i) - 1/(5 + 3i):

  • We combine to have a common denominator leading to a simplified form of 3/17 i.

Simultaneous Equations

Here, we solve for two variables (like w and z) using equations that involve them both:
1. Equations:

  • 4z + 3w = 23

  • z + iw = 6 + 8i

  • We can substitute one equation into the other.

  • Solving, we find w = 5 - 4i and z = 2 + 3i (showing they are complex).

Finding z!

This section addresses complex factorials: Given
ext{[} rac{1}{z!} = rac{1}{z"} + rac{1}{(z"z#)} ext{]}
If z" = 3 - 4i and z# = 5 + 2i, we find a common denominator and get 11/4 - (13/4)i.

Argand Diagram

This diagram helps visualize complex numbers on a coordinate system:
1. Example: For 4 + 3i:

  • The modulus is calculated by √(4² + 3²) = 5 (distance from the origin) and the argument is tan⁻¹(3/4) (angle).
    2. More examples: Include numbers like −3 + 5i where the modulus would be √34.

Evaluating Complex Products and Powers

When multiplying complex numbers in polar form, like z" = 2(8 cos(3π/4) + isin(3π/4)) and z# = 8(cos(π/6) + isin(π/6), you use polar multiplication: z"z# = r"r# (cos(θ" + θ#) + isin(θ" + θ#)).

General Polar Forms

You can convert complex numbers into polar forms (r, θ) and raise them to powers:
1. Example: For (8 + 8i)², convert to polar and square, resulting in 2 ∠ θ + kπ.
2. For (2 + 2i)^(3/2), it results in 4, ∠ θ + kπ (k = 0, 1, 2): values can be added based on so-called arguments.

General Exponential Form

This involves expressing complex numbers with an exponent: For (5 - 3i)^(3/5), convert to polar form, resulting in 34^(1/5)e^(iθ), which represents a complex number in exponential growth.

Solving Quadratic Equations

For z² - (3 + 5i)z + (8i - 5) = 0: use the quadratic formula again. This formula helps us find roots.
In this case, you'd set it into the quadratic formula leading to results indicated by the discriminant, revealing complex solutions.