This worksheet provides detailed solutions to common complex number operations. Understanding these operations is crucial for success in advanced mathematics, engineering, and physics. We'll cover addition, subtraction, multiplication, division, and finding the modulus and argument of complex numbers.
What are Complex Numbers?
Before diving into the answers, let's briefly review. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i² = -1). a is called the real part, and b is called the imaginary part.
Complex Number Operations: Solved Examples
Let's tackle some common operations with detailed explanations. Assume the following complex numbers for our examples:
- z₁ = 3 + 2i
- z₂ = 1 - i
1. Addition:
To add complex numbers, add their real parts and their imaginary parts separately.
z₁ + z₂ = (3 + 2i) + (1 - i) = (3 + 1) + (2 - 1)i = 4 + i
2. Subtraction:
Subtraction follows a similar pattern: subtract the real parts and the imaginary parts separately.
z₁ - z₂ = (3 + 2i) - (1 - i) = (3 - 1) + (2 - (-1))i = 2 + 3i
3. Multiplication:
Multiply complex numbers using the distributive property (FOIL method), remembering that i² = -1.
z₁ * z₂ = (3 + 2i)(1 - i) = 3(1) + 3(-i) + 2i(1) + 2i(-i) = 3 - 3i + 2i - 2i² = 3 - i - 2(-1) = 3 - i + 2 = 5 - i
4. Division:
Dividing complex numbers requires multiplying both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a + bi is a - bi.
z₁ / z₂ = (3 + 2i) / (1 - i)
First, find the complex conjugate of the denominator: 1 + i
Now multiply both numerator and denominator by the conjugate:
[(3 + 2i)(1 + i)] / [(1 - i)(1 + i)] = (3 + 3i + 2i + 2i²) / (1 + i - i - i²) = (3 + 5i - 2) / (1 + 1) = (1 + 5i) / 2 = ½ + (5/2)i
5. Modulus (Absolute Value):
The modulus of a complex number z = a + bi is its distance from the origin in the complex plane and is calculated as: |z| = √(a² + b²)
|z₁| = √(3² + 2²) = √(9 + 4) = √13
|z₂| = √(1² + (-1)²) = √(1 + 1) = √2
6. Argument (Angle):
The argument of a complex number is the angle it makes with the positive real axis in the complex plane. It's calculated using the arctangent function: arg(z) = arctan(b/a) Remember to consider the quadrant of the complex number.
arg(z₁) = arctan(2/3) ≈ 0.588 radians or ≈ 33.69° (z₁ lies in the first quadrant)
arg(z₂) = arctan(-1/1) = arctan(-1) = -π/4 radians or -45° (z₂ lies in the fourth quadrant)
Further Practice and Resources
This worksheet provides a foundation for understanding complex number operations. Further practice with more complex examples and different types of problems will solidify your understanding. Consider exploring online resources and textbooks for additional problems and explanations. Remember to always check your work carefully, paying close attention to the signs and the use of the imaginary unit, i. Mastering these operations is a crucial step in progressing through higher-level mathematics and related fields.
