Complex Number Operations And Expressions
#title: Complex Number Operations and Expressions
This article delves into complex number operations, specifically focusing on expressing the results in Cartesian form (a + bi, where a and b are real numbers). We will explore division, subtraction, and the use of conjugates to simplify complex expressions. Two examples will be provided to illustrate these concepts.
1. Expressing (z₂ / z₁) + [1 / (z₁ - z₂)] in Cartesian Form
In this section, we aim to express the complex expression (z₂ / z₁) + [1 / (z₁ - z₂)] in the standard Cartesian form. Given z₁ = 2 + 3i and z₂ = 4 - 4i, we will perform the necessary arithmetic operations to achieve this.
Step 1: Calculate z₂ / z₁
To divide complex numbers, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of z₁ = 2 + 3i is z₁̄ = 2 - 3i.
So, we have:
z₂ / z₁ = (4 - 4i) / (2 + 3i)
Multiply both numerator and denominator by the conjugate of z₁:
z₂ / z₁ = [(4 - 4i) * (2 - 3i)] / [(2 + 3i) * (2 - 3i)]
Expand the numerator:
(4 - 4i) * (2 - 3i) = 4 * 2 + 4 * (-3i) + (-4i) * 2 + (-4i) * (-3i)
= 8 - 12i - 8i + 12i²
Since i² = -1, we get:
= 8 - 20i - 12
= -4 - 20i
Expand the denominator. Multiplying a complex number by its conjugate results in the sum of the squares of the real and imaginary parts:
(2 + 3i) * (2 - 3i) = 2² + 3² = 4 + 9 = 13
Therefore:
z₂ / z₁ = (-4 - 20i) / 13
= -4/13 - (20/13)i
Step 2: Calculate z₁ - z₂
Subtract z₂ from z₁:
z₁ - z₂ = (2 + 3i) - (4 - 4i)
= 2 + 3i - 4 + 4i
= (2 - 4) + (3i + 4i)
= -2 + 7i
Step 3: Calculate 1 / (z₁ - z₂)
Now, we need to find the reciprocal of the complex number (-2 + 7i). Again, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of -2 + 7i is -2 - 7i.
1 / (z₁ - z₂) = 1 / (-2 + 7i)
Multiply both numerator and denominator by the conjugate of -2 + 7i:
1 / (z₁ - z₂) = (1 * (-2 - 7i)) / [(-2 + 7i) * (-2 - 7i)]
Expand the numerator:
1 * (-2 - 7i) = -2 - 7i
Expand the denominator:
(-2 + 7i) * (-2 - 7i) = (-2)² + 7² = 4 + 49 = 53
Therefore:
1 / (z₁ - z₂) = (-2 - 7i) / 53
= -2/53 - (7/53)i
Step 4: Calculate (z₂ / z₁) + [1 / (z₁ - z₂)]
Finally, we add the results from Step 1 and Step 3:
(z₂ / z₁) + [1 / (z₁ - z₂)] = (-4/13 - (20/13)i) + (-2/53 - (7/53)i)
To add these complex numbers, we add the real and imaginary parts separately. First, find a common denominator for the real parts (13 and 53), which is 13 * 53 = 689.
Real part: (-4/13) + (-2/53) = (-4 * 53) / 689 + (-2 * 13) / 689
= -212 / 689 - 26 / 689
= -238 / 689
Similarly, for the imaginary parts:
Imaginary part: (-20/13)i + (-7/53)i = [(-20 * 53) / 689]i + [(-7 * 13) / 689]i
= (-1060 / 689)i + (-91 / 689)i
= (-1151 / 689)i
Thus, (z₂ / z₁) + [1 / (z₁ - z₂)] = -238/689 - (1151/689)i
Therefore, the complex expression (z₂ / z₁) + [1 / (z₁ - z₂)] in Cartesian form is -238/689 - (1151/689)i.
2. Complex Number Operations with z₁ = -i and z₂ = 2 + i√3
In this example, we are given two complex numbers, z₁ = -i and z₂ = 2 + i√3. We will explore complex number operations, including squaring, finding conjugates, and evaluating complex fractions.
Part (a): Express z₁² and the Conjugate of z₂ in the Form a + bi
First, let's calculate z₁²:
z₁² = (-i)²
= (-1)² * i²
Since i² = -1, we have:
= 1 * (-1)
= -1
In the form a + bi, z₁² = -1 + 0i.
Next, let's find the conjugate of z₂. The conjugate of a complex number a + bi is a - bi. Therefore, the conjugate of z₂ = 2 + i√3 is:
z₂̄ = 2 - i√3
So, z₂̄ is already in the form a + bi.
In summary, z₁² = -1 + 0i and the conjugate of z₂ is 2 - i√3.
Part (b): Find W = (z₁²) / (z₂̄)
We are asked to find W = (z₁²) / (z₂̄). We already found z₁² = -1 and z₂̄ = 2 - i√3 in Part (a). Thus,
W = (-1) / (2 - i√3)
To express this in the form a + bi, we multiply the numerator and denominator by the conjugate of the denominator, which is 2 + i√3:
W = [(-1) * (2 + i√3)] / [(2 - i√3) * (2 + i√3)]
Expand the numerator:
(-1) * (2 + i√3) = -2 - i√3
Expand the denominator:
(2 - i√3) * (2 + i√3) = 2² + (√3)² = 4 + 3 = 7
Therefore:
W = (-2 - i√3) / 7
= -2/7 - (√3/7)i
Thus, W = (z₁²) / (z₂̄) = -2/7 - (√3/7)i.
Conclusion
This article has demonstrated how to perform various operations with complex numbers, including division, subtraction, squaring, and finding conjugates. We expressed the results in the standard Cartesian form (a + bi). By understanding these operations and using conjugates effectively, one can simplify complex expressions and solve problems involving complex numbers in various mathematical and engineering contexts. Mastering complex number manipulation is crucial for advanced topics in mathematics, physics, and electrical engineering. From finding the quotient and reciprocal of complex numbers to determining the real and imaginary parts of intricate expressions, the ability to operate with complex numbers is an invaluable skill.
#repair-input-keyword: Express (z₂ / z₁) + [1 / (z₁ - z₂)] in Cartesian form. Find W = (z₁²) / (z₂̄). Express z₁² and the conjugate of z₂ in the form a + bi. Given z₁ = 2 + 3i and z₂ = 4 - 4i, express (z₂ / z₁) + [1 / (z₁ - z₂)] in Cartesian form. Given z₁ = -i and z₂ = 2 + i√3.