Problem 51 - Precalculus with Calculus Previews

Question

Let \(z_{1}=a+b i\) and \(z_{2}=c+d i\). Show that $$ \overline{z_{1}+z_{2}}=\overline{z_{1}}+\overline{z_{2}} \text { and } \overline{z_{1}^{2}}=\bar{z}_{1}^{2} $$.

Step-by-Step Solution

Answer

Both identities are proven using properties of complex conjugates and algebra.

Step 1: Understand the notation

The notation \( \overline{z} \) represents the complex conjugate of \( z \). For a complex number \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \).

Step 2: Find \( \overline{z_1 + z_2} \)

Calculate \( z_1 + z_2 = (a + bi) + (c + di) = (a + c) + (b + d)i \). Then the conjugate is \( \overline{z_1 + z_2} = (a + c) - (b + d)i \).

Step 3: Find \( \overline{z_1} + \overline{z_2} \)

Calculate \( \overline{z_1} = a - bi \) and \( \overline{z_2} = c - di \). Thus, \( \overline{z_1} + \overline{z_2} = (a-c) + (b-d)i \).

Step 4: Prove \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \)

From Step 2, \( \overline{z_1 + z_2} = (a + c) - (b + d)i \). From Step 3, \( \overline{z_1} + \overline{z_2} = (a + c) - (b + d)i \). These expressions are equal, proving \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \).

Step 5: Find \( z_1^2 \)

Calculate \( z_1^2 = (a + bi)^2 = a^2 + 2abi - b^2 = (a^2 - b^2) + 2abi \).

Step 6: Compute \( \overline{z_1^2} \)

The conjugate is \( \overline{z_1^2} = (a^2 - b^2) - 2abi \).

Step 7: Compute \( \overline{z_1}^2 \)

Calculate \( \overline{z_1} = a - bi \). Then, \( \overline{z_1}^2 = (a - bi)^2 = a^2 - 2abi - b^2 = (a^2 - b^2) - 2abi \).

Step 8: Prove \( \overline{z_1^2} = \overline{z_1}^2 \)

Both \( \overline{z_1^2} \) from Step 6 and \( \overline{z_1}^2 \) from Step 7 are equal, showing that \( \overline{z_1^2} = \overline{z_1}^2 \), thus proving the identity.

Key Concepts

Complex Conjugate

The complex conjugate of a complex number plays a crucial role in understanding complex numbers. If you have a complex number, say \( z = a + bi \), its complex conjugate is noted as \( \overline{z} = a - bi \). Essentially, the complex conjugate changes the sign of the imaginary part while keeping the real part unchanged.

Understanding the concept of complex conjugates is pivotal because it helps in operations like division of complex numbers and finding the modulus. The conjugate is often used in calculations to simplify expressions in complex analysis. For instance, when you multiply a complex number by its conjugate, you end up with a real number, specifically \( a^2 + b^2 \), which is useful in many ways, such as simplifying fractions in complex number division.

Addition of Complex Numbers

Adding complex numbers is straightforward and mirrors adding real numbers but with separate consideration for real and imaginary parts. When you have two complex numbers, say \( z_1 = a + bi \) and \( z_2 = c + di \), the addition involves combining their respective parts:

• The real parts: \( a + c \)
• The imaginary parts: \( b + d \)

So the sum \( z_1 + z_2 \) results in \( (a+c) + (b+d)i \).

This addition operation is commutative and associative, much like how you would add real numbers, making it consistent and predictable. Understanding the addition of complex numbers lays down the foundation for more complex manipulations and operations with complex numbers, as seen in our proof about the complex conjugate of their sum matching the sum of their conjugates.

Multiplication of Complex Numbers

Multiplication of complex numbers extends the rules of distributive property and FOIL (First, Outer, Inner, Last) method when applied to binomials. When multiplying two complex numbers, such as \( z_1 = a + bi \) and \( z_2 = c + di \), calculate as follows:

• First: \( ac \)
• Outer: \( adi \)
• Inner: \( bci \)
• Last: \( bdi^2 \)

Remembering that \( i^2 = -1 \), you simplify to get \( (ac - bd) + (ad + bc)i \).

This process highlights how the real and imaginary components of the resulting complex number emerge from both individual products. Multiplication is foundational in complex number algebra, providing insights into their polar form, magnitude, and arguments. It is also essential for solving equations involving complex numbers and is widely used in fields such as physics and engineering.