Problem 114 - Calculus for Biology and Medicine

Question

Show \(\overline{z+w}=\bar{z}+\bar{w}\)

Step-by-Step Solution

Answer

Yes, \( \overline{z+w} = \overline{z} + \overline{w} \).

Step 1: Define the complex numbers

Let's assume that we have two complex numbers: \( z = a + bi \) and \( w = c + di \), where \( a, b, c, \) and \( d \) are real numbers and \( i \) is the imaginary unit.

Step 2: Express the sum \( z + w \)

Find the sum of the two complex numbers: \( z + w = (a + bi) + (c + di) = (a+c) + (b+d)i \).

Step 3: Find the conjugate \( \overline{z + w} \)

The conjugate of a complex number \( x + yi \) is \( x - yi \). Thus, the conjugate of \( z + w = (a+c) + (b+d)i \) is \( \overline{z + w} = (a+c) - (b+d)i \).

Step 4: Calculate \( \overline{z} + \overline{w} \)

The conjugate of \( z = a + bi \) is \( \overline{z} = a - bi \) and the conjugate of \( w = c + di \) is \( \overline{w} = c - di \). Thus, \( \overline{z} + \overline{w} = (a - bi) + (c - di) = (a+c) - (b+d)i \).

Step 5: Compare the results

From Step 3, we have \( \overline{z + w} = (a+c) - (b+d)i \). From Step 4, we have \( \overline{z} + \overline{w} = (a+c) - (b+d)i \). These two expressions are identical, so \( \overline{z+w} = \overline{z} + \overline{w} \).

Key Concepts

Complex Conjugate

A complex conjugate is a fundamental concept when dealing with complex numbers. If we have a complex number expressed as \( x + yi \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit (i.e., \( i^2 = -1 \)), the complex conjugate of this number is \( x - yi \).
This operation essentially reflects the complex number over the real axis in the complex plane, changing the sign of the imaginary part while keeping the real part unchanged.

• This reflection is important in various mathematical proofs and simplifies computations, such as simplifying division of complex numbers.
• It also plays a crucial role in determining the magnitude of a complex number as it's used in defining and verifying certain properties.

Understanding conjugates helps simplify expressions and prove equalities, as seen in our example with the expression \( \overline{z+w} = \bar{z} + \bar{w} \).

Addition of Complex Numbers

When adding complex numbers, we treat the real and imaginary parts separately. Consider two complex numbers:\( z = a + bi \) and \( w = c + di \).

The sum \( z + w \) is calculated by adding the real parts together, and the imaginary parts together, giving us \((a+c) + (b+d)i\).
This follows directly from how real numbers are added, by grouping like terms.

• This method allows complex numbers to be added in a straight-forward, intuitive way.
• By keeping the real and imaginary components distinct yet manipulable, it mirrors the method by which we add polynomials.

In our example, we see how adding complex numbers simplifies expressions, laying the groundwork for evaluating further relations, like those involving complex conjugates.

Properties of Complex Numbers

Complex numbers boast several properties that make them a powerful tool in mathematics. They extend the concept of numbers beyond real numbers, illustrating that each number comprises two parts: a real part and an imaginary part.

• Closure under Addition and Multiplication: Complex numbers, like real numbers, are closed under both addition and multiplication. This means when you add or multiply two complex numbers, the result will also be a complex number.
• Associative and Commutative Properties: Addition and multiplication of complex numbers follow the associative and commutative properties, enabling rearrangement of terms or factors.
  
• Distributive Property: This property applies in much the same way as for real numbers: \( z(w+x) = zw + zx \).
• Existence of Additive and Multiplicative Identities: The complex number 0 (or \( 0 + 0i \)) acts as an additive identity, and 1 acts as a multiplicative identity.

These properties assure us that many arithmetic techniques familiar from real numbers apply equally to complex numbers, facilitating their use in equations and functions. In our exercise, we see such properties at play, as the commutative property of addition helps prove \( \overline{z+w} = \bar{z} + \bar{w} \).