Math Problem Statement

Given that z and w are complex numbers, prove that: (a) (z - w)* = z* - w* (b) (z/w)* = z*/w*

Solution

Let's go through the problem step by step. We are given two complex numbers zz and ww, and asked to prove the following properties involving complex conjugates:

Part a) (zw)=zw(z - w)^* = z^* - w^*

  1. Complex Conjugate Property: The complex conjugate of a sum or difference of two complex numbers is the same as the sum or difference of their individual conjugates.

    (zw)=zw(z - w)^* = z^* - w^*

    This follows from the general property of the complex conjugate: (a+bi)=abi(a + b i)^* = a - b i So, when you subtract two complex numbers and take their conjugate, it behaves in the same way.

Thus, (zw)=zw(z - w)^* = z^* - w^* is directly a consequence of this property. Hence, the statement is true.

Part b) (zw)=zw\left( \frac{z}{w} \right)^* = \frac{z^*}{w^*}

  1. Complex Conjugate Property for Division: The complex conjugate of a quotient of two complex numbers is equal to the quotient of their individual conjugates.

    (zw)=zw\left( \frac{z}{w} \right)^* = \frac{z^*}{w^*}

    To verify this:

    • Let z=a+biz = a + bi and w=c+diw = c + di, where a,b,c,da, b, c, d are real numbers.
    • The quotient zw\frac{z}{w} can be expressed as: zw=(a+bi)(c+di)(cdi)(cdi)=(a+bi)(cdi)c2+d2\frac{z}{w} = \frac{(a+bi)}{(c+di)} \cdot \frac{(c-di)}{(c-di)} = \frac{(a+bi)(c-di)}{c^2 + d^2}
    • The complex conjugate of zw\frac{z}{w} will be the conjugate of the numerator divided by the conjugate of the denominator, which follows from basic algebraic rules of conjugates.

Therefore, we confirm that (zw)=zw\left( \frac{z}{w} \right)^* = \frac{z^*}{w^*}.

Both statements have been proven based on standard properties of complex conjugates.


Would you like more details on any of these proofs or related topics? Here are five follow-up questions that expand on this:

  1. How do you compute the conjugate of a product of two complex numbers?
  2. What is the geometric interpretation of a complex conjugate?
  3. How does the modulus of a complex number relate to its conjugate?
  4. What happens to the argument (angle) of a complex number when you take its conjugate?
  5. Can the complex conjugate be applied to functions of complex numbers, like f(z)f(z)?

Tip: The conjugate of a complex number is useful for simplifying expressions, especially when dividing by a complex number.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Complex Conjugates
Algebra

Formulas

(z - w)* = z* - w*
(z/w)* = z*/w*

Theorems

Complex Conjugate Property for Subtraction
Complex Conjugate Property for Division

Suitable Grade Level

College-level mathematics (or advanced high school)