Problem 78 - Precalculus Mathematics for Calculus

Question

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z-\bar{z}\) is a pure imaginary number.

Step-by-Step Solution

Answer

Since \(z - \bar{z} = 2bi\), it is purely imaginary.

Step 1: Express the Conjugate

Given the complex number \(z = a + bi\), its complex conjugate is \(\bar{z}= a - bi\). The conjugate of a complex number involves changing the sign of the imaginary part.

Step 2: Expand the Expression

Consider the expression \(z - \bar{z}\). Substitute \(z\) and \(\bar{z}\) into the expression: \((a + bi) - (a - bi)\).

Step 3: Simplify the Expression

Simplify the expression by subtracting the terms: \((a + bi) - (a - bi) = a + bi - a + bi = 2bi\).

Step 4: Determine the Nature of the Result

The result \(2bi\) has no real part (since the real part is zero), and only an imaginary part. Therefore, \(z - \bar{z}\) is a pure imaginary number, confirming the statement.

Key Concepts

Complex Conjugate

A complex conjugate is a concept linked to complex numbers, which are numbers that include a real part and an imaginary part. The imaginary part is typically denoted by the letter 'i', which represents the square root of -1. Given a complex number, say \( z = a + bi \), its complex conjugate is expressed as \( \bar{z} = a - bi \). Essentially, this operation involves flipping the sign of the imaginary component of the complex number.

• The real part, 'a', remains unchanged.
• The imaginary part, 'bi', changes to '-bi'.

By using the conjugate, we can explore various properties of complex numbers, such as simplifying expressions or determining magnitudes. Conjugates are especially useful in rationalizing complex denominators and understanding the geometry of complex numbers upon the complex plane.

Pure Imaginary Number

A pure imaginary number is a type of complex number where the real part is zero, and its entire value is derived from the imaginary component alone. In mathematical terms, a pure imaginary number is typically denoted as \( bi \), where 'b' is a real number and 'i' signifies the imaginary unit.

• No real part present, so we can say \( a = 0 \).
• Entire value is expressed in the form of \( bi \).

Going back to our exercise, when you take the expression \( z - \bar{z} \), you end up with \( 2bi \). Notice here that the real parts cancel each other out, leaving only the imaginary parts. This conclusion validates that \( z - \bar{z} \) is indeed a pure imaginary number, since it only contains \( 2bi \), with no real terms involved.

Algebraic Expressions

Algebraic expressions with complex numbers can at times look intimidating, but they follow similar principles to traditional algebra. The main thing to remember is how to manage the imaginary unit 'i'. To make sense of algebraic manipulations with complex numbers, let's consider a few points:

• Grouping like terms: Real terms with other reals, and imaginary terms with others involving 'i'.
• Simplification: Aim to simplify each component separately, working on real and imaginary sections distinctly.
• Operations: Addition and subtraction are performed by combining like terms; multiplication requires the distributive property and \( i^2 = -1 \).

By mastering these operations with complex numbers, you can build on basic algebraic manipulation skills as seen in the solution, where the expression \( (a + bi) - (a - bi) \) simplifies neatly to \( 2bi \). Simplifying complex expressions is a key skill that can translate into solving broader mathematical problems involving complex numbers.