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Chapter 8: Problem 97

Consider the following complex numbers, and work in order. $$ w=-1+i \quad \text { and } \quad z=-1-i $$ Multiply \(w\) and \(z\) using their rectangular forms and the FOIL method. Leave the product in rectangular form.

Short Answer

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Step by step solution

01

Write down the complex numbers

The given complex numbers are \( w = -1 + i \) and \( z = -1 - i \).
02

Apply the FOIL method

FOIL stands for First, Outer, Inner, Last. Use the FOIL method to multiply the two complex numbers: First: \( -1 \cdot -1 = 1 \) Outer: \( -1 \cdot (-i) = i \) Inner: \( i \cdot -1 = -i \) Last: \( i \cdot (-i) = -i^2 \)
03

Simplify the expression

Combine all the terms obtained from the FOIL method: \( 1 + i - i - i^2 \)
04

Simplify \( -i^2 \)

Recall that \( i^2 = -1 \). Therefore, \( -i^2 = -(-1) = 1 \) The expression now becomes: \( 1 + i - i + 1 \)
05

Combine like terms

Combine the real parts and the imaginary parts of the expression: The real part: \( 1 + 1 = 2 \) The imaginary part: \( i - i = 0 \) Therefore, the product is \( 2 + 0i \) or simply \( 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular Form
When working with complex numbers, one common way to represent them is using the rectangular form. In rectangular form, a complex number is expressed as the sum of a real part and an imaginary part. A complex number in rectangular form looks like this:
  • The real part: a number without an imaginary unit, usually noted as 'a'
  • The imaginary part: a number with the imaginary unit, usually noted as 'bi'
Here, 'i' is the imaginary unit. So, for example, the complex number a + bi = 3 + 4i is in rectangular form. In our exercise, the numbers w and z are given as w = -1 + i and z = -1 - i, which are already in rectangular form.
FOIL Method
The FOIL method is a technique to multiply two binomials. FOIL stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms in the binomials. Here's how it works for multiplying complex numbers:
- **First**: Multiply the first terms of each binomial.
- **Outer**: Multiply the outer terms.
- **Inner**: Multiply the inner terms.
- **Last**: Multiply the last terms.
Let鈥檚 repeat this process for our complex numbers w and z as follows:
  • First: (-1) * (-1) = 1
  • Outer: (-1) * (-i) = i
  • Inner: (i) * (-1) = -i
  • Last: (i) * (-i) = -i^2
Combine these results to get: 1 + i - i - i^2. Now simplify the terms.
Imaginary Unit
The imaginary unit, denoted as 'i', is a fundamental concept in complex numbers. The key property of the imaginary unit is that it's the square root of -1:
i^2 = -1.
In the context of our exercise, this is crucial for simplifying the expression we get from the FOIL method. When we have the term -i^2, we recall that i^2 = -1, so we have: -i^2 = -(-1) = 1
Then the expression 1 + i - i - i^2 simplifies to 1 + i - i + 1 = 2. Thus, the product of (-1 + i) and (-1 - i) results in a real number, 2, as the imaginary parts cancel each other out.

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