Problem 83 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (-1+2 i)(-1-2 i) $$

Step-by-Step Solution

Answer

The standard form of the product is 5.

Step 1: Identify the expression to be expanded

The given expression is \((-1 + 2i)(-1 - 2i)\). This expression is a product of a complex number and its conjugate.

Step 2: Apply the formula for the product of a complex number and its conjugate

The product of a complex number \((a + bi)\) and its conjugate \((a - bi)\) is given by \((a^2 + b^2)\).

Key Concepts

Complex Conjugate

The complex conjugate is a fundamental concept when working with complex numbers. Complex numbers are expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The complex conjugate of this number is \(a - bi\). You simply change the sign of the imaginary component while keeping the real part unchanged.

By using a conjugate, we can simplify expressions and perform certain operations more easily. For instance, the conjugate helps to eliminate the imaginary part when a complex number is divided by another complex number.

In our exercise, the complex number is \(-1 + 2i\). Its conjugate is \(-1 - 2i\), which can be used to find the product as demonstrated in the solution. Using conjugates, we can solve problems and ensure our results can be presented in a more manageable form.

Standard Form of Complex Numbers

When expressing the result of operations involving complex numbers, the standard form is typically preferred. The standard form of a complex number is given by \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part while \(b\) represents the coefficient of the imaginary part, \(i\).

In the context of our exercise, converting the product of complex numbers into standard form means obtaining a solution without any imaginary unit part causing the value to remain within the real number line. After finding the product, the process involves simplifying the expression into the form \( c + 0i \), where \(c\) is a real number, illustrating that the imaginary parts effectively cancel each other out and clarify computations into real numbers alone.

For instance, if our result were obtained to have an imaginary coefficient with no real counterpart, we rearrange it into terms of \( a+bi \). However, when multiplied with its conjugate, the result is purely real, signifying a cancellation of imaginary components.

Product of Complex Numbers

Calculating the product of complex numbers follows specific arithmetic operations akin to those with binomials. When multiplying two complex numbers, like \((-1 + 2i)\) and \((-1 - 2i)\), these terms can be expanded using the distributive property, treating \(i\) as a variable.

• First, multiply the real parts: \((-1) \times (-1) = 1\)
• Multiply the imaginary parts: \((2i) \times (-2i) = -4i^2\)
• Combine mixed terms: \((-1) \times (-2i) + (2i) \times (-1) = 2i - 2i\)

However, in this case as suggested by the solution, a special property arises when complex conjugates are multiplied. The product simplifies to \(a^2 + b^2\). Applying this rule directly negates the need to handle the individual real, imaginary, and crossover terms explicitly, immediately offering the simplified form.

Therefore, knowing these operations not only aids in solving complex multi-step problems but also enhances our understanding regarding how real numbers emerge from calculated results. This approach reinforces comprehension of how completing multiplication with conjugates results in canceling out imaginary values.