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Chapter 8: Problem 44

Multiply. $$ (5 i)(125 i) $$

Short Answer

Expert verified
The result is \(-625\).

Step by step solution

01

Multiply the coefficients

First, multiply the numerical coefficients of the terms. Here we have 5 and 125. \(5 \times 125 = 625\).
02

Multiply the imaginary units

Next, multiply the imaginary units. Remember that \(i \times i = i^2\). Thus, \(i \times i = i^2\).
03

Substitute the value of \(i^2\)

Recall that \(i^2 = -1\). So, \(i^2\) can be replaced by \(-1\).
04

Multiply the results from the previous steps

Now we have 625 multiplied by -1. Therefore, \(625 \times -1 = -625\).

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Key Concepts

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

Imaginary Unit
The imaginary unit is a fundamental concept in dealing with complex numbers. It is often denoted by the letter 'i'. The imaginary unit 'i' is defined by the property that its square is equal to -1. This can be written as: \(i^2 = -1\). This definition is essential in understanding and working with complex numbers, where 'i' represents the square root of -1, something not possible with real numbers alone.
Multiplication of Imaginary Numbers
Multiplying imaginary numbers follows the same rules as multiplying real numbers, but with an added complexity due to the presence of the imaginary unit 'i'. Here's how multiplication works:
  • Multiply the numerical coefficients.
  • Multiply the imaginary units using their properties.
For example, if you multiply two imaginary numbers like \(5i\) and \(125i\), first multiply the numbers: \(5 \times 125 = 625\). Then, multiply the imaginary units: \(i \times i = i^2\).
Complex Number Properties
Complex numbers have unique properties that differentiate them from real numbers. A complex number is usually written in the form \(a + bi\), where 鈥榓鈥� is the real part and 鈥榖鈥� is the imaginary part. Important attributes include:
  • Addition and Subtraction: Combine real parts with real parts, and imaginary parts with imaginary parts.
  • Multiplication: Multiply each part separately, remembering that \(i^2 = -1\).
  • Division: Involves multiplying by the complex conjugate to clear the imaginary unit from the denominator.
Understanding these properties makes it easier to perform operations with complex numbers.
i Squared Equals Negative One
One of the most significant aspects of imaginary numbers is the relationship \(i^2 = -1\). This simple yet powerful equation allows us to connect the imaginary unit 'i' with real numbers. Whenever you see an expression involving \(i^2\), you can replace it with -1. For example, when we multiply \(5i\) and \(125i\), we get \(625i^2\). Knowing that \(i^2 = -1\), we substitute -1 for \(i^2\), giving us \(625 \times -1 = -625\).

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Most popular questions from this chapter

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