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Chapter 10: Problem 26

Multiply. Write the product in the form \(a+b i .\) $$ -2 i \cdot-11 i $$

Short Answer

Expert verified
The product is \(-22 + 0i\).

Step by step solution

01

Multiply the coefficients

First, multiply the numerical coefficients of the complex numbers. Here, the coefficients are -2 and -11. So we multiply: \[-2 \times -11 = 22\]
02

Multiply 'i' terms

Now, multiply the 'i' terms. Because we are multiplying \(-2i\) and \(-11i\), the result is:\[i \times i = i^2\]
03

Substitute \(i^2\) with -1

According to the definition of the imaginary unit, \(i^2 = -1\). So, replace \(i^2\) with -1 in the context of our product:\[22 \cdot i^2 = 22 \cdot (-1) = -22\]
04

Write the expression in standard form

Express the product in the form \(a + bi\). Since we only have a real part and no imaginary part, it simplifies to:\[a + bi = -22 + 0i\]

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Key Concepts

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

Imaginary Unit
In the world of complex numbers, the 'imaginary unit' is a fundamental concept. It's represented by the letter \(i\). But what exactly is it? Simply put, \(i\) is defined as the square root of \-1. This is special because we know from real numbers that square roots of negative numbers don't exist in real terms. However, with complex numbers, \(i\) makes that possible!

Here's something important to remember about \(i\):
  • \(i^2 = -1\)
  • \(i^3 = i imes i^2 = -i\)
  • \(i^4 = i^2 imes i^2 = 1\)
These properties make calculations with \(i\) quite interesting and somewhat cyclic in nature. This characteristic plays a crucial role when you're dealing with imaginary numbers, particularly in calculations involving multiplication.
Multiplying Complex Numbers
When we talk about multiplying complex numbers, it involves combining their real and imaginary parts through multiplication. The key to simplifying this process is to remember how to handle the imaginary unit \(i\) during multiplication.

Let's break it down:
  • First, multiply the coefficients. For \(-2i\) and \(-11i\), multiplying the coefficients \(-2\) and \(-11\) gives us \(22\).
  • Next, multiply the imaginary parts. \(i\) times \(i\) equals \(i^2\), which as we discussed, is \(-1\).
  • Multiply the resulting coefficient by \-1 from \(i^2\) to get the final result.
Problem-solving simplifies when you apply these straightforward steps. Keep practicing to master the art of combining real and imaginary parts effectively in multiplication.
Complex Number Standard Form
A complex number is typically represented in standard form as \(a + bi\), where \(a\) is the real part, and \(bi\) represents the imaginary part.
After performing calculations like multiplication, reshaping the result to this form helps in understanding its components. Let's see how this applies to our earlier exercise:

- Our multiplication gave us \(-22\). To express this in standard form, consider this:
  • The real part \(a\) is \(-22\).
  • There is no imaginary part, so \(b = 0\).
Thus, the result in standard form is \(-22 + 0i\), simplifying to just \(-22\). Remember, presenting results clearly in standard form can significantly aid comprehension and ease further computations.

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