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Chapter 9: Problem 16

Add or subtract as indicated. $$ (12-3 i)-8 $$

Short Answer

Expert verified
\(4 - 3i\)

Step by step solution

01

Identify the Components

Identify the components of the expression. The given expression is \( (12 - 3i) - 8 \). Here, \(12 - 3i\) is a complex number, and \(8\) is a real number.
02

Distribute the Subtraction

Distribute the subtraction sign to the \(8\). This means subtract \(8\) from the real part of the complex number. Thus, the expression becomes \(12 - 3i - 8\).
03

Combine Like Terms

Combine the real parts and the imaginary parts separately. For the real part: \(12 - 8 = 4\). The imaginary part remains \(-3i\).
04

Write the Final Expression

Combine the real part and the imaginary part to get the final answer: \(4 - 3i\).

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Key Concepts

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

Subtraction of Real Numbers
Subtraction is a fundamental concept in mathematics. It involves taking one number away from another. When subtracting real numbers, you simply take the value of one number and decrease it by the value of the other number. For example, in the expression given in the exercise, you have to subtract 8 from 12. This can be visualized as taking 8 steps back from 12 on a number line.
This gives us:
  • 12 - 8 = 4.

By understanding and mastering the subtraction of real numbers, you build a foundation for more complex mathematical operations.
Complex Numbers
Complex numbers are numbers that include both a real part and an imaginary part. A standard form for a complex number is written as: often denoted as:
  • a + bi.

where:
  • The 'a' represents the real part.
  • The 'b' represents the coefficient of the imaginary part 'i', where 'i' is the imaginary unit and satisfies i^2 = -1.

In the given exercise (12 - 3i) - 8, 12 - 3i is the complex number. Understanding complex numbers is vital for solving equations beyond the standard real number system.
Distribution in Algebra
Distribution, or the distributive property in algebra, involves multiplying a single term by every term within a parenthesis. However, it also applies to subtraction by distributing the subtraction operation across the terms inside the parenthesis. In our exercise, we apply distribution as follows:
  • The initial expression is (12 - 3i) - 8.
  • We distribute the subtraction sign to each term.
  • This gives us 12 - 8 - 3i.

By distributing the subtraction across the terms, you break down the problem into simpler parts. This method is useful as it allows you to handle each component individually, making the combining process straightforward.

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Most popular questions from this chapter

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ 2 r^{2}+3 r+5=0 $$

Jeff solved the equation in Exercise 47 and wrote his answer as \(\left\\{\frac{5+\sqrt{30}}{2}, \frac{5-\sqrt{30}}{2}\right\\} .\) Linda solved the same equation and wrote her answer as \(\left\\{\frac{-5+\sqrt{30}}{-2}, \frac{-5-\sqrt{30}}{-2}\right\\} .\) The teacher gave them both full credit. Explain why both students were correct.

Use a calculator with a square root key to solve each equation. Round your answers to the nearest hundredth. \((r-3.91)^{2}=9.28\)

We can use a graphing calculator to illustrate how the graph of \(y=x^{2}\) can be transformed through arithmetic operations. In the standard viewing window of your calculator, graph the following pair of parabolas on the same screen. $$ Y_{1}=x^{2} \quad Y_{2}=-x^{2} $$ Describe how the graph of \(\mathrm{Y}_{2}\) can be obtained from the graph of \(\mathrm{Y}_{1}\)

Solve each problem. Find two numbers whose sum is 80 and whose product is a maximum. (Hint: Let \(x\) represent one of the numbers. Then \(80-x\) represents the other. A quadratic function represents their product.)
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