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Chapter 2: Problem 6

Add or subtract as indicated and write the result in standard form. $$7-(-9+2 i)-(-17-i)$$

Short Answer

Expert verified
The result of \(7-(-9+2 i)-(-17-i)\) expressed in standard form is \(33 - i\)

Step by step solution

01

- Identify real and imaginary parts

Separate the real numbers from the imaginary numbers. Doing so, we can rewrite the expression as follows: \[7 - (-9) - 2i - (-17) - (-i)\] This can be further broken down to \[7 + 9 + 17 - (2i - i)\]
02

- Simplify Real and Imaginary Parts

Perform addition and subtraction operations separately for the real part and the imaginary part.The real part simplifies to \[7 + 9 + 17 = 33\] and the imaginary part simplifies to \[- (2i - i) = -1i\]Thus, the expression simplifies to \[33 - i\]
03

- Convert to Standard Form

The standard form of a complex number is a + bi. Thus, the final answer is: \[33 - i\] which is already in standard form

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Key Concepts

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

Real Numbers
Real numbers are the backbone of mathematics. They consist of all the numbers you can think of that aren't imaginary. These include:
  • Positive numbers like 1, 2, 3...
  • Negative numbers like -1, -2, -3...
  • Zero (0).
  • Fractions and decimals, such as 1/2 or 0.75.
Real numbers can be visualized on a number line that extends infinitely in both directions.
They are part of everyday counting, measuring, and calculations.In the context of complex numbers, the real part is simply all the numbers without the imaginary unit, \(i\).
For example, in the expression from our exercise, \(7\), \(-9\), and \(-17\) are all real numbers.
When we subtract these values appropriately, they combine to form a single real number in the standard form of a complex number.
Imaginary Numbers
Imaginary numbers offer a fascinating extension to the number system by introducing the element \(i\), which represents the square root of \(-1\).
While initially it might sound odd, they provide solutions to equations that don't have answers in the realm of real numbers.
Every imaginary number can be written as \(bi\), where \(b\) is a real number.
  • \(i = \sqrt{-1}\)
  • \(2i\) or \(-3i\) are typical examples of imaginary numbers.
In complex numbers, imaginary parts incorporate these units. In our exercise, \(2i\) and \(-i\) are the imaginary components.
By performing operations like addition or subtraction on these terms separately, we handle them effectively without interfering with the real numbers.
Standard Form
Standard form is a neat way to express complex numbers, allowing for easy interpretation and calculation.
It presents complex numbers as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.This format is important because:
  • It clearly shows both components of a complex number.
  • Makes calculations easier, particularly when adding, subtracting, multiplying, or dividing complex numbers.
  • Facilitates understanding of the geometric representation on a plane called the complex plane.
In our exercise, after performing the required arithmetic, the expression simplifies to \(33 - i\).
Here, \(33\) is the real part and \(-i\) is the imaginary part.
This example fits the standard form perfectly, making it simple to analyze or use in further calculations.

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Most popular questions from this chapter

Use a graphing utility to graph $$ f(x)=\frac{x^{2}-4 x+3}{x-2} \text { and } g(x)=\frac{x^{2}-5 x+6}{x-2} $$ What differences do you observe between the graph of \(f\) and the graph of \(g\) ? How do you account for these differences?

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose y-coordinate is the same as the given point. $$f(x)=3(x+2)^{2}-5 ; \quad(-1,-2)$$

Describe in words the variation shown by the given equation. \(z=\frac{k \sqrt{x}}{y^{2}}\)

If you have difficulty obtaining the functions to be maximized in Exercises \(73-76,\) read Example 2 in Section \(1.10 .\) A car rental agency can rent every one of its 200 cars at \(\$ 30\) per day. For each \(\$ 1\) increase in rate, five fewer cars are rented. Find the rental amount that will maximize the agency's daily revenuc. What is the maximum daily revenue?

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=0.01 x^{2}+0.6 x+100$$
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