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Chapter 7: Problem 50

Add or subtract as indicated. Give answers in standard form. $$[(7+2 i)+(-4-i)]-(2+5 i)$$

Short Answer

Expert verified
1 - 4i

Step by step solution

01

Simplify inside the brackets

First, simplify the expression inside the brackets: [(7+2i)+(-4-i)] Combine the like terms: Real parts: 7 + (-4) = 3 Imaginary parts: 2i + (-i) = i Therefore, [(7+2i)+(-4-i)] becomes (3 + i).
02

Simplify the entire expression

Now subtract (2 + 5i) from (3 + i) (3 + i) - (2 + 5i) Subtract the real parts: 3 - 2 = 1 Subtract the imaginary parts: i - 5i = -4i Therefore, (3 + i) - (2 + 5i) becomes (1 - 4i).
03

Write in standard form

In standard form, the answer is: 1 - 4i

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Key Concepts

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

Adding Complex Numbers
Complex numbers can be added by combining their real parts and their imaginary parts separately. For example, if you have two complex numbers (7 + 2i) and (-4 - i), you would add the real parts (7 + (-4)) and the imaginary parts (2i + (-i)) independently. Once you combine the real parts, you get 3, and combining the imaginary parts gives you i. Therefore, (7 + 2i) + (-4 - i) simplifies to (3 + i). Always keep in mind that the result of adding complex numbers is also a complex number.
Subtracting Complex Numbers
Subtracting complex numbers is very similar to adding them. You still handle the real and imaginary parts separately. For example, if you need to subtract (2 + 5i) from (3 + i), you start by subtracting the real parts: 3 - 2 = 1. Then you subtract the imaginary parts: i - 5i = -4i. So, (3 + i) - (2 + 5i) simplifies to (1 - 4i). Just like with addition, subtraction of complex numbers results in another complex number. The key is to treat the real and imaginary components individually.
Standard Form of Complex Numbers
The standard form of a complex number is written as a + bi, where a and b are real numbers. In this form, 'a' is the real part, and 'bi' is the imaginary part. So if your simplified expression is (1 - 4i), it is already in standard form. The standard form makes it easy to identify and work with the real and imaginary parts separately. Simply ensure 'a' (the real part) comes first, followed by 'bi' (the imaginary part). This format is crucial for clarity in complex number operations.

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Most popular questions from this chapter

The illumination \(I\), in foot-candles, produced by a light source is related to the distance \(d\), in feet, from the light source by the equation $$ d=\sqrt{\frac{k}{I}} $$ where \(k\) is a constant. If \(k=640,\) how far from the light source will the illumination be 2 foot-candles? Give the exact value, and then round to the nearest tenth of a foot.

Find the distance between each pair of points. \((\sqrt{2}, \sqrt{6})\) and \((-2 \sqrt{2}, 4 \sqrt{6})\)

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{27}}{3-\sqrt{3}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{5}+\sqrt{6}}{\sqrt{3}-\sqrt{2}} $$

Rationalize each denominator. Assume that all radicals represent real numbers and that no denominators are \(0 .\) $$ \frac{5}{\sqrt{m-n}} $$
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