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Chapter 31: Problem 1008

Find the values of the following expressions: a. \((2+3 i)+(6-2 i)\) b. \((2-i)(1+3 i)\) c. \(i-(2+3 i)\)

Short Answer

Expert verified
The short answer for the given expressions are: a. \(8 + 1i\) b. \(5 + 5i\) c. \(-2 - 2i\)

Step by step solution

01

Identify the given complex numbers

The given complex numbers are: \(2 + 3i\) and \(6 - 2i\).
02

Add the corresponding real and imaginary parts

To add the complex numbers, simply add their real parts and imaginary parts separately: Real part: \(2 + 6 = 8\) Imaginary part: \(3i - 2i = 1i\)
03

Write the result as a complex number

Combine the real and imaginary parts into a single complex number: \(8 + 1i\). #b. Multiplying Complex Numbers#
04

Identify the given complex numbers

The given complex numbers are: \(2 - i\) and \(1 + 3i\).
05

Use the distributive property to multiply the complex numbers

To multiply the complex numbers, apply the distributive property and then perform the multiplication: \((2 - i)(1 + 3i) = 2(1) + 2(3i) - i(1) - i(3i)\)
06

Evaluate expressions involving i虏

Remember that \(i^2 = -1\). Using this, we can simplify the above expression: \(2 + 6i - i - 3i^2 = 2 + 6i - i + 3\), since \(-3i^2 = -3(-1) = 3\)
07

Write the result as a complex number

Combine the simplified real and imaginary parts into a single complex number: \(5 + 5i\). #c. Subtracting Complex Numbers#
08

Identify the given complex numbers

The given complex numbers are: \(i\) and \(2 + 3i\).
09

Change the subtraction to addition of the negation

To subtract the complex numbers, change the subtraction to the addition of the negation: \(i - (2 + 3i) = i + (-2 - 3i)\)
10

Add the corresponding real and imaginary parts

To add the complex numbers, simply add their real parts and imaginary parts separately: Real part: \(0 + (-2) = -2\) Imaginary part: \(1i + (-3i) = -2i\)
11

Write the result as a complex number

Combine the real and imaginary parts into a single complex number: \(-2 - 2i\).

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Most popular questions from this chapter

Multiply \((3+4 i)\) by \((5+2 i)\)

Find the product \((2+3 i)(-2-5 i)\).

Show that \((\mathrm{a}+\mathrm{bi})+(\mathrm{c}+\mathrm{di})=(\mathrm{c}+\mathrm{di})+(\mathrm{a}+\mathrm{bi})\)

What is the conjugate of \(3-2 \mathrm{i}\) and the conjugate of \(5+7 \mathrm{i}\) ?

Find the real numbers a and \(\mathrm{b}\) such that \((a+b i)+(2-3 i)=2(-2+i)\)
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