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Chapter 2: Problem 23

Perform the operation and write the result in standard form. $$(7+i)+(3-4 i)$$

Short Answer

Expert verified
The sum of \(7+i\) and \(3-4i\) is \(10-3i\)

Step by step solution

01

Identify the real and imaginary parts of the complex numbers

For the complex number \(7+i\), the real part is 7 and the imaginary part is 1. For the complex number \(3-4i\), the real part is 3 and the imaginary part is -4.
02

Add the real parts

Add the real parts of the two complex numbers: \(7+3 = 10\).
03

Add the imaginary parts

Add the imaginary parts of the two complex numbers: \(1+(-4) = -3\).
04

Write the sum in standard form

Combine the sum of the real parts and the sum of the imaginary parts to write the result in standard form: \(10+(-3)i\), which can be written as \(10-3i\).

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Key Concepts

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

Standard Form of Complex Numbers
The standard form of a complex number provides a clear way to represent the unique combination of a real number and an imaginary number. It's written as \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part, with \(i\) being the imaginary unit, defined by \(i^2 = -1\).

When dealing with complex numbers, it's crucial to keep them in this format because it makes operations like addition, subtraction, multiplication, and division straightforward. In our exercise, we performed an addition of two complex numbers, which resulted in a sum that also conforms to this standard format.
Real Parts of Complex Numbers
The real part of a complex number is the 'a' in the standard form \(a + bi\). It's simply a real number and represents the 'horizontal' component on the complex plane. For the complex number \(7 + i\), the real part is 7, and for \(3 - 4i\), the real part is 3.

When we add complex numbers, as in the exercise provided, we add the real parts together. This operation is similar to adding two normal numbers and hence is quite intuitive. In our case, adding the real parts 7 and 3 gives us 10, which forms the real part of the resulting complex number after the addition.
Imaginary Parts of Complex Numbers
The imaginary part of a complex number is the 'b' in the standard form \(a + bi\), where 'b' is multiplied by the imaginary unit \(i\). Unlike the real part, the imaginary part defines the 'vertical' component in the complex plane. In the given complex numbers, \(7 + i\) has an imaginary part of 1, and \(3 - 4i\) has an imaginary part of -4.

During addition, the imaginary parts are also added together, similar to how real parts are added. However, remember to consider the sign in front of each imaginary part. For our example, when adding the imaginary parts 1 and -4, we get -3, which gives us the final imaginary component of the complex number in our solution.

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Most popular questions from this chapter

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