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Chapter 6: Problem 16

Find the absolute value of each complex number $$ 3.6+4.8 i $$

Short Answer

Expert verified
6.00

Step by step solution

01

Identify the Real and Imaginary Parts

The given complex number is \(3.6 + 4.8i\). The real part is 3.6 and the imaginary part is 4.8.
02

Use the Formula for Absolute Value

The absolute value of a complex number \(a + bi\) is given by \(\sqrt{a^2 + b^2}\).
03

Substitute the Values

Substitute \(a = 3.6\) and \(b = 4.8\) into the formula: \ \[|3.6 + 4.8i| = \sqrt{3.6^2 + 4.8^2}\].
04

Compute the Squares

Calculate the squares of 3.6 and 4.8: \ \[3.6^2 = 12.96\] \ \[4.8^2 = 23.04\].
05

Sum the Squares

Add the results of the squares: \ \[12.96 + 23.04 = 36.00\].
06

Take the Square Root

Find the square root of 36.00: \ \[\sqrt{36.00} = 6.00\].
07

Write the Final Answer

The absolute value of the complex number \(3.6 + 4.8i\) is 6.00.

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Key Concepts

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

Real Part
In complex numbers, the real part is the component without the imaginary unit 'i'. If you consider a complex number in the form of \(a + bi\), the real part is represented by 'a'. For example, in the complex number \(3.6 + 4.8i\), the real part is 3.6. The real part signifies a horizontal shift on the complex plane. Handling this part of the number is just like dealing with any other real number, meaning it's affected by basic arithmetic operations like addition, subtraction, multiplication, and division.
Imaginary Part
The imaginary part of a complex number is paired with the imaginary unit 'i', which is defined as the square root of -1. In a complex number written as \(a + bi\), the imaginary part is 'b'. For the complex number \(3.6 + 4.8i\), the imaginary part is 4.8.
Imaginary numbers allow us to perform operations involving the square roots of negative numbers. In this scenario:
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Most popular questions from this chapter

Find the product and quotient of each pair of complex numbers using trigonometric form. Write your answers in \(a+\) bi form. $$ z_{1}=2+i, z_{2}=3-2 i $$

Convert the polar coordinates of each point to rectangular coordinates. $$ \left(-4,30^{\circ}\right) $$

Convert the polar coordinates of each point to rectangular coordinates. $$ \left(\sqrt{5}, 230^{\circ}\right) $$

Graph each pair of polar equations on the same screen of your calculator and use the trace feature to estimate the polar coordinates of all points of intersection of the curves. Check your calculator manual to see how to graph polar equations. $$ r=\sin \theta, r=\sin 2 \theta $$

Convert the rectangular coordinates of each point to polar coordinates. Use radians for \(\theta .\) $$ (-3 \sqrt{2}, 3 \sqrt{2}) $$
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