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Chapter 7: Problem 1

Find the absolute value. $$|3-4 i|$$

Short Answer

Expert verified
The absolute value is 5.

Step by step solution

01

Understanding Absolute Value of a Complex Number

The absolute value (or modulus) of a complex number \( a + bi \) is given by the formula: \( \sqrt{a^2 + b^2} \). In the expression \( |3 - 4i| \), \( a = 3 \) and \( b = -4 \).
02

Calculating the Real Component Squared

Square the real component \( a \): \( 3^2 = 9 \).
03

Calculating the Imaginary Component Squared

Square the imaginary component \( b \): \( (-4)^2 = 16 \).
04

Adding the Squares of the Components

Add the squares of the real and imaginary components: \( 9 + 16 = 25 \).
05

Taking the Square Root

Take the square root of the sum obtained in the previous step: \( \sqrt{25} = 5 \).

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Key Concepts

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

absolute value
The absolute value of a complex number is a way to express its size or magnitude. It is similar to the absolute value of real numbers, which shows the distance from zero on the number line.
However, for a complex number, it indicates the distance from the origin on the complex plane.
To find the absolute value of a complex number like (3 - 4i), we use the formula: \( \sqrt{a^2 + b^2} \). Here, \(a\) and \(b\) are the real and imaginary parts, respectively.
  • For the real part, \(a = 3\).
  • For the imaginary part, \(b = -4\).
Using the formula, we calculate \(\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
Thus, the absolute value of the complex number \(3 - 4i\) is 5.
complex plane
The complex plane is a two-dimensional plane that helps us visualize complex numbers. Each point on this plane represents a complex number. The horizontal axis is the real axis, while the vertical axis represents the imaginary part.
Visualizing our complex number \(3 - 4i\), we can plot it by moving 3 units along the real axis (horizontal) and then 4 units downwards, as the imaginary component is negative.
In this graphical representation:
  • The x-axis (horizontal line) indicates real numbers.
  • The y-axis (vertical line) represents imaginary numbers.
  • The origin corresponds to complex number \(0 + 0i\), or zero.
Placing complex numbers on the plane makes it easier to grasp their relationships and perform operations such as addition or multiplication visually.
modulus of a complex number
The modulus of a complex number is essentially another term for its absolute value. It represents the same concept: the distance from the origin on the complex plane.
To determine the modulus, we use the formula \( \sqrt{a^2 + b^2} \), just as with absolute values. For the complex number \(3 - 4i\), we have:
  • Real part \( (a) = 3 \)
  • Imaginary part \( (b) = -4 \)
Calculating the modulus involves computing \( \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \). Thus, the modulus of \(3 - 4i\) is 5, confirming it measures the same as the absolute value.
This consistency highlights the interchangeable use of these terms in complex number analysis.

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Most popular questions from this chapter

Find \(\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}, \mathbf{4 a}+\mathbf{5 b}, \mathbf{4 a}-\mathbf{5 b},\) and \(\|\mathbf{a}\|\) $$\mathbf{a}=-3 \mathbf{i}+\mathbf{j}, \quad \mathbf{b}=-3 \mathbf{i}+\mathbf{j}$$

Find \(\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}, \mathbf{4 a}+\mathbf{5 b}, \mathbf{4 a}-\mathbf{5 b},\) and \(\|\mathbf{a}\|\) $$\mathbf{a}=-\langle 7,-2\rangle, \quad \mathbf{b}=3\langle 0,-2\rangle$$

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Find the magnitude of the vector a and the smallest positive angle \(\boldsymbol{\theta}\) from the positive \(\boldsymbol{x}\) -axis to the vector \(O P\) that corresponds to a. $$\mathbf{a}=\langle\mathbf{0}, 10\rangle$$
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