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Chapter 6: Problem 8

Calculate the given expression. \(|-3+2 i|\)

Short Answer

Expert verified
\( \sqrt{13} \)

Step by step solution

01

Understand the Concept of Magnitude

The magnitude or modulus of a complex number, written as \(|a + bi|\), is calculated using the formula \[ \sqrt{a^2 + b^2} \].
02

Identify Components

Identify the real part \(-3\) and the imaginary part \(+2i\). In this case, \(a = -3\) and \(b = 2\).
03

Plug into Formula

Substitute the values into the magnitude formula: \[ |-3 + 2i| = \sqrt{(-3)^2 + (2)^2} \].
04

Simplify the Expression

First, calculate the squares of \(-3\) and \(2\): \((-3)^2 = 9\) and \(2^2 = 4\). Sum these values: \(9 + 4 = 13\).
05

Compute the Square Root

Finally, take the square root of \(13\): \[ \sqrt{13} \].

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Key Concepts

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

Complex Number Components
A complex number is written in the form 'a + bi', where 'a' is the real part and 'b' is the imaginary part. These two components are crucial for understanding and performing operations with complex numbers.
- The Real Part: The 'a' component represents the real part of the complex number. This can be any real number.
- The Imaginary Part: The 'bi' component represents the imaginary part. 'b' is a real number, but when multiplied by 'i' (the imaginary unit), it indicates an imaginary value.
When analyzing or performing arithmetic with complex numbers, always separate and clearly understand these components. This makes it easier to apply formulas and comprehend the number's position in the complex plane.

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Most popular questions from this chapter

Prove the given identity for all complex numbers. \(\left|\frac{z_{1}}{z_{2}}\right|=\frac{\left|z_{1}\right|}{\left|z_{2}\right|}\)

Solve the given equation (in radians). \(2 \cos \theta+1=0\)

Calculate the given expression. \(i^{3}\)

Prove the given identity for all complex numbers. \(\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|\)

Write the given equation in polar coordinates. \( x^{2}-y^{2}=1\)
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