Problem 6 Determine whether each complex n... [FREE SOLUTION]

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Trigonometry

Mark Dugopolski

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 6: Problem 6

Determine whether each complex number is real or imaginary and write it in the standard form $a+b i$. $$ -3 i+\sqrt{6} $$

Short Answer

Expert verified

$\sqrt{6} - 3i$; General complex number

Step by step solution

Identify the Real Part

The real part of a complex number is the term without the imaginary unit $i$. Look at the given complex number $-3i + \sqrt{6}$. The real part here is \sqrt{6}\.

Identify the Imaginary Part

The imaginary part of a complex number is the term with the imaginary unit $i$. In the given complex number $-3i + \sqrt{6}$, the imaginary part is $-3$, since it is multiplied by $i$.

Write in Standard Form

A complex number in standard form is written as $a + bi$, where \a\ and \b\ are real numbers. From the previous steps: \a = \sqrt{6} \ and \b = -3.\ Hence, the standard form is \[ \sqrt{6} - 3i \].

Determine if Real or Imaginary

A complex number is purely real if its imaginary part $b$ is zero and purely imaginary if its real part $a$ is zero. Since \b \ is \ -3 \ (not zero) and \a \ is \sqrt{6}\ (not zero), the given number is neither purely real nor purely imaginary; it is a general complex number.

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Key Concepts

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

Real Part of a Complex Number

For any complex number, the real part is the term that does not involve the imaginary unit $i$. When we look at a complex number written as $a + bi$, where $a$ and $b$ are real numbers, the real part is simply $a$. To identify the real part in the complex number $-3i + \sqrt{6}$, we need to find the term that does not include $i$. Here, it is clearly $\sqrt{6}$. The real part helps us to understand how the complex number behaves on a real number line, without involving the imaginary component.
Examples can be helpful:

In $4 + 5i$, the real part is 4.
In $-2.7 + 3.1i$, the real part is -2.7.

This understanding allows us to easily pick out the real components in more complicated complex numbers.

Imaginary Part of a Complex Number

The imaginary part of a complex number is the term that includes the imaginary unit $i$. For a complex number written in the form $a + bi$, the imaginary part is simply $b$, which is the coefficient of $i$. To identify the imaginary part in the complex number $-3i + \sqrt{6}$, we look for the term attached to $i$. Here, it's $-3$ since this term is multiplied by $i$.
This negative sign indicates the direction of the imaginary part along the imaginary axis.
Examples for clarity:

In $4 + 5i$, the imaginary part is 5.
In $3 - 2i$, the imaginary part is -2.
In $-3i + \sqrt{6}$, the imaginary part is -3.

The imaginary part is crucial for plotting the complex number on the complex plane and understanding its imaginary component.

Standard Form of a Complex Number

The standard form of a complex number is expressed as $a + bi$, where $a$ is the real part and $b$ is the imaginary part. Writing a complex number in this form ensures clarity, making both its real and imaginary parts evident.
Using our example, $-3i + \sqrt{6}$:

The real part $a$ is $\sqrt{6}$.
The imaginary part $b$ is $-3$.

Hence, the number can be written as $\sqrt{6} - 3i$. This format simplifies understanding and operations involving complex numbers.
Remember, the standard form is consistent: even if either $a$ or $b$ is zero, it's still written in the form $a + bi$. For instance:

If a complex number is purely real, like $5$: It is written as $5 + 0i$.
If it is purely imaginary, like $-2i$: It appears as $0 - 2i$.

This framework helps in performing mathematical operations, comparisons, and visualizing complex numbers easily.

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91影视

Determine whether each complex number is real or imaginary and write it in the standard form \(a+b i\). $$ -3 i+\sqrt{6} $$

Short Answer

Step by step solution

Identify the Real Part

Identify the Imaginary Part

Write in Standard Form

Determine if Real or Imaginary

Key Concepts

Real Part of a Complex Number

Imaginary Part of a Complex Number

Standard Form of a Complex Number

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