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Chapter 1: Problem 40

Determine the real and imaginary parts of the complex number. $$-\frac{4}{7}$$

Short Answer

Expert verified
The real part is \(-\frac{4}{7}\) and the imaginary part is 0.

Step by step solution

01

Identify the Complex Number

The given number is \(-\frac{4}{7}\). It is a purely real number with no imaginary part.
02

Determine the Real Part

The real part of the complex number is simply the non-imaginary component. For \(-\frac{4}{7}\), the real part is \(-\frac{4}{7}\).
03

Determine the Imaginary Part

The imaginary part is the coefficient of the imaginary unit \(i\). Since \(-\frac{4}{7}\) has no imaginary component, the imaginary part is 0.

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Key Concepts

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

Understanding the Real Part of a Complex Number
In a complex number, the real part is the component that does not involve the imaginary unit, denoted as \(i\). For example, in the complex number \(a + bi\), the real part is \(a\). It represents a value on the real number line.
If you consider the given example \(-\frac{4}{7}\), it is evident that the value has no imaginary component and lies entirely on the real number line. Therefore, the real part of \(-\frac{4}{7}\) is simply \(-\frac{4}{7}\).
Always remember, to find the real part, identify the portion of the complex number that lacks the imaginary unit \(i\). This makes understanding complex numbers a tad easier.
Deciphering the Imaginary Part
The imaginary part of a complex number involves the imaginary unit \(i\). In a complex number of the form \(a + bi\), the term \(bi\) is the imaginary part.
The coefficient \(b\) before the \(i\) is what we consider the imaginary part. If a complex number has the form \(a\), without an \(i\), the imaginary part is zero. For instance, in the given exercise, the complex number is \(-\frac{4}{7}\), which does not have an \(i\). Hence, the imaginary part is zero.
Identifying the imaginary part is straightforward: Look for the term multiplied by \(i\). If none exists, the imaginary part is zero.
Defining a Purely Real Number
A purely real number is a complex number with an imaginary part of zero. This means it does not contain the imaginary unit \(i\). Purely real numbers lie entirely on the real number line and can be represented as real numbers in everyday mathematics.
In the context of the provided problem, \(-\frac{4}{7}\), since there is no \(i\) component, it is a purely real number. Every real number we typically encounter in basic math is just a special case of a complex number where the imaginary part equals zero.
Just remember, whenever you see a number without an imaginary component, you are dealing with a purely real number.
Coefficient of the Imaginary Unit
The coefficient of the imaginary unit \(i\) determines the imaginary part of a complex number. In the general form \(a + bi\), \(b\) is the coefficient of \(i\).
Let鈥檚 break this down: If you see a number like \(3 + 2i\), here, the coefficient of \(i\) (or the imaginary part) is 2.
However, in our example, \(-\frac{4}{7}\), there is no \(i\) present. This means the coefficient of the imaginary unit is 0.
Thus, identifying the coefficient of \(i\) is simply looking at the constant multiplied by \(i\). If there isn鈥檛 an imaginary unit, the coefficient is zero.

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