Problem 6 - Algebra for College Students

Question

Label each statement true or false. The imaginary part of the complex number 7 is 0 .

Step-by-Step Solution

Answer

True

Step 1: Understanding Complex Numbers

A complex number is expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) being the imaginary unit where \( i^2 = -1 \). In a complex number, \( b \) represents the coefficient of the imaginary unit \( i \).

Step 2: Identify the Parts of the Given Number

The given number is \( 7 \). To compare it with the general form \( a + bi \), we can consider \( 7 = 7 + 0i \). From this, \( a = 7 \) and \( b = 0 \). Thus, \( b \), which is the coefficient of the imaginary part, is 0.

Step 3: Evaluate the Statement

Since the imaginary part of the complex number 7 is represented by the coefficient \( b \), and we identified \( b = 0 \), the statement "The imaginary part of the complex number 7 is 0" is true.

Key Concepts

Imaginary Part

The imaginary part of a complex number determines the component of that number which combines with the imaginary unit, often denoted as \( i \). Complex numbers take the form \( a + bi \), where:

• \( a \) is the real part
• \( b \) is the coefficient of the imaginary part

For any complex number, \( b \) is critical in specifying the imaginary part.
It differentiates complex numbers from real numbers, providing depth to calculations.
When a number doesn't follow the \( a + bi \) format, like the number 7 in the example, it's important to express it as a full complex number. Hence, 7 can be written as \( 7 + 0i \). The imaginary part is then identified as 0 because \( b = 0 \).
Since imaginary components have the imaginary unit \( i \), they become significant in distinguishing complex numbers from purely real numbers.

Real Part

The real part of a complex number is essentially the familiar number line value, excluding any influence from the imaginary unit. In the expression \( a + bi \), \( a \) is the real part. This familiarization comes from arithmetic operations we commonly perform.
Even though real parts look identical to everyday numbers, they contribute jointly with an imaginary part to form complex numbers.

In our example, the real number 7 is expressed as \( 7 + 0i \), confirming that its real part is 7. When observing a number on its own, without an explicit \( i \) term, it's easily interpretable as having no imaginary influence. Thus, the whole number is its real part because the coefficient of \( i \) in the imaginary part is zero.

Imaginary Unit

The imaginary unit, often symbolized by \( i \), is the backbone of complex numbers. It distinguishes the unique property that \( i^2 = -1 \). This property is counterintuitive because no real number squared equals a negative.

The involvement of \( i \) converts ordinary arithmetic into complex calculations, which can solve problems non-resolvable by real numbers alone.
It provides solutions to equations involving square roots of negative numbers, expanding the scope of mathematical analysis.

• In equations, \( i \) stands firm as a fundamental piece of complex numbers.
• When combined with a real number, it forms the structural base of the imaginary part.

In practical terms, the imaginary unit empowers complex functions by broadening their applications within mathematics, engineering, and physics. It enables exploration beyond the boundaries of real number constraints, which is crucial for understanding and solving advanced mathematical problems.