Problem 6 - Intermediate Algebra

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: Understand Complex Numbers

A complex number is typically written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

Step 2: Analyze the Given Complex Number

The given complex number is 7, which can be written as \( 7 + 0i \). Here, 7 is the real part and 0 is the coefficient of \( i \), making the imaginary part 0.

Step 3: Determine the Imaginary Part

Since a complex number of the form \( a + bi \) has an imaginary part \( bi \), and for \( 7 \) written as \( 7 + 0i \), the imaginary part is indeed 0.

Step 4: Conclusion

Based on the analysis, the statement "The imaginary part of the complex number 7 is 0" is true.

Key Concepts

Real Part

The concept of the real part of a complex number is fundamental when dealing with complex numbers. Complex numbers are written in the form \( a + bi \), where \( a \) is known as the real part.

• Definition: In a complex number, the real part is simply the number without any imaginary component, represented by \( a \) in the expression \( a + bi \).
• Example: Consider the complex number \( 3 + 4i \), here the real part is \( 3 \).

When looking at a pure real number, like 7 in earlier examples, it can still be considered a complex number with an imaginary part of 0, written as \( 7 + 0i \).
Thus, in this format, the real part remains 7.

Imaginary Part

The imaginary part of a complex number complements the real part and is crucial for the full expression of complex numerals. It is written with the imaginary unit \( i \), commonly known as \( bi \) in the expression \( a + bi \).

• Imaginary Unit: \( i \) is defined as the square root of \( -1 \). Thus, \( i^2 = -1 \).
• Imaginary Part: In a complex number like \( 2 + 5i \), the imaginary part is \( 5 \). It can be considered as the coefficient of \( i \).

Importantly, even if a number doesn't seem to have an imaginary component, such as 7, it means it has an imaginary part of 0, \( 7 = 7 + 0i \). This makes it easier to understand and manipulate when performing operations with other complex numbers.

Complex Number Notation

Complex number notation is how we represent these numbers to utilize both real and imaginary parts efficiently. This includes writing numbers in the format \( a + bi \).

• Standard Form: The real part \( a \) is written first, followed by the imaginary part \( bi \). For instance, \( 3 + 4i \) where \( 3 \) is real and \( 4i \) is imaginary.
• Special Cases: Purely real numbers are noted as \( a + 0i \), like 7 can be thought of as \( 7 + 0i \), highlighting the absence of an imaginary part.
• Complex Conjugates: Often used in operations, the conjugate \( a - bi \) is derived from a complex number \( a + bi \).

This notation helps in performing arithmetic operations and finding solutions to problems involving complex numbers with much ease and clarity.