Problem 56 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The complex number \(a+0 i\) is the real number \(a\).

Step-by-Step Solution

Answer

The statement 'The complex number \(a+0 i\) is the real number \(a\)' is true.

Step 1: Understand the concept of real and complex numbers

Complex numbers are expressed in the form \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\)^2 = -1. In the context of this exercise, the imaginary part of the complex number (\(b\)) is 0, rendering it \(a+0i\). On the other hand, a real number is a value that represents a quantity along a continuous line, which includes all numbers without imaginary components.

Step 2: Confirm if a complex number with no imaginary part is a real number

Because the imaginary part (\(b\)) here is 0, the complex number becomes \(a+0i\). This essentially equals \(a\), because adding 0 to any number doesn't affect its value. Therefore, a complex number with no imaginary part is identical to a real number.

Step 3: Conclusion

After analyzing, we can confirm that the statement 'The complex number \(a+0 i\) is the real number \(a\)' is indeed true. This is so because when the imaginary component (\(b\)) of a complex number is zero, the complex number is equivalent to a real number.

Key Concepts

Real Numbers

Real numbers are fundamental in mathematics, representing all possible magnitudes on a continuous line. This concept includes both positive and negative numbers, as well as zero. Essentially, any number that doesn't have an imaginary part can be categorized as a real number. This means:

• Integers like -2, 0, and 5
• Fractions such as 3/4
• Decimals including 0.75
• Irrational numbers like √2

Real numbers form the backbone of many mathematical concepts and are used extensively in everyday calculations. When a complex number has no imaginary component (i.e., its imaginary part is zero), it simplifies to a real number. Hence, complex numbers that are written as \(a + 0i\) are merely real numbers \(a\). This is because the absence of the imaginary component leaves us with the real portion alone.
The continuous nature of real numbers allows them to be graphed along a number line, making them useful in defining all possible positions in both positive and negative directions. Thus, understanding real numbers is crucial for further exploration into mathematics.

Functions

Functions are mathematical relations where every input has a distinct output. Often denoted as \(f(x)\), a function can be defined by various rules, including algebraic equations. A key characteristic of functions is that each input corresponds to precisely one output, which sets them apart from other types of relationships.
Consider a real-world example: calculating the cost of apples based on their weight. If the cost per kilogram is fixed at \(3, then buying 2 kg would always cost \)6, establishing a predictable function \(f(x) = 3x\). Here:

• Input (\(x\)): weight of apples in kg
• Output (\(f(x)\)): total cost

This unique one-to-one mapping is what defines a function. When dealing with complex numbers, functions can still be applied, adding dimensions in which they operate. But remember that for a relation to qualify as a function, each input must map to a single output. This critical understanding is foundational in not just mathematics, but also in various scientific and engineering applications.

Imaginary Component

The imaginary component of a number is what distinguishes between real and complex numbers. In complex numbers, we express them as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary component multiplied by \(i\). Here, \(i\) is defined as the square root of -1, often making it an abstract concept to grasp at first.
Despite its elusive nature, the imaginary component is essential for handling problems involving two dimensions, such as in electrical engineering or quantum physics. While real numbers represent quantities along a continuous line, the imaginary part opens up a whole new dimension. Together, they allow complex numbers to describe rotations and oscillations more effectively than real numbers alone.
When dealing with exercises that involve determining the nature of numbers, remember that if the imaginary component \(b\) in \(a + bi\) equals zero, the complex number simplifies to the real number \(a\). This insight helps in recognizing when a number, initially expressed in complex form, actually lies on the real number line, devoid of an imaginary dimension.