Problem 4 - Precalculus Student Solutions Manual 5th

Question

Determine whether each statement is true or false. If it is false, tell why. A number can be both real and complex.

Step-by-Step Solution

Answer

The statement is true. Real numbers are a subset of complex numbers.

Step 1: Understand definitions

Review the definitions of real numbers and complex numbers. A real number can be any number on the number line, while a complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with \(i^2 = -1\).

Step 2: Identify overlap

Observe that complex numbers include all real numbers when the imaginary component \(b\) is zero. Thus every real number is also a complex number with an imaginary part of zero.

Step 3: Determine validity of the statement

Recognize that a number can be real (if \(b=0\)) and still be considered a complex number. Therefore, the statement is true.

Key Concepts

Real Numbers

Real numbers are the numbers you often deal with in everyday life. They include:

• Natural numbers (like 1, 2, 3)
• Whole numbers (like 0, 1, 2, 3)
• Integers (like ...-2, -1, 0, 1, 2,...)
• Rational numbers (like fractions, 1/2, 3/4)
• Irrational numbers (like square roots and pi, \(\pi\))

All these types of numbers can be placed on a number line. This is a good way to think about real numbers—they have a specific place on a continuous line that stretches infinitely in both directions.

Imaginary Numbers

Imaginary numbers are a little bit trickier. They stem from the need to solve equations where taking the square root of a negative number is required. A common example is the square root of -1. Normally, we can't find such a number among the set of real numbers. To address that, mathematicians created an imaginary unit 'i', where \(i^2 = -1\).
Imaginary numbers are represented as multiples of 'i'. For instance:

• \(2i\)
• \(3.5i\)
• \(-5i\)

These numbers can't be placed on the same number line as real numbers. Instead, they are visualized on a plane called the complex plane.

Number Classification

Number classification helps in organizing different types of numbers under broad categories. A number can belong to more than one category:

• Real Numbers: These can be further split into rational and irrational numbers.
• Complex Numbers: These are of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

Notice how a real number is also technically a complex number. This is because if we set \(b = 0\) in the equation \(a + bi\), we simply get \(a\), a real number. Therefore, all real numbers can be considered complex numbers with no imaginary part. This makes the original statement 'A number can be both real and complex' true.