Problem 116 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some real numbers are not rational numbers.

Step-by-Step Solution

Answer

The statement 'some real numbers are not rational numbers' is true.

Step 1: Define Real Numbers

Real numbers include all the numbers on the number line, both rational and irrational numbers. These are numbers that can be either positive, negative, or zero and they can be written as a decimal that may either terminate or recur, or not terminate or recur (those are irrational numbers).

Step 2: Define Rational Numbers

Rational numbers are a subset of the real numbers. Rational numbers can be numbers, fractions or decimals that can be expressed in the form \(\frac{a}{b}\), where 'a' and 'b' are integers and 'b' does not equal 0. Rational numbers, when written in decimal form, are either terminating or recurring.

Step 3: Compare and Evaluate the Statement

On comparing the definitions of real numbers and rational numbers, it becomes clear that while all rational numbers are real numbers, not all real numbers are rational. There is a set of real numbers, called irrational numbers, which cannot be expressed as the ratio of two integers. Therefore, the statement 'some real numbers are not rational numbers' is indeed correct and therefore true.

Key Concepts

Rational Numbers

Rational numbers are a fascinating part of the number universe. They include integers, fractions, and decimals that can be expressed as a ratio of two integers. For example, the number \( \frac{1}{2} \) is rational because it can be expressed as a ratio where the numerator is 1 and the denominator is 2.

When written in decimal form, rational numbers either terminate or repeat after a certain point. For example, 0.75 is a terminating decimal, while 0.333... (which equals \( \frac{1}{3} \)) is a repeating decimal.

To summarize, the characteristics of rational numbers include:

• They can be positive, negative, or zero.
• They must be expressible as \( \frac{a}{b} \), where 'a' and 'b' are integers, and \( b eq 0 \).
• Decimal representations are either terminating or repeating.

Irrational Numbers

Irrational numbers might seem mysterious at first, but they're simply real numbers that cannot be written as a simple fraction or ratio of two integers. Examples include \( \pi \) (pi) and \( \sqrt{2} \), which cannot be exactly expressed as a ratio of two integers.

Unlike rational numbers, the decimal form of an irrational number neither terminates nor repeats. This makes them unique and intriguing, as they continue indefinitely without a predictable pattern.

Key points about irrational numbers include:

• They cannot be expressed as \( \frac{a}{b} \), where 'a' and 'b' are integers.
• Their decimal forms go on forever without repeating.
• They fill in the gaps between rational numbers on the number line.

Number Line

The number line is a visual representation of all real numbers, both rational and irrational. It extends infinitely in both directions and includes every conceivable number you can imagine.

On the number line, every point corresponds to a real number. Rational numbers are neatly placed at specific points, while irrational numbers often fill in the gaps, ensuring there are no voids.

Understanding the number line helps us see:

• All real numbers have a place on the line, including both rational and irrational numbers.
• Integers like -3, 0, and 4 are also marked as specific points.
• The line is a continuous representation, reflecting the endless possibilities of real numbers.

This concept aids in realizing how vast and encompassing the world of numbers truly is.