Problem 119 - Intermediate Algebra

Question

True or false: The real numbers is the set of all decimals.

Step-by-Step Solution

Answer

True, the real numbers include all decimals.

Step 1: Define Real Numbers

Real numbers include all rational numbers, such as integers and fractions, and all irrational numbers, which do not terminate or repeat, for example, \( \pi \, \sqrt{2} \, \text{etc.} \).

Step 2: Define Decimal Numbers

Decimal numbers can be either terminating, like 0.5, or non-terminating but repeating, like 0.333... However, decimal representation can also express irrational numbers, such as \( \pi \, \text{which is approximately} \, 3.14159... \).

Step 3: Compare Real Numbers and Decimals

Every real number can be expressed as a decimal, though not all decimals are simple rational numbers. Real numbers encompass both simple decimals (like 1.2) and complex non-repeating decimal forms (like \( \pi \). Hence, the set of real numbers indeed includes all possible decimals.

Step 4: Evaluate the Statement

The statement "The real numbers is the set of all decimals" holds true because any real number has a decimal representation, whether it is simple, non-terminating, repeating, or non-repeating.

Key Concepts

Decimal Numbers

Decimal numbers are numbers expressed in a format that features a decimal point. This format separates the whole number part from the fractional part. Decimal numbers can be of two main types:

• **Terminating Decimals**: These are decimals that come to an end. For example, 5.75 is a decimal that stops at two decimal places. Here, there is no infinity in the representation.
• **Repeating Decimals**: These numbers extend infinitely with a repeating pattern. An example is 0.666..., where the digit '6' repeats forever.

Despite these two categories, there are also non-terminating, non-repeating decimals. Notably, such numbers are associated with irrational numbers because their decimal form never ends or settles into a repeat pattern. For instance, the number π (pi) seems to go on forever without repeating, making it an example of a non-repeating decimal.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In mathematical terms, if you can write a number as \( \frac{a}{b} \) (where 'a' and 'b' are integers and 'b' is not zero), it is considered a rational number.

• **Example of Rational Numbers**: The number 3 is rational because it can be expressed as \( \frac{3}{1} \). Similarly, 0.5 is also rational because it can be represented as \( \frac{1}{2} \).
• **Decimal Representation**: Rational numbers can be either terminating or repeating when expressed as decimals. For instance, \( \frac{1}{4} = 0.25 \) which is terminating, while \( \frac{1}{3} = 0.333... \) is repeating.

Thus, any decimal number that either terminates or repeats can be considered rational.

Irrational Numbers

An irrational number is a number that cannot be expressed as a simple fraction, which means its decimal form is neither terminating nor repeating. This makes them quite special as they cannot be written exactly, only approximated.

• **Examples**: Famous irrational numbers include \( \pi \) (pi), which is approximately equal to 3.14159, and the square root of 2, written as \( \sqrt{2} \), roughly approximating to 1.41421.
• **Characteristics of Irrational Numbers**: Unlike rational numbers, their decimal expansions go on forever without finding a pattern to repeat. This unique trait makes identifying irrational numbers straightforward.

Irrational numbers fill the gaps between rational numbers on the number line, contributing to the completeness of the real number system.