Problem 63 - Prealgebra

Question

Identify all sets to which each number belongs. $$0 . \overline{8}$$

Step-by-Step Solution

Answer

The number belongs to the sets of rational and real numbers.

Step 1: Understand the Representation

The number \(0.\overline{8}\) means the digit 8 repeats indefinitely after the decimal point. It can be written as \(0.8888\ldots\). This is a non-terminating repeating decimal.

Step 2: Identify as a Rational Number

Any non-terminating repeating decimal can be expressed as a fraction, making \(0.\overline{8}\) a rational number. Specifically, \(0.\overline{8}\) can be expressed as the fraction \(\frac{8}{9}\).

Step 3: Identify as a Real Number

Since rational numbers are a subset of real numbers, \(0.\overline{8}\) is also considered a real number. It can be represented on the real number line.

Step 4: Exclude from Integers, Whole Numbers, and Natural Numbers

Integers include whole numbers and their negatives, whole numbers include zero and natural numbers include positive counting numbers. Since \(0.\overline{8}\) is a decimal, it does not belong to the sets of integers, whole numbers, or natural numbers.

Key Concepts

Real Numbers

Real numbers are all the numbers that we can locate on an infinite number line. They include both rational and irrational numbers. Rational numbers are numbers that can be expressed as fractions of two integers, like \( \frac{8}{9} \) for example, while irrational numbers cannot be expressed in such a form, like \( \sqrt{2} \).

Both these types of numbers combined make up the real numbers, which encompass almost every number you encounter in everyday life.When we talk about real numbers, we include:

• Positive numbers, like 3 or 4.5.
• Negative numbers, like -1 or -0.25.
• Zero (0).
• Decimals, like 0.8888…, especially non-terminating ones like \(0.\overline{8}\).
• Special constants, such as \( \pi \) and \( e \).

Even though irrational numbers are part of the real numbers, all rational numbers such as \(0.\overline{8}\) can also be classified as real numbers.

Decimals

Decimals are a way to represent numbers that are not whole. They make use of a point called the decimal point to separate the whole part of a number from its fractional part. In the numeral system, the position of digits after the decimal point represents powers of ten. For instance, in the number 0.8888, the decimal point tells us we are dealing with parts of a whole.

Decimals can be:

• Terminating decimals: These decimals come to an end. e.g., 0.75 or 1.25.
• Non-terminating repeating decimals: Here, digits repeat infinitely. e.g., \(0.\overline{8}\), where 8 repeats in perpetuity.
• Non-terminating non-repeating decimals: These decimals go on forever without any repeating pattern, like \( \pi \).

Understanding decimals is key because they help in measuring things that aren't whole, like parts of a meter or a liter. The repeated decimal \(0.\overline{8}\) is an excellent example of creating a fraction from a repeating decimal sequence, which results in \( \frac{8}{9} \).

As decimals fit into the set of real numbers, they help us provide precise measurements beyond whole numbers.

Number Sets

Number sets are ways to classify and organize different kinds of numbers based on their properties. Each set has its unique characteristics, some of which may overlap with others. This helps in understanding where a given number belongs. Here's a quick breakdown of the main number sets:

• Natural Numbers: These are counting numbers, including 1, 2, 3, etc. They do not include zero or any decimals.
• Whole Numbers: This set includes all natural numbers plus zero, so 0, 1, 2, 3, etc.
• Integers: Integers expand whole numbers to include their negative counterparts, such as -3, -2, -1, 0, 1, 2, etc.
• Rational Numbers: These numbers can be expressed as fractions, like \(\frac{2}{3}\) or \(\frac{-5}{4}\), and include both terminating and repeating decimals like \(0.\overline{8}\).
• Irrational Numbers: These cannot be expressed as simple fractions; examples include \(\sqrt{3}\) and \(\pi\).
• Real Numbers: Combining both rational and irrational numbers, this is the most inclusive set often used in everyday calculations.

It's important to understand that each number has a specific place or places in these sets. A number like \(0.\overline{8}\) is not just a rational and real number, but it doesn't qualify as a natural, whole, or integer number due to its decimal nature. So recognizing its position in several but not all sets gives more insights into how that number functions and is classified.