Problem 15 - Algebra 2

Question

Name the sets of numbers to which each number belongs. $$ -4.55 $$

Step-by-Step Solution

Answer

-4.55 is a rational number.

Step 1: Identify the Type of Number

The number given is \(-4.55\). This is a decimal number, not a whole number or integer.

Step 2: Determine if It is a Rational Number

A rational number is any number that can be expressed as the quotient of two integers. Since \(-4.55\) can be expressed as \(-\frac{455}{100}\), it is a rational number.

Step 3: Check Against Other Sets

Now that we know\(-4.55\) is a rational number, it is not a natural number, whole number, or integer because natural numbers and whole numbers are non-negative, and integers do not have a fractional part.

Key Concepts

Rational Numbers

Rational numbers are a special group of numbers that can be written as a fraction, where both the numerator and the denominator are integers. The numerator should be an integer, and the denominator cannot be zero. This means any number that fits this form is considered rational.
For example, the number 0.5 is a rational number because it can be written as \( \frac{1}{2} \). Similarly, the number \(-4.55\) is also rational because it can be expressed as \( \frac{-455}{100} \). 
This shows that rational numbers include integers, fractions, and terminating or repeating decimals. To check if a number is rational, see if you can write it as a simple fraction with integers.

Integers

Integers are a set of numbers that include all positive whole numbers, negative whole numbers, and zero. Essentially, if a number doesn't have a decimal or fractional part, it qualifies as an integer.
Examples of integers include -3, 0, and 7. This set does not include numbers like 0.5 or -4.55 because these have decimal places.
So, remember that while every integer is also a rational number (since it can be expressed as a fraction with a denominator of 1), not every rational number is an integer.

Whole Numbers

Whole numbers are a set of non-negative numbers that include zero and extend to positive numbers without stopping. The most obvious examples of whole numbers are 0, 1, 2, 3, and so on.
Whole numbers do not include fractions, decimals, or negative numbers. Therefore, while numbers like 5 or 0 are whole numbers, \(-4.55\) or even \(-2\) are not considered whole numbers.
Whole numbers are included in the set of integers and, by extension, in the set of rational numbers, because they can be represented as fractions.

Natural Numbers

Natural numbers are often referred to as the counting numbers. They are a part of whole numbers but without zeros. This makes them the set of positive whole numbers such as 1, 2, 3, and so on.
Natural numbers do not include negative numbers, decimals, fractions, or zero. Therefore, numbers like 5 or 102 are natural numbers, whereas \(-4.55\) is not.
Natural numbers are considered a subset of integers, so they are rational numbers as well, because we can represent them with fractions like \( \frac{4}{1} \). However, not every rational number is natural.