Problem 3 - Elementary and Intermediate Algebra

Question

In each of Exercises \(1-8\) fill in the blank using one of the following terms: natural number, whole number, integer, rational number, tern ting irrational. If a number is a(n)__________ , it is either a whole number or the opposite of a whole number.

Step-by-Step Solution

Answer

Integer

Step 1 - Understanding the Question

Identify the given statement: 'If a number is a(n) __________, it is either a whole number or the opposite of a whole number.'

Step 2 - Define Whole Numbers

Whole numbers are the numbers {0, 1, 2, 3, ...}. They do not include negative numbers or fractions.

Step 3 - Opposite of Whole Numbers

The opposite of a whole number means taking the negative of the whole number set. This includes numbers like {-1, -2, -3, ...}.

Step 4 - Identify the Answer

A number that is either a whole number or the opposite of a whole number includes both non-negative whole numbers and their negative counterparts. This set of numbers is known as integers.

Key Concepts

Whole Number

A whole number is essentially any number from the set {0, 1, 2, 3, ...}.
Whole numbers are non-negative and do not include any decimal or fractional parts.
For example, 5 is a whole number, but 5.5 or 5/2 are not.

Whole numbers are useful in various real-life situations, such as counting objects or representing exact quantities. They also serve as the building blocks for more complex mathematical concepts.

It’s important to note that whole numbers are also a part of larger sets of numbers:

• All whole numbers are also natural numbers (excluding 0), which are used mainly for counting and ordering.
• Whole numbers fall under the category of integers, which we’ll explore further.

Rational Number

A rational number is any number that can be expressed as the ratio of two integers.
This means it can be written in the form \(\frac{a}{b}\) where both a and b are integers and b is not zero.
Rational numbers include:

• Whole numbers and integers (since they can be written as \(\frac{n}{1}\), where n is an integer)
• Fractions (like \(\frac{3}{4}\) or \(\frac{-2}{5}\))
• Terminating decimals (such as 0.75, which can be written as \(\frac{3}{4}\))
• Repeating decimals (like 0.333..., which can be expressed as \(\frac{1}{3}\))

Understanding rational numbers broadens our ability to deal with various numerical values in both theoretical math and practical scenarios, like dividing a pizza among friends.

Opposite Number

The opposite of a number, often called its additive inverse, is the number that, when added to the original number, results in zero.
For any whole number n, its opposite is -n.
For instance:

• The opposite of 3 is -3.
• The opposite of -7 is 7.
• The opposite of 0 is still 0; 0 is its own opposite.

Understanding opposite numbers helps in solving equations and understanding symmetry in mathematics.
In our exercise, the concept of the opposite of a whole number helps to define integers.
The set of integers includes both the whole numbers and their opposites.
This is why if a number is either a whole number or the opposite of a whole number, it must be an integer.