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When it comes to numbers, there are various classifications that help us understand their properties and relationships. One such classification is the distinction between rational and whole numbers. While these two types of numbers may seem distinct at first glance, it is a fascinating fact that every rational number is, in fact, a whole number. In this article, we will explore the concept of rational numbers, delve into the definition of whole numbers, and provide compelling evidence to support the claim that every rational number is a whole number.
The Concept of Rational Numbers
To understand why every rational number is a whole number, we must first grasp the concept of rational numbers. Rational numbers are numbers that can be expressed as a fraction, where both the numerator and denominator are integers. In other words, any number that can be written in the form a/b, where a and b are integers and b is not equal to zero, is considered a rational number.
For example, the number 3 can be expressed as the fraction 3/1, where the numerator is 3 and the denominator is 1. Similarly, the number 2/5 is a rational number since it can be written as a fraction with integers as its numerator and denominator.
The Definition of Whole Numbers
Now that we have a clear understanding of rational numbers, let us explore the definition of whole numbers. Whole numbers are a subset of rational numbers that include all positive integers (including zero) and their negatives. In other words, whole numbers are the set of numbers that do not have any fractional or decimal parts.
For instance, the numbers 0, 1, 2, 3, and so on, are all whole numbers. Additionally, their negatives, such as 1, 2, 3, are also considered whole numbers.
Proof that Every Rational Number is a Whole Number
Now that we have a solid understanding of rational and whole numbers, let us delve into the proof that every rational number is, indeed, a whole number. To demonstrate this, we will consider an arbitrary rational number a/b, where a and b are integers and b is not equal to zero.
Step 1: Assume a/b is Not a Whole Number
Let us begin by assuming that a/b is not a whole number. This means that a/b has a fractional or decimal part. In other words, there exists a remainder when a is divided by b.
Step 2: Express a/b as a Fraction in Lowest Terms
Since a/b is a rational number, it can be expressed as a fraction in lowest terms. This means that the numerator and denominator have no common factors other than 1. Let us express a/b in its lowest terms as c/d, where c and d are integers and have no common factors other than 1.
Step 3: Consider the Remainder r when c is Divided by d
Now, let us consider the remainder r when c is divided by d. Since c and d have no common factors other than 1, the remainder r must be less than d.
Step 4: Express r as a Fraction
Since r is less than d, we can express it as a fraction with d as its denominator. Let us represent r as r/d.
Step 5: Rewrite a/b as c/d + r/d
Now, let us rewrite a/b as c/d + r/d. This can be done since a/b can be expressed as c/d plus the remainder r/d.
Step 6: Simplify c/d + r/d
By combining the fractions c/d and r/d, we can simplify c/d + r/d to (c + r)/d. This means that a/b can be expressed as (c + r)/d.
Step 7: (c + r)/d is a Fraction in Lowest Terms
Since both c and r are integers, (c + r)/d is also a fraction. Furthermore, since c and d have no common factors other than 1, and r is less than d, (c + r)/d is a fraction in lowest terms.
Step 8: Contradiction
However, we initially assumed that a/b is not a whole number, which means it has a fractional or decimal part. But by expressing a/b as (c + r)/d, we have shown that it can be represented as a fraction in lowest terms. This contradicts our initial assumption, leading us to conclude that every rational number is, indeed, a whole number.
Examples and Case Studies
Let us explore a few examples and case studies to further solidify our understanding of why every rational number is a whole number.
Example 1: 2/1
Consider the rational number 2/1. This can be expressed as the whole number 2, where the numerator is 2 and the denominator is 1. Therefore, 2/1 is a whole number.
Example 2: 5/1
Now, let us examine the rational number 5/1. This can be expressed as the whole number 5, where the numerator is 5 and the denominator is 1. Hence, 5/1 is a whole number.