Yes, the difference between two rational numbers is always a rational number. To understand why, let’s first define what a rational number is.
A rational number is any number that can be expressed as the quotient or fraction a/b, where a is an integer and b is a non-zero integer. For example, 1/2, -3/4, and 5 (which can be written as 5/1) are all rational numbers.
Now, let’s denote two rational numbers as follows:
- rational number 1 (a/b)
- rational number 2 (c/d)
Here, a, b, c, and d are integers, and b and d are not zero. The difference between these two rational numbers can be calculated as:
Difference = (a/b) – (c/d)
To subtract these two fractions, we need a common denominator. The common denominator of b and d is bd. Therefore, we rewrite the fractions as:
Difference = (a*d)/(b*d) – (c*b)/(d*b)
Now, we can perform the subtraction:
Difference = (ad – cb) / (bd)
In this result, ad – cb is an integer because both ad and cb are products of integers. Additionally, the denominator bd is also an integer and non-zero (since b and d are not zero).
Thus, we can conclude that the result of the subtraction is in the form of integer/integer, meaning that the difference is a rational number!
This property of rational numbers is essential in various fields, including mathematics, science, and even finance, where precise calculations and the ability to understand relationships between values are crucial.
In summary, whenever you subtract one rational number from another, you always end up with another rational number, ensuring that the realm of rational numbers remains closed under subtraction.