any number that can be expressed as the ratio of 2 integers a/b, where b does not equal 0, is a rational number
when you divide the numerator of a fraction by the denominator you always get either a decimal that terminates or a decimal that repeats a digit or block of digits again and again
To understand why any number that can be expressed as the ratio of two integers is a rational number, we need to first understand what a rational number is.
A rational number is any number that can be written as a fraction with an integer numerator (the number on the top) and a non-zero integer denominator (the number on the bottom). In other words, it is a number that can be expressed as the ratio of two integers, where the denominator is not zero.
So, let's consider a number that can be expressed as the ratio of two integers, a/b, where b is not equal to zero. Since both a and b are integers, we know that the numerator and denominator are whole numbers.
Now, if we divide a by b, we will get a decimal representation. This decimal might be finite or it might be an infinite repeating decimal. However, regardless of its decimal representation, we can always express it as a fraction by writing it as a/b.
For example, let's consider the number 2.5. We can express it as a fraction by writing it as 5/2. Similarly, the number 0.75 can be written as 3/4, and so on. In each case, we are expressing a decimal number as a ratio of two integers.
Therefore, any number that can be expressed as the ratio of two integers (a/b) where b is not equal to zero is a rational number.