Although we have noticed this pattern using only the examples above, it is true for all rationals of the form `p/q \ \ (q ≠ 0).`
On division of p by q, two main things happen either the remainder becomes zero or never becomes zero and we get a repeating string of remainders. Let us look at each case separately.
`color{fuchsia}color({fuchsia}("Case (i) : The remainder becomes zero")`
In the examples of `1/3` and `1/7,` we notice that the remainders repeat after a certain stage forcing the decimal expansion to go on for ever. In other words, we have a repeating block of digits in the quotient. We say that this expansion is non-terminating recurring. For example `color{green}(1/3 = 0.3333..)` and `color{green}(1/7 = 0.142857142857142857...)`
The usual way of showing that 3 repeats in the quotient of `1/3` is to write it as `bar0.3` Similarly, since the block of digits 142857 repeats in the quotient of `1/7` we write `1/7` as `bar(0.142857)` where the bar above the digits indicates the block of digit Also 3.57272... can be written as `bar(3.572)` So, all these examples give us non-terminating recurring (repeating) decimal expansions.
Thus, we see that the decimal expansion of rational numbers have only two choices: either they are terminating or non-terminating recurring.
Now suppose, on the other hand, on your walk on the number line, you come across a number like `3.142678` whose decimal expansion is terminating or a number like `1.272727...` that is, `bar(1.27)` , whose decimal expansion is non-terminating recurring, can you conclude that it is a rational number? The answer is yes!
We will not prove it but illustrate this fact with a few examples. The terminating cases are easy.
Although we have noticed this pattern using only the examples above, it is true for all rationals of the form `p/q \ \ (q ≠ 0).`
On division of p by q, two main things happen either the remainder becomes zero or never becomes zero and we get a repeating string of remainders. Let us look at each case separately.
`color{fuchsia}color({fuchsia}("Case (i) : The remainder becomes zero")`
In the examples of `1/3` and `1/7,` we notice that the remainders repeat after a certain stage forcing the decimal expansion to go on for ever. In other words, we have a repeating block of digits in the quotient. We say that this expansion is non-terminating recurring. For example `color{green}(1/3 = 0.3333..)` and `color{green}(1/7 = 0.142857142857142857...)`
The usual way of showing that 3 repeats in the quotient of `1/3` is to write it as `bar0.3` Similarly, since the block of digits 142857 repeats in the quotient of `1/7` we write `1/7` as `bar(0.142857)` where the bar above the digits indicates the block of digit Also 3.57272... can be written as `bar(3.572)` So, all these examples give us non-terminating recurring (repeating) decimal expansions.
Thus, we see that the decimal expansion of rational numbers have only two choices: either they are terminating or non-terminating recurring.
Now suppose, on the other hand, on your walk on the number line, you come across a number like `3.142678` whose decimal expansion is terminating or a number like `1.272727...` that is, `bar(1.27)` , whose decimal expansion is non-terminating recurring, can you conclude that it is a rational number? The answer is yes!
We will not prove it but illustrate this fact with a few examples. The terminating cases are easy.