Revisiting Rational Numbers and Their Decimal Expansions
● You studied that rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion. In this section, we are going to consider a rational number, say `p/q ( q ne 0)` , and explore exactly when the decimal expansion of `p/q` is terminating and when it is non-terminating repeating (or recurring). We do so by considering several examples.
● Let us consider the following rational numbers :
(i) 0.375 (ii) 0.104 (iii) 0.0875 (iv) 23.3408.
Now (i) `0.375 = 375/1000 = 375/(10)^3` , (ii) `0.104 = 104/1000 = 104/(10)^3`
(iii) `0.0875 = 875/10000 = 875/(10)^4` (iv) `23.3408 = (233408)/(10000) = (233408)/(10)^4`
● As one would expect, they can all be expressed as rational numbers whose denominators are powers of 10. Let us try and cancel the common factors between the numerator and denominator and see what we get :
(i) `0.375 = 375/(10)^3 = (3xx5^3)/(2^3 xx 5^3) = 3/2^3`
(ii) `0.104 = 104/(10)^3 = (13xx2^3)/(2^3 xx 5^3) = 13/5^3`
(iii) `0.0875 = 875/(10)^4 = 7/(2^4 xx 5)`
(iv) `23.3408 = (233408)/(10)^4 = (2^2xx7xx521)/(5^4)`
● It appears that, we have converted a real number whose decimal expansion terminates into a rational number of the form `p /q` where `p` and `q` are coprime, and the prime factorisation of the denominator (that is, q) has only powers of 2, or powers of 5, or both. We should expect the denominator to look like this, since powers of 10 can only have powers of 2 and 5 as factors.
● Even though, we have worked only with a few examples, you can see that any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10. Also the only prime factors of 10 are 2 and 5. So, cancelling out the common factors between the numerator and the denominator, we find that this real number is a rational number of the form `p/ q` where the prime factorisation of q is of the form `2^n 5^m`, and `n, m `are some non-negative integers.
Let us write our result formally:
`color{blue}{"Theorem 1.5 : Let x be a rational number whose decimal expansion terminates."}`
`=>` Then x can be expressed in the form `p/q` where p and q are coprime, and the prime factorisation of q is of the form `2^n 5^m`, where n, m are non-negative integers.
● You are probably wondering what happens the other way round in Theorem 1.5. That is, if we have a rational number of the form `p/q` and the prime factorisation of q is of the form `2^n 5^m`, where n, m are non negative integers, then does `p/q` have a terminating decimal expansion?
●Let us see if there is some obvious reason why this is true. You will surely agree that any rational number of the form `a/b` where b is a power of 10, will have a terminating decimal expansion. So it seems to make sense to convert a rational number of the form `p/q` where q is of the form `2^n 5^m`, to an equivalent rational number of the form `a/b` where b is a power of 10. Let us go back to our examples above and work backwards.
(i) `3/8 = 3/2^3 = (3 xx 5^3)/(2^3 xx 5^3) = 375/10^3 = 0.375`
(ii) `13/125 = 13/5^3 = (13xx2^2)/(2^3 xx 5^3) = 104/10^3 = 0.104`
(iii) `7/80 = 7/(2^4 xx 5) = (7 xx 5^3)/(2^4 xx 5^4) = 875/10^4 = 0.0875`
(iv) `14588/625 = (2^2 xx 7 xx521)/5^4 = ( 2^6 xx 7 xx 521)/(2^4 xx 5^4) = (233408)/(10^4) = 23.3408`
● So, these examples show us how we can convert a rational number of the form `p/q` where q is of the form `2^n 5^m`, to an equivalent rational number of the form `a/b` where b is a power of 10. Therefore, the decimal expansion of such a rational number terminates. Let us write down our result formally.
`color{blue}{"Theorem 1.6 :"}` Let `x = p/q` be a rational number, such that the prime factorisation of q is of the form `2^n 5^m`, where n, m are non-negative integers. Then x has a decimal expansion which terminates.
`=>` We are now ready to move on to the rational numbers whose decimal expansions are non-terminating and recurring. Once again, let us look at an example to see what is going on. We refer to Example 5, Chapter 1, from your Class IX textbook, namely `1/7` Here, remainders are `3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1, . . .` and divisor is` 7`.
`=>` Notice that the denominator here, i.e., 7 is clearly not of the form `2^n 5^m`. Therefore, from Theorems 1.5 and 1.6, we
know that `1/7` will not have a terminating decimal expansion. Hence, 0 will not show up as a remainder (Why?), and the remainders will start repeating after a certain stage. So, we will have a block of digits, namely, 142857, repeating in the
quotient of `1/7`. What we have seen, in the case of `1/4` is true for any rational number not covered by Theorems 1.5 and 1.6. For such numbers we have :
`color{blue}{"Theorem 1.7 :"}` Let `x = p/q` be a rational number, such that the prime factorisation of q is not of the form `2^n 5^m`, where n, m are non negative integers. Then, `x` has a decimal expansion which is non-terminating repeating (recurring).
`=>` From the discussion above, we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.
.GIF)
● Let us consider the following rational numbers :
(i) 0.375 (ii) 0.104 (iii) 0.0875 (iv) 23.3408.
Now (i) `0.375 = 375/1000 = 375/(10)^3` , (ii) `0.104 = 104/1000 = 104/(10)^3`
(iii) `0.0875 = 875/10000 = 875/(10)^4` (iv) `23.3408 = (233408)/(10000) = (233408)/(10)^4`
● As one would expect, they can all be expressed as rational numbers whose denominators are powers of 10. Let us try and cancel the common factors between the numerator and denominator and see what we get :
(i) `0.375 = 375/(10)^3 = (3xx5^3)/(2^3 xx 5^3) = 3/2^3`
(ii) `0.104 = 104/(10)^3 = (13xx2^3)/(2^3 xx 5^3) = 13/5^3`
(iii) `0.0875 = 875/(10)^4 = 7/(2^4 xx 5)`
(iv) `23.3408 = (233408)/(10)^4 = (2^2xx7xx521)/(5^4)`
● It appears that, we have converted a real number whose decimal expansion terminates into a rational number of the form `p /q` where `p` and `q` are coprime, and the prime factorisation of the denominator (that is, q) has only powers of 2, or powers of 5, or both. We should expect the denominator to look like this, since powers of 10 can only have powers of 2 and 5 as factors.
● Even though, we have worked only with a few examples, you can see that any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10. Also the only prime factors of 10 are 2 and 5. So, cancelling out the common factors between the numerator and the denominator, we find that this real number is a rational number of the form `p/ q` where the prime factorisation of q is of the form `2^n 5^m`, and `n, m `are some non-negative integers.
Let us write our result formally:
`color{blue}{"Theorem 1.5 : Let x be a rational number whose decimal expansion terminates."}`
`=>` Then x can be expressed in the form `p/q` where p and q are coprime, and the prime factorisation of q is of the form `2^n 5^m`, where n, m are non-negative integers.
● You are probably wondering what happens the other way round in Theorem 1.5. That is, if we have a rational number of the form `p/q` and the prime factorisation of q is of the form `2^n 5^m`, where n, m are non negative integers, then does `p/q` have a terminating decimal expansion?
●Let us see if there is some obvious reason why this is true. You will surely agree that any rational number of the form `a/b` where b is a power of 10, will have a terminating decimal expansion. So it seems to make sense to convert a rational number of the form `p/q` where q is of the form `2^n 5^m`, to an equivalent rational number of the form `a/b` where b is a power of 10. Let us go back to our examples above and work backwards.
(i) `3/8 = 3/2^3 = (3 xx 5^3)/(2^3 xx 5^3) = 375/10^3 = 0.375`
(ii) `13/125 = 13/5^3 = (13xx2^2)/(2^3 xx 5^3) = 104/10^3 = 0.104`
(iii) `7/80 = 7/(2^4 xx 5) = (7 xx 5^3)/(2^4 xx 5^4) = 875/10^4 = 0.0875`
(iv) `14588/625 = (2^2 xx 7 xx521)/5^4 = ( 2^6 xx 7 xx 521)/(2^4 xx 5^4) = (233408)/(10^4) = 23.3408`
● So, these examples show us how we can convert a rational number of the form `p/q` where q is of the form `2^n 5^m`, to an equivalent rational number of the form `a/b` where b is a power of 10. Therefore, the decimal expansion of such a rational number terminates. Let us write down our result formally.
`color{blue}{"Theorem 1.6 :"}` Let `x = p/q` be a rational number, such that the prime factorisation of q is of the form `2^n 5^m`, where n, m are non-negative integers. Then x has a decimal expansion which terminates.
`=>` We are now ready to move on to the rational numbers whose decimal expansions are non-terminating and recurring. Once again, let us look at an example to see what is going on. We refer to Example 5, Chapter 1, from your Class IX textbook, namely `1/7` Here, remainders are `3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1, . . .` and divisor is` 7`.
`=>` Notice that the denominator here, i.e., 7 is clearly not of the form `2^n 5^m`. Therefore, from Theorems 1.5 and 1.6, we
know that `1/7` will not have a terminating decimal expansion. Hence, 0 will not show up as a remainder (Why?), and the remainders will start repeating after a certain stage. So, we will have a block of digits, namely, 142857, repeating in the
quotient of `1/7`. What we have seen, in the case of `1/4` is true for any rational number not covered by Theorems 1.5 and 1.6. For such numbers we have :
`color{blue}{"Theorem 1.7 :"}` Let `x = p/q` be a rational number, such that the prime factorisation of q is not of the form `2^n 5^m`, where n, m are non negative integers. Then, `x` has a decimal expansion which is non-terminating repeating (recurring).
`=>` From the discussion above, we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.