Depending on the value of `n`, the decimal for `1/n` can take one of three forms:
Type A: It may terminate after a certain number of digits
for example `1/5 = 0.2 \ \ \ \ \ \ \ \ \ 1/16=0.0625 \ \ \ 1/25=0.04`
Type B: It may recur after a certain number of digits
for example `1/3 = 0.bar(3) \ \ \ \ \ \ \ \ 1/11=0.bar(09) \ \ \ \ \ \ 1/7=0.bar(142857)`
Type C: It may be "hybrid" ie start with some non-recurring digits, then continue with a certain number of recurring digits
for example `1/12 = 0.08bar(3) \ \ \ 1/55=0.0bar(18) \ \ \ \ 1/296=0.003bar(378)`
The question is, for a given value of n, how can we tell whether `1/n` will be of type A, type B or type C?
Here are a couple of tools to help you investigate this.
Below those are some questions with hints and answers to guide you.
Decimal for `1/n`: Type in n:
Prime factorisation Type in a whole number: ` `
The youngest group I have used this with was a class of very able Year 9s (13-14 year olds).
Some of the individual questions (eg Q3, Q5) could also be used as extension material after teaching the relevant topic.')
If you prefer to use a printed worksheet, a print-friendly version of the questions (without the hints and answers) is here: recdecs.pdf and a sheet listing the decimal expansion of `1/n` for values of `n` from 2 to 60 is here: recdeclist.pdf. Like nearly all the worksheets on this website, these are "hybrid" pdfs: if you open them with LibreOffice or OpenOffice you can edit them to meet your needs. If you use this material in any form, I would be interested to know how it goes - it is quite a challenging investigation. You can email me at firstname.lastname@example.org
1) What can you say about `n` if `1/n` is Type A (ie it terminates)? Can you explain your answer?
Think about the standard method for converting a decimal such as 0.025 into a fraction.
2) By factorising `n`, can you tell how many digits `1/n` will have before it terminates?
Think about the prime factorisation of 10, 100, 1000 etc.
3) Some specific questions for which you can check your answers:
(a) There are 7 values of `n` for which `1/n` terminates after 3 digits. Can you find them all?
8, 40, 200, 125, 250, 500, 1000
(b) For how many values of `n` will `1/n` terminate after 30 digits?
61 values of n
4) What type of number must `n` be for `1/n` to be of Type B?
Think about the standard method for converting a recurring decimal such as 0.090909... to a fraction.
5) Some more questions for which you can check your answers:
(a) For how many values of n does `1/n` consist of just 3 recurring digits? What are these values?
Five values: 27, 37, 111, 333, 999
(b) There are 6 values of n for which `1/n` consists of just 4 recurring digits. What are these values?
101, 303, 909, 1111, 3333, 9999
6) What is the maximum number of recurring digits which `1/n` can have? Explain why this is.
Think about the method by which you convert a fraction like 1/13 to a decimal. How do you know when to stop because it is recurring?
7) List the values of n up to 20 which achieve this maximum value.
7, 17 and 19
(a) What type of number must `n` be for `1/n` to achieve its maximum number of digits?
(b) Do all values of n of this type achieve this maximum?
(c) If not, what can you say about the number of recurring digits?
For all three parts of this question, it is not too difficult to spot the answers. But proving them is more difficult - probably best to come back to that after you have done the rest of the questions.
9) What type of number must `n` be for `1/n` to be of Type C?
How can we describe numbers which are not of Type A or Type B?
10) There are 9 values of `n` for which `1/n` has 1 non-recurring digit followed by 2 recurring digits. Can you find them all?
22, 55, 110, 66, 165, 330, 198, 495, 990
11) If `1/m` is Type B with `x` recurring digits, and `1/n` is Type B with `y` recurring digits, what can you say about the number of recurring digits in `1/(mn)`? Explain why.
Try a few examples such as `1/7` and `1/101`, or `1/7` and `1/73`
(a) Go back to question 8 and prove the answers you gave there.
(b) For which primes `p` does `1/p` achieve the maximum number of recurring digits?
This question has never been answered completely - if you answer it you will become famous!