Maths IA · Algebra and functions · topic 7 of 40

Repeating decimals: what patterns are in the digits of fractions? · IB Maths IA idea: guide, sample graph and step-by-step plan

abstract, but beautiful if you stay concrete

1/7 gives 0.142857142857... and then it repeats, forever. Why that block, why 6 digits, and what would 1/13 do? You're going to find out why, not only what.

Difficulty●●● Stretch
KitPaper, then a spreadsheet or short program
TimeAbout 12 to 15 hours
FitsAA HL
  • ✓ you like patterns in numbers
  • ✓ you enjoy proof and abstract ideas
  • ✓ you can write a short loop to test conjectures

The big idea

← conjecture, then prove

Long division is a machine: multiply the remainder by 10, divide by n, write the digit, keep the new remainder. Because there are only finitely many remainders, the digits must eventually repeat. Your question: how long is the repeating block, and what decides it? You will link the length to modular arithmetic and to the powers of 10. At HL this is natural HL-level maths if you prove your claims: conjecture, test, then prove. Plenty of this is beyond the syllabus, so teach yourself carefully and show your understanding in your own words.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Block length for primes

Tabulate the length of the repeating block for the reciprocals of many primes, then look for what controls it.

→ Your data suggests conjectures about divisors of n − 1, which you then try to prove.
Change the base

Write fractions in base 2, 3 or 12 and find the length of the repeating block.

→ The same remainder machine with 10 replaced by the base. Tests whether your explanation is general or just a fact about base ten.
Full-length blocks

Find which denominators give the longest possible block, and explain what has to be true about 10 modulo n.

→ You move from data to a precise claim that needs modular arithmetic or group ideas.
Cyclic numbers

Multiply the repeating block of 1/7 by 2, 3, 4 and so on. What happens to the digits?

→ The remainder cycle explains a surprising pattern, which makes your proof concrete.
Which fractions terminate?

Prove exactly which denominators give a finite decimal, using a geometric series for the repeating ones.

→ A clean theorem, written as a proof. Good for a first section that is fully rigorous.
Change the context

Calendars (mod 7), clocks (mod 12), or check digits on a barcode.

→ A real cycle in a different place, with the same modular arithmetic at the centre.

Your Maths IA research question: run the check

one question, explained deeply

Write your own question first. Then make sure it ticks every box:

  • Says exactly which fractions you study (unit fractions, denominators prime, a base other than 10) and why.
  • Names the quantity you will explain (length of the repeating block, the digits themselves, or which fractions terminate).
  • Can be answered by a proof, or part-proof, that you write yourself, not only by tables.
  • States which ideas you will need beyond the syllabus (such as modular arithmetic) and how you will teach yourself them.
  • Is narrow enough for 12 to 20 pages: one question explained deeply.

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The model

use YOUR n

Long division as a loop. Draw yours with your own n, and the remainders you actually get from the first few steps.

remainder rstart with r = 1multiply by 1010rdivide by nquotient and remaindernew remainderr = 10r mod ndigit d = ⌊10r / n⌋r < n, so only n − 1 remainderssame remainder twice: it repeats!

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
A fraction that surprised you, your question in your own words, and your conjecture before the calculations.
Terminating or repeating?~2 pages
A proof, using a geometric series, of which fractions give finite decimals and which give a repeating block.
The remainder machine~2 pages
Long division as a loop, with a clear argument that the digits must repeat, and the block length bounded.
Data and conjecture~2 pages
A table of block lengths for the cases you chose, how you generated it, and the pattern you now believe.
Modular arithmetic~3 pages
The ideas you need, in your own words, and how the block length links to powers of 10 modulo n.
Proof of your main claim~3 pages
A proof written as a logical chain, with each step justified. Say which steps use results you did not prove.
A second base or a limiting case~1.5 pages
Test whether the explanation holds in another base or for a case where the theorem should fail.
Reflection and evaluation~1.5 pages
What the proof needed, where your first conjecture was wrong, and what you could not prove.
Conclusion~1 page
Back to your aim and your conjecture: what is now proved, and what is only observed.
Referencesnot counted
Number theory sources. Direct statements of results must be cited.

What your Maths IA graph could look like

Made-up illustration of the form of a strong graph, not the answer. Here, the remainders of long division for one fraction. example only

012345670246810121416Step number k of the long divisionRemainder r after k steps (a whole number)remainders, for 1/13step 6 gives step 0'sremainder again: why?ILLUSTRATIVE · MADE-UP DATA
  • Axis titles say what k and r mean. Remainders are whole numbers, so only points are drawn, with no line joining them.
  • No error bars: this is exact arithmetic. Say so, so the reader does not look for them.
  • The pattern is visible at a glance. Your text must say what to notice, then prove it.
  • Choose a denominator whose block is not obvious, so the graph shows something.
  • Another graph to consider: for many primes p, plot the block length of 1/p against p, with the line p − 1 for comparison.

Your game plan

prove the easy theorem first!

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Play and predictDivide 1 by several n by hand. Predict how long the block will be for a prime you have not done, and why.
  2. Trial run (1 lesson)Write a short loop that prints the remainders and digits. Check it against your hand work. Decide how to keep large numbers exact.
  3. Terminating or notProve which fractions terminate. This is a good first, fully rigorous result.
  4. Collect block lengthsTabulate the lengths for a set of denominators you chose and justify. Look for patterns, and write them as conjectures.
  5. Learn the modular ideasRead about congruences and the order of 10 modulo n. Write the ideas in your own words and check them on small examples.
  6. Link themShow how the block length is a statement about powers of 10 modulo n.
  7. Prove what you canWrite the proof of your main claim as a chain of justified steps. Mark clearly any step you use from a source.
  8. Test generalityTry another base or a case that should break your claim. Does your explanation survive?
  9. ReflectWhich conjectures were wrong? What would you still need to prove? What would you try next?
  10. Write, cut, checkWrite in the layout order. Put long tables in one summary table. Run the checklist, then get your question checked before your teacher's draft.

Maths you'll need

Traps to avoid in this Maths IA

  • Tables of block lengths with no proof. At HL this is the pattern-spotting that criterion E does not reward.
  • Quoting a theorem from a website and applying it without showing you understand why it holds.
  • Computer output that is rounded: a calculator hides the digits you need. Use exact division or a program.
  • Mixing up the number of digits in the block with the period of the remainders.
  • Ignoring denominators with factors of 2 or 5, so the pattern seems cleaner than it is.
  • Jumping into abstract algebra the writer cannot explain. Stay within what you can defend.
  • History of recurring decimals instead of your own conjectures.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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