On this blog’s homepage I state “I’ve been in love with numbers for as long as I can remember.” Even when I was a toddler I never wanted to practice reciting the alphabet – I preferred reciting numbers.
Yet, one number has always stood above the rest. It is my favourite number – 7.
I think I decided that 7 ought to be my favourite number when I was relatively young. I was 7 years old when my sister was born (technically, 6yrs 10mos 23days 17hrs and 39mins, and yes I did do that calculation the day she was born itself). 7 was the first jersey number I had for soccer. 7 is the sum of the digits of my birth date.
However, I remember recognizing quite early on, no later than the age of 8, that the reciprocal of 7 was the most interesting reciprocal of all of the numbers less than 12.
- 1/2 = 0.5
- 1/3 = 0.333333…
- 1/4 = 0.25
- 1/5 = 0.2
- 1/6 = 0.166666…
- 1/7 = 0.142857142857….
- 1/8 = 0.125
- 1/9 = 0.1111111…
- 1/10 = 0.1
- 1/11 = 0.090909…
- 1/12 = 0.083333…
The reciprocals of 2, 4, 5, 8, and 10 all have finite decimal expansions. The reciprocals of 3, 6, 9, and 12 all end with a single digit repeating ad infinitum while the reciprocal of 11 ends with a repeating two-digit sequence. Yet 1/7 was in a class of its own. I didn’t know why 7 should have such a unique decimal expansion at the time, but I was captivated by it.
A growing fancy
A few years later I realized that the pattern went deeper. Comparing 1/3 (0.333…) to 2/3 (0.666…), the decimal expansions have the same form, in that it is a single digit that repeats, but the digit that repeats is different. This holds for the fractions of 6, 9, 11, and 12 as well. However, the fractions of 7 do something entirely different:
- 1/7 = 0.142857142857…
- 2/7 = 0.2857142857…
- 3/7 = 0.42857142857…
- 4/7 = 0.57142857…
- 5/7 = 0.7142857…
- 6/7 = 0.857142857…
I’ve aligned the decimal expansions to help identify the pattern. Each of the fractions has the same six numbers in the same order, just with a different starting point! 7 was simply outlapping the other numbers in terms of mystique. Why were the same numbers repeating for each fraction, and why were they in the same order?
I began my search for other numbers that had this pattern, but, only using pen and paper, or calculators that only displayed 8 to 10 digits, proved limiting in my search.
Getting serious with number theory
I held on to this intrigue with the number 7 into college, and approached the professor of my number theory class with the question ‘why does 1/7 have such unique patterns?’. He then showed me a whole new dimension to the number 7. It was as if I was peering through Lewis Carroll’s looking glass into a hitherto unknown world of exotic beauty.
He explained that the pattern occurred because we operate in the numeric base of 10, and 10 is a primitive root modulus 7. That means that a series of 6 9’s in a row is divisible by 7, (important since 6 = 7-1) i.e. 999999/7 is an integer, and no other smaller series of 9’s divided by 7 is so (9/7 is not an integer, 99/7 is not an integer, etc.). Any prime number that has this property is called a full repetend prime, and all full repetend primes exhibit the same properties that I described above for the number 7. The first five full repetend primes are 7, 17 (meaning that a series of 16 9’s in a row is the smallest series of 9’s that is divisible by 17), 19, 23, and 29.
He then showed me the property of 9’s, also called Midy’s Theorem. If we recall the repetend of 1/7, i.e. the part that repeats, or 142857:
- 1+4+2+8+5+7 will be divisible by 9 (it equals 3*9)
- 14+28+57 will be divisible by 99 (it equals 99)
- 142+857 will equal 999
Similarly, for the repetend of 1/17, 0588235294117647:
- 0+5+8+8+2+3+5+2+9+4+1+1+7+6+4+7 will be divisible by 9 (it equals 8*9)
- 05+88+23+52+94+11+76+47 will be divisible by 99 (it equals 4*99)
- 0588+2352+9411+7647 will be divisible by 9999 (it equals 2*9999)
- 05882352+94117647 will equal 99999999
All full repetend primes have this property. We went on to discuss many other things, including discrete logarithms and other properties of cyclic numbers and prime reciprocals, and I went on to discover and play with subclasses of the full repetend primes, but I never lost any love or interest for my favourite number, 7.