Last-digit cycles
June 24th 2010 13:46
Quick maths thing I've been fiddling with... I'm sure this has been very well studied, but, hey, it's new to me...
Say you're dealing with subtraction or addition, and you're working with a base 10 number system (that is, the normal system). What you'll find is that if you keep subtracting or adding the same number, the last digits of the results will repeat in a steady cycle.
Not sure if this is of much practical use, beyond a method of error-checking.
The most obvious example is the number 10. I don't know straight away what "1,346,035 - (2747 x 10)" is, but I know that the answer ends in 5, because when you're subtracting 10, the last digit will repeat in each instance. Similarly, I don't know straight away what "394,392 plus (4,657,347,397 x 35)" is, but I know the last digit will be 7.
10 and 0 have cycles of 1 before the last digit starts to repeat.
5 has a cycle of 2.
2, 4, 6, 8 have cycles of 5.
1, 3, 7, 9 have cycles of 10.
And these cycles also seem to work for all numbers greater than 10. All you need to do is look at the final digit of the number you're adding or subtracting. For instance, 20 will also have a cycle of 1, 25 will also have a cycle of 2, 23 will also have a cycle of 10, etc.
The big exception is when the cycle lands on a 1, 2, 3 or 4 -- the last digit is then not always as easily predictable.
Some remaining questions: (1) the precise rules governing passing through 1, 2, 3 or 4; (2) multiplication, division, and other functions seem to have more complex pattern-rules, so these remain to be outlined; (3) there must be a way to prove whether an addition/subtraction cycle does repeat ad infinitum for a particular number (given certain restrictions); (4) there must be a way to prove whether all numbers have addition/subtraction cycles (given certain restrictions); (5) how do matters stand for non-base 10 systems?
Notes
-- Friday 25 June 2010: If you have a sum like "53,332 - (36 x 5) = x", it's true that you can use multiplication as a quick form of error-checking. For instance:
-- Look at the 36 x 5.
-- Just focus on the last digits, ie 6 and 5.
-- Multiply these. 6 x 5 = 30
-- Take the last digit of that number (in this case, 0).
-- Look at the last digit of 53,332, and subtract 0. So, 2 - 0 is still 2, and you know that the last digit of x is going to be 2.
But thinking in terms of cycles is occasionally easier than multiplication. For instance, when you're error-checking a nested equation like "533,332 - {48 x [25 x (36 x 5)]}". In this case, you can quickly see that whatever 5 is being multiplied by is still going to be divisible by 2, so the last digit of x is still going to be 2.
-- Friday 25 June 2010: Apart from error-checking, cycles-thinking is potentially useful when you only care about the last number. For instance, it may be that a physical state (like a light switch being on or off) coincides with a particular last digit.
Say you're dealing with subtraction or addition, and you're working with a base 10 number system (that is, the normal system). What you'll find is that if you keep subtracting or adding the same number, the last digits of the results will repeat in a steady cycle.
Not sure if this is of much practical use, beyond a method of error-checking.
The most obvious example is the number 10. I don't know straight away what "1,346,035 - (2747 x 10)" is, but I know that the answer ends in 5, because when you're subtracting 10, the last digit will repeat in each instance. Similarly, I don't know straight away what "394,392 plus (4,657,347,397 x 35)" is, but I know the last digit will be 7.
10 and 0 have cycles of 1 before the last digit starts to repeat.
5 has a cycle of 2.
2, 4, 6, 8 have cycles of 5.
1, 3, 7, 9 have cycles of 10.
And these cycles also seem to work for all numbers greater than 10. All you need to do is look at the final digit of the number you're adding or subtracting. For instance, 20 will also have a cycle of 1, 25 will also have a cycle of 2, 23 will also have a cycle of 10, etc.
The big exception is when the cycle lands on a 1, 2, 3 or 4 -- the last digit is then not always as easily predictable.
Some remaining questions: (1) the precise rules governing passing through 1, 2, 3 or 4; (2) multiplication, division, and other functions seem to have more complex pattern-rules, so these remain to be outlined; (3) there must be a way to prove whether an addition/subtraction cycle does repeat ad infinitum for a particular number (given certain restrictions); (4) there must be a way to prove whether all numbers have addition/subtraction cycles (given certain restrictions); (5) how do matters stand for non-base 10 systems?
***
Notes
-- Friday 25 June 2010: If you have a sum like "53,332 - (36 x 5) = x", it's true that you can use multiplication as a quick form of error-checking. For instance:
-- Look at the 36 x 5.
-- Just focus on the last digits, ie 6 and 5.
-- Multiply these. 6 x 5 = 30
-- Take the last digit of that number (in this case, 0).
-- Look at the last digit of 53,332, and subtract 0. So, 2 - 0 is still 2, and you know that the last digit of x is going to be 2.
But thinking in terms of cycles is occasionally easier than multiplication. For instance, when you're error-checking a nested equation like "533,332 - {48 x [25 x (36 x 5)]}". In this case, you can quickly see that whatever 5 is being multiplied by is still going to be divisible by 2, so the last digit of x is still going to be 2.
-- Friday 25 June 2010: Apart from error-checking, cycles-thinking is potentially useful when you only care about the last number. For instance, it may be that a physical state (like a light switch being on or off) coincides with a particular last digit.
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