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The easiest way I know to explain it is fractions. 1 / 3 = 0.33333.... 2 / 3 = 0.66666.... So what's 3 / 3? Some people don't like that one. They might

The easiest way I know to explain it is fractions. 1 / 3 = 0.33333.... 2 / 3 = 0.66666.... So what's 3 / 3? Some people don't like that one. They might like this one better: 1 / 11 = 0.0909090909... What's 10 times that? 10 * 0.0909090909... = 0.90909090... So, let's do some addition and let the values zipper together because a nine will always line up with a zero: 10 * 0.0909090909... + 0.0909090909... = 0.90909090... + 0.0909090909... = 0.9999999999... However, 10 * 1 / 11 = 10 / 11. And 10 / 11 + 1 / 11 = 11 / 11. So 11 / 11 must be the same as 0.99999.... This works for any repeating fraction. You can do it with 1/7 and 6/7. You add the decimal representations of the numbers up and the value will be 0.99999... Technically, it works for any repeating fraction in any base. This is great because a lot of fractions are only repeating fractions in certain bases. So if 0.1 in base 10 is a repeating decimal in base 2 (it is) then you can show that (in decimal) 0.1 + 9 * 0.1 will represent (in binary) 0.11111...., which is equal to 1. The issue is that 1 / 11 + 10 / 11 (in decimal) must still equal 1 in ALL bases. Well, guess what? In Base 11 the decimal looks like: 0.1 + 0.A = 1.0 And 1.0 in base 11 is 1.0 in any base.

war1025 7y ago
That 11ths thing is actually really clever. I had never seen that argument before.
xyzzyz 7y ago
This is basically how we were taught to convert decimal fractions with periodic decimal expansions to regular fractions in school. You can even do that without that lining up of zeroes and nines, just multiply by the period: if x = 0.090909..., then 100x = 09.090909... (we shift the decimal point by two positions), and since the stuff after the point is the same, after subtracting it cancels out: 100x - x = 9.0, and so 99x = 9, from which we obtain x = 1/11.

war1025 7y ago

That 11ths thing is actually really clever. I had never seen that argument before.

xyzzyz 7y ago

This is basically how we were taught to convert decimal fractions with periodic decimal expansions to regular fractions in school. You can even do that without that lining up of zeroes and nines, just multiply by the period: if x = 0.090909..., then 100x = 09.090909... (we shift the decimal point by two positions), and since the stuff after the point is the same, after subtracting it cancels out: 100x - x = 9.0, and so 99x = 9, from which we obtain x = 1/11.