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I stumbled upon a very bizarre result when playing with wolfram alpha. According to it, the last digit of $\frac{1}{3} = 7$. Is that true? If yes, is there a mathematical argument for such a result?

The reason I require an explanation for such a statement is that the impression is that the theoretical definition of infinite decimal expansion suggests that the last digit should be 3. Moreover, if we force a different number, (mostly used by calculators) to terminate the decimal expansion in the manner $0.333333....4$, since it's more "sensible" for a calculator, even such an argument motivates the use of $4$ as the last digit. So why $7$?

Xander Henderson
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mathnoob123
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    In its usual decimal expansion $1/3$ has no last digit. Looks like WA is not programmed to answer "does not exist" :-) My guess is that $7$ pops out because it is the modular inverse of $3$ (modulo ten). – Jyrki Lahtonen Jun 25 '18 at 20:41
  • Never trust a software with such things :p – Eben Kadile Jun 25 '18 at 20:43
  • Yes, there is no “last” digit of $1/3$. Granted, it’s decimal respresentation, being $0.\bar{3}$, can be rounded to $0.333\dots 4$ at an arbitrary decimal place if one wants to write it out in a finite number of places. But, I am not sure why Wolfram would yield seven. – Crosby Jun 25 '18 at 20:44
  • Oh okay, thanks. Can anyone of you provide a small answer to this question which I can accept and mark closure to this question? Or should I just delete it? – mathnoob123 Jun 25 '18 at 20:46
  • @Crosby, the decimal representation of $1/3$ is $0.333...$ – Green Jun 25 '18 at 20:46
  • @Green Oh, I need to go back to elementary school lol. – Crosby Jun 25 '18 at 20:48

2 Answers2

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If the last digit of $n$ is $d$, then this means $n\equiv d \pmod{10}$.

Now $3\cdot 7 = 21 \equiv 1 \pmod{10}$, therefore it makes sense to say $1/3\equiv 7 \pmod{10}$ (note that $\gcd(3,10)=1$, therefore it actually makes sense to speak of an inverse of $3$ modulo $10$).

Interpreting “last digit of” as “the single digit number which the number is equivalent to module $10$” then gives the Wolfram result.

celtschk
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It's likely Wolfram Alpha computed the modular inverse of $3 \bmod 10$, which is $7$.

orlp
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