Reversing a number of infinitely many digits

Question

Lets say we have a function that gets as input a real number and returns its reverse

e.g. 123.12 -> 21.321

So what happens when the input is a number α that has infinitely many digits. Does then reverse(α) exist as a number? Is this number well defined?

if you were to type $\pi$ in by hand, you would never finish. If you type $\dfrac{1}{7}$ into your computer, the computer would never finish. So you can't input a number with $\infty$ digits. Full stop. — JMP, Feb 23 '15 at 12:20
you know, ratio of the circumference of a circle to its radius, roughly 22/7, starts 3.14159265358979323846.... — JMP, Feb 23 '15 at 12:24
@JonMarkPerry Well we can define a function that is f(x) = 2x. Then f(π) = 2π. I talk in the same sense as this example. Is the reverse(π) an actual number?That was my question — Lan Pac, Feb 23 '15 at 12:28
it would be infinite, because there are an infinite number of digits after the decimal point. Irrationals have no last digit. — JMP, Feb 23 '15 at 12:30

score 3 · Accepted Answer · answered Feb 23 '15 at 12:04

3

If the input is real numbers, then no. There is no real number which has an integral part with infinitely many digits.

But it's worse, do you map $\frac12$ to $5$ or the $\ldots9994$?

answered Feb 23 '15 at 12:04

Asaf Karagila

OK just saw about this. As for the second part isn't it 1/2 = 0.5 reverse(0,5) = 50.I didn't get you on that part. – Lan Pac Feb 23 '15 at 12:17
I thought that $0{.}5$ is reversed to $5{.}0$ which is equal to $5$ the last time I checked. – Asaf Karagila Feb 23 '15 at 12:19
Ops sorry my mistake.It should be 5.0. Can you elaborate on the ..9994? – Lan Pac Feb 23 '15 at 12:23
Well, $\frac12=0{.}5=0{.}4999\ldots$. So which one do you take? – Asaf Karagila Feb 23 '15 at 12:25
Thanks.One last thing you said if then input is real numbers then no.In which other set it may work? Will it be fine if the inputs were to be natural numbers? – Lan Pac Feb 23 '15 at 12:31
1

Yes, if the input is a natural number the result is a rational number. You can always consider other systems like the $p$-adic numbers, or $10$-adic numbers which are slightly different but can be thought as infinite strings of digits, and so on. There's much much much more to mathematics than just the real numbers! – Asaf Karagila Feb 23 '15 at 12:43

score 0 · Answer 2 · answered Feb 23 '15 at 12:07

0

There is an idea like that, called the $p$-adics. The $p$ is because it works better when the base is prime instead of ten.

answered Feb 23 '15 at 12:07

Empy2

2 Answers2