Can # be used to mean a repeating digit?

Question

Is this an accepted way to represent larger repeated numbers: 4#40 + 3#25 = ? Is there a function for these maths?

Answer 1

Do you mean $$\underbrace{444...44}_{40 \text{ digits}} + \underbrace{333...33}_{25 \text{ digits}}?$$ I have never seen that notation, however you could always define it as such if it would be useful. You could also write

$$4\sum\limits_{k=0}^{39}10^k + 3 \sum\limits_{k=0}^{24}10^k.$$

On the other hand, if you mean

$$ \underbrace{444...44}_{40 \text{ digits}}\underbrace{333...33}_{25 \text{ digits}}$$

then you could write $$4\sum\limits_{k=25}^{40+25-1}10^k + 3 \sum\limits_{k=0}^{24}10^k.$$

Answer 2

Most professional mathematicians and even some amateurs eventually lose interest in problems that depend on the base 10 representation of numbers. For that reason there are no accepted standard notations for repeated digits. So you have to define the notation and give an unambiguous example.

The first thing I thought when I saw "4#40" is that the "#" character is a wildcard character representing an arbitrary amount of 4s. But your comment on another answer suggests to me my first hunch was wrong.

It sounds like you're talking about repdigits. A notation that is sometimes used for repunits (repdigits in which the repeated digit is 1) is $R_n$. For example, $R_2 = 11$, $R_3 = 111$, $R_4 = 1111$, etc. The formula is then $$R_n = \frac{10^n - 1}{9}.$$ Then "4#40" could be $4R_{40}$.

But then the "+" in your question is ambiguous. I first thought it meant addition, so $4R_{40} + 3R_{25} = 4444444444444447777777777777777777777777$ (a divisor of this number is $41 \times 271 \times 761 \times 8537$).

But what if instead of addition you mean concatenation? That can be confusing, especially if you don't use quotation marks to make it clearer that "+" means string concatenation. Better to leave "+" to mean addition only.

You can still express that concatenation with addition: $(4 \times 10^{25})R_{40} + 3R_{25} = $

$44444444444444444444444444444444444444443333333333333333333333333$. I don't know if this number is prime or composite.

Answer 3

4R sub 40 + 3R sub 25 I had typed out other mathematical system like this to other person and made this interesting complex math numbers as short as possible using/replacing with other symbols, but I did something like this in just adding/subtracting, then was willing to show to others. I put repeated digits to a number sign. Then for that math question- if I made-out that question, I would put 4#15&3#25 as if it was the answer I'll put in - Just as counting that part of digits, then count the rest of the digits.

Answer 4

If you define the notation consistently and coherently, and use it clearly, you can use anything to represent anything.