# Why is the digit-swap interval 11?
**URL:** https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384
**Category:** Factual Questions
**Created:** [December 27, 2017, 3:53pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384 "2017-12-27T15:53:17Z")
**Posts on this page:** 13
**Page:** 1
### Author:  [@eschereal](https://boards.straightdope.com/u/eschereal)
#### Post date: [December 27, 2017, 3:53pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/1 "2017-12-27T15:53:17Z")
My mother was 27 when I was born. 27, as we know, is a multiple of 9, and the thing about multiples of 9 is that for any number of two different digits, call it AB, the difference between AB and BA will always be a multiple of 9.
So, when I turned (zero) 3, she was thirty; then, eleven years later, when I turned 14, she was 41; and when – on _ad infinitum_ (until it kind of breaks down when I turn seventy and she is 107, though the lower two digit pattern seems to continue).
My question is, why is it 11 years? I know that the reciprocal of nine is point-one-one-bar, but that seems reaching. Is there a mathematical way to describe this, or is it just what it is?
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### Author:  [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)
#### Post date: [December 27, 2017, 3:57pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/2 "2017-12-27T15:57:33Z")
If you add 11 to a number, you’re adding 1 to the tens digit and 1 to the ones digit.
If you add 11 to two different numbers that are already reverses of one another (like 14 and 41), it preserves this.
I think that’s what you’re asking about, right?
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### Author:  [@Mangetout](https://boards.straightdope.com/u/Mangetout)
#### Post date: [December 27, 2017, 4:42pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/3 "2017-12-27T16:42:59Z")
One way to visualise this is to look at what the reversed digits of each number is doing for every case between 01 and 99 - here’s a quick and dirty graph:
[https://1drv.ms/i/s!Ar4eOUAx-yGwh\_UrVYm5ZOiWdrGEqg](https://1drv.ms/i/s!Ar4eOUAx-yGwh_UrVYm5ZOiWdrGEqg)
As you can see, the condition where the digits are the same (22, 33, 44 etc) occurs with an interval of 11 - because, whereas the original numbers (blue dots) are just rising on a line, the reverse-digit numbers (orange dots) are completing a series of increments of 10, then resetting back to 1 more than where they started.
Any line you plot with the gradient of the blue dots, if it intersects any orange dots at all, will only do so every 11th increment.
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### Author:  [@Mangetout](https://boards.straightdope.com/u/Mangetout)
#### Post date: [December 27, 2017, 4:52pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/4 "2017-12-27T16:52:47Z")
… so if we also plot your age plus 27 as a series of green dots, it intersects the pattern of orange dots every 11 years. If your mother was 28 or 26 years older than you, there would be no collisions at all.
[https://1drv.ms/i/s!Ar4eOUAx-yGwh\_U6t-M41ON-2b6djA](https://1drv.ms/i/s!Ar4eOUAx-yGwh_U6t-M41ON-2b6djA)
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### Author:  [@Keeve](https://boards.straightdope.com/u/Keeve)
#### Post date: [December 28, 2017, 2:31am UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/5 "2017-12-28T02:31:24Z")
The reason 9 and 11 are special is that we are using a base ten number system.
If one would use a base twelve system, the same things would happen with eleven and thirteen.
If one would use a base eight system, the same things would happen with seven and nine.
and so on…
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### Author:  [@eschereal](https://boards.straightdope.com/u/eschereal)
#### Post date: [December 28, 2017, 4:00am UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/6 "2017-12-28T04:00:54Z")
Well, yes, I realize that it is pretty silly. A lot of stuff is pretty silly.
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### Author:  [@Mangetout](https://boards.straightdope.com/u/Mangetout)
#### Post date: [December 28, 2017, 7:19am UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/7 "2017-12-28T07:19:05Z")
I don’t think it’s silly at all - these are the same sorts of questions (granted this one is very low level) that drive mathematical research.
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### Author:  [@ftg](https://boards.straightdope.com/u/ftg)
#### Post date: [December 29, 2017, 4:13pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/8 "2017-12-29T16:13:29Z")
> [@Keeve](#):
>
> The reason 9 and 11 are special is that we are using a base ten number system.
>
> If one would use a base twelve system, the same things would happen with eleven and thirteen.
> If one would use a base eight system, the same things would happen with seven and nine.
> and so on…
Notationally …
For base 12 it would be C and 11.
For base 8 it would be 7 and 11.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [December 30, 2017, 9:17pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/9 "2017-12-30T21:17:48Z")
…where in each case “11” is to be interpreted in that base (so, for instance, in base 12 it means “one times a dozen plus one”).
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### Author:  [@eschereal](https://boards.straightdope.com/u/eschereal)
#### Post date: [December 30, 2017, 9:23pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/10 "2017-12-30T21:23:22Z")
> [@ftg](#):
>
> Notationally …
>
> For base 12 it would be C and 11.
> For base 8 it would be 7 and 11.
Actually, base 12 would go to B, not C (assuming you are describing hex-like notation).
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### Author:  [@ftg](https://boards.straightdope.com/u/ftg)
#### Post date: [December 30, 2017, 9:42pm UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/11 "2017-12-30T21:42:02Z")
> [@eschereal](#):
>
> Actually, base 12 would go to B, not C (assuming you are describing hex-like notation).
Ah! Yes, 12 is C in hex notation but you have to subtract to get one under… :smack:
But I got the octal right. Partial credit???
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### Author:  [@eschereal](https://boards.straightdope.com/u/eschereal)
#### Post date: [December 31, 2017, 12:08am UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/12 "2017-12-31T00:08:57Z")
> [@ftg](#):
>
> But I got the octal right. Partial credit???
Octal is useful for something, I suppose. But you need hex for your DEADBEEF and FECECAFE and stuff.
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### Author:  [@filmore](https://boards.straightdope.com/u/filmore)
#### Post date: [December 31, 2017, 12:45am UTC](https://boards.straightdope.com/t/why-is-the-digit-swap-interval-11/805384/13 "2017-12-31T00:45:58Z")
…