## Reflected Numbers – An Idea

think about this

look at **pi**, its digits go on forever

3.14159265…

each successive digit moving to the right is an inverse power of 10. you could speak it as

3 units

1 tenth

4 hundredths

1 thousandth

5 ten thousandths

(…)

consider a mathematical operation I will call “reflection”

to make it easier I will start with pi-3

**0.14159265358979…**

reflection is like you place a mirror at the decimal point, so the tenth part is moved to the units column, the hundredth part to the tens column, the thousandth part to the hundreds column and so on. With this method we construct a new number, reflection(pi-3).

reflection(pi-3) has some interesting properties. While the digit expansion of pi-3 goes on forever to the right, reflection(pi-3) has digits that go on to infinity to the left. Thus each added digit makes the number larger.

(…)97985356295141

so reflection(pi-3) is clearly a number that is infinite in size. But it is not infinity because reflection(pi-3) -1 is different to reflection(pi-3). consider

(…)97985356295141 -1 = (…)97985356295140

whereas infinity -1 = infinity

or by adding 1 we can show that for any reflected number

x-1 < x < x+1

look at a reflected number consisting only of 2s

(…)2222222222222

trivially here

2 * (…)2222222222222 = (…)4444444444444

and

(…)2222222222222 / 2 = (…)1111111111111

so also we can assume

x / a < x < x * a

as long as a > 1

to be logical of course the reflection process could also be done to any non terminating decimal like the rational number 1/7 = 0.14285714285714285714(…). forming

(…)41758241758241758241

we can also use other ways of constructing reflected numbers, like reflecting the number

0.1234567891011121314151617181920(…)

to get

(…)0291817161514131211101987654321

so a reflected number is infinite in its size or magnitude, but it is not infinity. Is this then a possible way to argue that infinity itself is *not* a number, rather a convenient concept.

How many reflected numbers are there? Is their cardinality that of the Reals?

What other properties do they have?

can you express the “reflection operation” as an arithmetic operation or function? Using sigma notation?

Here is a pseudocode function that does the job of reflection, I know the main loop doesnt terminate but it is still logical in a way 🙂

def ref(x): for z in range(infinity): dig = x*10-x output+= 10^z * dig x = x*10 - int(x*10) return output

of course this next one is really cheating:

def ref(x): return int(str(x)[::-1])

**But there’s more, lad !**

what about this number:

**reflected(pi-3) + pi-3**

Let’s call it w.

This number has an infinite digit expansion to the left and to the right of its decimal point!

…So surely for every real number between 0 and 1 we could add one of the multitude of reflected numbers to form a two-way infinite number like w.

This surely means that for the uncountable number of reals between 0 and 1 each can form a two-way number with each reflected number.

Yet for each single distinct real number between 0 and 1 there is a corresponding reflected number.

does this not then suggest that if the cardinality of the reflected numbers is

called Aleph 1

then the cardinality ofthe two way numbers is

*

is this number larger than Aleph 1 ?

=== update ===

My latest thought about this is the following:

The “*place-value notation*” or “*positional notation*” is a way of representing numbers, using columns where digits in a column show the number of occurrences of the power of 10 for the given column. This, however, can mislead us to think that a number represented thus actually **is the number** in itself in *some absolute way*. This can be seen as wrong if one understands that, actually, place-value is just a **convention** and quite arbitrary. It is the *most successful convention* but this is only because it is the most convenient, contrasted say with Roman Numerals like i, ii, iv, x etc which make arithmetic harder.

So when we play our trick of constructing these reflected numbers I think it’s possible that we may be “spoofing” the place-value system in a way that involves paradox or some kind of logical inconsistency. It makes me wonder what the real nature of a number is, kind of a *true identity* that is beyond any one convention of how it is represented. What if we think of a number as something more like a cloud of dots, where each dot is a unit? From this angle we clearly see that any reflected number is simply infinite in size, without any real difficulty.

To a set theorist any whole number is really a set. But how can we get a handle on what a set is, really? what does a set look like, feel like? Maybe this is an impossible question because a set is such a pure and ineffable concept that it is too slippery for the human mind. But that is defeatist !!

Comments more than welcome…

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