A new way to count all the rational numbers!

Philosophy Forums Science, Technology, and Math
1

I found a new way to count all of the rational numbers and I figured out that Cantors diagonal argument is meaningless as a proof that you cannot correspond all of the reals to counting numbers.
I’ll show you!

First you count the first ten numbers and their negatives (and 0)
0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, -9

This realization came to me when I figured out that if you mirror all of the counting numbers, you have every possible decimal expansion, and when this converges at infinity, you have all of the reals in the set, however, I haven’t yet figured out how to correspond all of them to counting numbers, or whether they can be, which is a proof I’m currently working on by using an infinite amount of lists with an infinite amount of dimension per list, Cantor only proves that you cannot count all numbers in a single list with one dimension, however, when dealing with multiple dimensions, it’s exceedingly easy to count all of the diagonals… so Cantor’s disproof that the reals cannot all be counted is false. I’m trying to find the limit for single lists in single dimensions, I think I’ve found it, but haven’t quite proved it yet. Anyways, aside from that work, a small portion of this is a new way to list all the rationals… and we’ll continue with that!

Anyways, when you hit the number 10, the mirroring starts. The mirror of 10 is 01. After you mirror any number, you move the decimal point in one place… so the next part of the sequence after -9 is…

10, -10, 0.1, -0.1, 0.(1 repeating), -0.(1 repeating)
11, -11, 1.1, -1.1, 1.(1 repeating), -1.(1 repeating)
12, -12, 2.1, -2.1, 2.(1 repeating), -2.(1 repeating)
etc…

Until you hit 100, then a new rule begins. The mirror of 100 is 001. When the number that’s about to be mirrored ends in a zero, you can only move the decimal point in ONE place after you mirror it, otherwise there will be infinite overlap as the numbers continue to expand. 00.1 means the same as 0.1, which already occurred in the mirror of 10 and will occur an infinite number of times as you count up and move the decimal point in as the zeros expand. So 100 looks like this:

100, -100, 0.01, -0.01, 1.0(1 repeating), -0.0(1 repeating), 0.(01 repeating), -0.(01 repeating)

However 101, the next number does not end in a zero, and it looks like this:
101, -101, 1.01, -1.01, 1.0(1 repeating), -1.0(1 repeating), 1.(01 repeating), -1.(01 repeating) [Then you march the decimal point in one more time!!!] 10.1, -10.1, 10.(1 repeating), -10.(1 repeating)

If you march the decimal point in again, you’ll have 101 again, which you already counted, so stop before the last digit. If you count this sequence, you will count every rational number with no overlap!!

102 looks like this…

102, -102, 2.01, -2.01, 2.0(1 repeating), -2.0(1 repeating), 2.(01 repeating), -2.(01 repeating), 20.1, -20.1, 20.(1 repeating), -20.(1 repeating)

etc…

That was my discovery, a new way to count all the rational numbers. There are so many real numbers, that I’m having trouble finding a proof or disproof for whether they can be counted in infinite lists with infinite dimensions per list, but I’m still working on it. The numbers get VERY exotic, and you may have to apply a dimension for each place a number is in as well!

2

I know nothing of math, I was completely lost. Maybe you can try to explain it to me as if I were a half-wit, just in case I find it interesting enough to read it closely a couple times.

3

Well, a rational number is every number that can be expressed as a fraction. All the numbers that cannot be expressed as a fraction are considered irrational and there are classifications of irrationals from algorithmically simple, algebraic and even maximally algorithmically complex. The thing is, it’s considered common knowledge in math that you cannot correspond all of the real numbers with counting numbers…

counting numbers are 1,2,3,4,5,6,7,8,9,10,11,12,13 etc…

When I made my lists, it’s implied that a counting number goes beside each one where a comma divides it:

1.) 0
2.) 1
3.) -1
4.) 2
5.) -2
6.) 3
7.) -3

etc…

Well what Cantor showed is that while you can line up all the rational numbers with counting numbers (place them in 1:1 correspondence), there are actually more real numbers than rational numbers through a method which showed that you cannot count them all using this method. However, the method he uses to prove this only shows that you cannot count all of the reals in a single list with a single dimension per list, and also his proof doesn’t show the LIMIT for how many numbers overcrowd a single list in a single dimension, so actually he didn’t really figure out very much about what he was studying.

You might want to Wikipedia http://en.wikipedia.org/wiki/Rational_number and http://en.wikipedia.org/wiki/Georg_Cantor
http://en.wikipedia.org/wiki/Cantor’s_diagonal_argument

Hope that helps :question:

4

Hi Commentary,

Irrational numbers can not be represented by a decimal expansion.

However one of the properties of the Reals, either by assumption or definition, is that the limits of all bound Cauchy sequences, sequences whose terms get arbitrarily close to each other as n gets large and (the bound part) whose terms are neither greater than or less than a given positive number or a given negative number, exist as Real numbers. This property is called, confusingly enough, “Completeness”. Thus decimal expansions, which by themselves can not generally represent irrational numbers, can converge to any irrational number.

Additionally, infinity is not a Real number. There are logical extensions of the Reals that include infinity but infinity itself is not part of the Real numbers.

Mathematical limits have specific definitions. For a sequence Sn to have a limit L it must satisfy the following condition, and in this specific order:

For every given positive Real number a, there must be a positive integer N (which depends on a) such that if n is an Integer which is greater than N then |Sn – L| (the distance between Sn and L) is less than a.

Using this definition, we should be able to see that the limit of the sequence Sn = n does not exist. I.e. the sequence 1, 2, 3, …, n does not have a limit of infinity because |n - infinity| = infinity (or it could simply be said to be undefined) for all n. This means that the distance between Sn and a theoretical L never gets small.

Something to think about:

If you are unfamiliar with bijections you should skip to Comments about Bijections, and then continue here.

I will construct a 1-1 and onto map (sometimes called a bijection) from C, the Counting numbers to Z X Z, where Z is the set of Integers, and simply state that the Counting numbers have the same cardinality as Z X Z … X Z n times.

This is intened to show that adding dimensionality does not necessarily change the Cardinality of a set.

Define f1such that C is mapped onto the Odds by f1(n) = 2n – 1. f1 defined this way can be shown to be a bijection. This means that f3 = Inv(f1), Inv is used to denote the inverse function, exists and is also a bijection.

Define f2 such that C is mapped onto the Evens by f2(n) = 2n. By the same reasoning f2 is a bijection, and f4 = Inv(f2) exists and is also a bijection.

Define f5 such that C is mapped to the non-positive Integers by f5(n) = -(n+1). Again f5 is a bijection and f5(1) = 0, f5(2) = -1, f5(3) = -2, … Then f5 is such that C is mapped onto Z – C (the non-positive Integers).

Define g1such that g1(n) = f3(n) if n is odd and f5(f4(n)) if n is even. Then g1 is a bijection and maps C onto Z.

Now define g2 such that C is mapped to Z X Z by (g1(f3(n)), g1(f4(m))). Here n is an arbitrary odd number and m is an independent and arbitrary even number. Again g2 is a bijection.

It should be clear that for each additional Z, C can be mapped to Z X Z X Z …X Z (n times), because gn can be defined analogously and C can be mapped to Z^n.

Comments about Bijections:

A function is said to be a bijection from set A onto set B, if and only if every element of set A is mapped uniquely to every element of set B. That is to say for any element x of set A f(x) = y and f(x) = z can only be true if y = z. Further the function f must be such that for every y an element of B then y = f(x) for some x in A. If there is a bijection from set A to set B then by definition the two sets are said to have the same Cardinality.

From this it can be shown that if f is a bijection from A to B, then the inverse function Inv(f) is a bijection from B to A.

Additionally, it should be clear that if f is a bijection and g is a bijection then the function h(x) = f(g(x)) is also a bijection.

Ed

5

Basically the cardinality of infinity is variable if you accept that ZFT is axiomatically true, which all mathmos do.

So diagonalising all the numbers ads up to itself and is 1 to 1, where as all the fractions add up to itself and are 1 to 1, and hence all the decimals are larger than both sets because they cannot be finitely represented, pi for example is transcendental because it’s extent cannot be physically expressed, in other words what’s more circular than a circle, a circle obviously? :wink:

What’s 2 x pi?

What’s (E^i.pi^2)-1=0?

What’s i^2? equal to is it 1 or -1 or both?

What’s equivalent to i? What is rational and irational? is a number real?

What axis are the reals on? What axis are the imaginary numbers on? Are they representable by a lorentz transform, what angle do they need to be to represent pi?

The answer is an identity called Euler’s identity:

so if f(x)=g(x) then f’(x) = g’(x) hence the rules of integration are axiomatically true.

An integral is the area under the line of a graph and the gradient of the graph is a differential. So if you integrate a differential you get a number, and if you differentiate that you get c or a constant. Maths it’s mathtastic. :wink:

d/dx x = x
d^2/d^2x = x’
d/dx of x is x
d/dx of c is 0

d^2/d^2x is a second order differential of distance, called acceleration, which is why s=u+1/2at^2 which is an integral ultimately of gravity or f=ma or f of g = gravity. Trivially if you integrate one side of the graph you end up with speed=acceleration which is what Newton did basically to produce the equation .

Force(g)=gravitational constant(gk)xmass 1.mass2 divided by distance aka the inverse square law for obvious reasons.

6

1.) 0.01234567891011121314151617181920212223…
2.) 0.12345678910111213141516171819202122232…
3.) 0.23456789101112131415161718192021222324…
4.) 0.34567891011121314151617181920212223242…
5.) 0.45678910111213141516171819202122232425…
etc…

Each one of those is a different irrational number being expressed by a decimal. The sequence is easy, it’s the counting numbers.

7

Yeah I already said that, but no fucker ever reads what I Say. which is fine but holy madre de dios I can’t believe it’s not butter. :stuck_out_tongue:

That post I just fucking made right, it explained the whole thing right? Do I actually think anyone anywhere is ever going to read it? Probably fucking not, probably fucking never. #-o

Fuck me! #-o Differ fucking rentional calculus, it’s fucking different. :wink:

1 to 1, 1 to many 1 to the big old cock! Shit.

so f(x) or d/dx or the differential of x in relation to x is? fucked if you care, obviously maths is fucking shit. :stuck_out_tongue:

8

Hi Commentary,

It looks to me like you are simply summing rational numbers. Sums of rational numbers are rational numbers.

The only way that you can get to an irrational is to look at the limit of the series.

My contention is that the numbers that you have written are not irrational.

Look at it this way:

.x1x2x3x4x5…xn… = x1/10 + x2/100 + x3/1000 +x4/10000 + x5/100000 + … + xn/10^n + …

Each of the numbers in the series is rational. Therefore the sum is rational.

Thanks Ed

9

Quite sums of irrational numbers are also rational numbers, if you think about why it’s very obvious.

Let’s take a limit for example Ed and I don’t mean to be patronising but this will of course demonstrate the difference between x and >x.

if we put x/>X at which point does x = 0? Well even a tardlord can see that the point at which x is equal to nothing is a finite and quantifiable number only if x is > x, and is not a constant. It always amazes me how dim people are with maths, but that’s not a bad thing, after all it makes us autisitc tards look clever. :wink:

If x > x then what is the solution to d/dx?

10

Hi to All,

Helanderhighwater made the claim that the sum of Irrational numbers was Rational.

I would like to prove that claim is not generally true.

Claim: There exists unique Irrational numbers, j1 and j2, such that j1 + j2 is Irrational.

Proof: Let j be any Irrational number.

Since j + j = 2j, I would like to show is that 2j is Irrational.

Suppose not.

Then 2j = p/q where p and q are Integers. Dividing by 2 we get j = p/2q. However 2q is again an Integer, which we will define as q’, and thus j is of the form p’/q’ where p’= p and q’ are Integers. This implies that j is Rational which is a contridiction. Therefore 2j must be Irrational.

Thus if j is Irrational then 2j is Irrational.

Now we let 3j = 2j + j. Then since 2j is not equal to j and both 2j and j are Irrational, their sum is 3j. Using the same argument that 2j is Irrational we can show that 3j is Irrational.

Now we have three unique Irrational numbers, j, 2j, and 3j such that j + 2j = 3j.

End of Claim

In general if j is Irrational and r1 is Rational (not equal to 0), then numbers of the form r1j are Irrational. Additionally, r1j + r2j is Irrational, providing that r1 and r2 are Rational numbers not equal to 0 (we also need to assume that r1 is not equal to -r2).

11

I learned it wrong (I learn math on the internet) I thought they just had to not repeat at some point was how I saw the definition, I wasn’t aware that you searched the limit to determine this. I’ll certainly be thinking about this Ed - thx