Aww... More math! Just what I need.

I am more of an integer nerd The sum of all integers up to infinity, is -1/12 (using Ramanujan Summation).

Looking at your steps, it seems you have a lot of repetition of numbers, like 0.01 and 0.010 (both being the same number, but in different steps). While searching for all real numbers, it would be such a waste of time to include the same number over and over as that time adds up, and since we only have finite time to work with (universe has only existed for a finite time, or will only exist.... Well, we don't know if it has an end. All we know is that we are running out of time here).

step 0:
0.0

step 1:
0.1

step 2:
0.10
0.11

step 3:
0.001
0.011
0.101
0.111

step 4:
0.0001
0.0011
0.0101
0.0111
0.1001
0.1101
0.1011
0.1111

step 5:
0.00011
0.00111
0.00101
0.01001
0.10001
0.11101
0.11011
0.10111
0.01111
0.11111

step 6:
0.000001
0.000111
0.001101
0.011001
0.110001
0.001111
0.011101
0.111001
0.111101
0.111011
0.110111
0.101111
0.011111
0.111111

step 7:
0.0000001
0.1000001
0.0000011
0.1100001
0.1000011
0.0000111
0.1110001
0.1100011
0.1000111
0.0001111
0.1111001
0.1110011
0.1100111
0.1001111
0.0011111
0.1111101
0.1111011
0.1110111
0.1101111
0.1011111
0.0111111
0.1111111

Oh, this is a challenge. I sense a pattern!

EDIT: I had to add some more because it was fun.

@Splashee I meant that we keep only the numbers from the current step. If we're going to keep those from the previous steps, then we only need to add those with a '1' at the end.

As for the infinity thing, it often happens in math that we imagine to do something an infinite number of times without actually doing it (because, obviously, it's impossible). This kind of operations is called "supertasks" and can lead to some interesting paradoxes.

7 hours ago, PawelS said:

then we only need to add those with a '1' at the end.

Yea, but the challenge is to find the new ones without using one from the previous step (there are new ones, not just new ones of the old ones with an added '1' at the end, so that also requires some work, which takes time unless there is an easier pattern to follow). I added a few more steps because it was fun.

Thinking, the actual count in each step might be an interesting number as well. Not on a prime level, but still

I didn't know about supertasks. Fun names used as well, such as hypertask, ultratask, and so on.

@Splashee Isn't the count in each step just the powers of 2? (unless I misunderstood what you mean here) And if you want to generate the new ones without using the old ones, you can use the following pattern:

0.[N]1

Where [N] means all the integers from 0 to 2^(S-1) - 1, converted to binary form with S-1 digits, and S is the step number.

Step 1: (alternatively 0 can be moved to step 0, so step 1 obeys the same rules as the subsequent ones)
0
0.1

Step 2:
0.01
0.11

Step 3:
0.001
0.011
0.101
0.111

Step 4:
0.0001
0.0011
0.0101
0.0111
0.1001
0.1011
0.1101
0.1111

(the digits in red are those that belong to [N])

Graphically, we're just putting dots on the number line with distances between them that halve in each step, and the initial position shifts so the new ones are halfway between the old ones. After an infinite number of iterations, we fill the entire [0,1] interval (closed on both sides, because 0.(1) = 1 in binary, just like 0.(9) = 1 in decimal).

While I can see my approach ended up being just binary (with an added bit for each step), I fail to see what @PawelS is trying to prove by using binary in the first place? Maybe I need an example with decimal instead? Or maybe I am just not qualified to understand Real numbers.

For the binary part however, if you treat the non-changing part 0. as the sign bit in a binary number, you can range not only from 0.0 to 0.infinity, but from 0.0 to -1.0 as well with just a flip of that bit. Or are we just using 0. as a demonstration rather than information since we don't care about 1.0?

I was also thinking a little bit about Run Length encoding, and thought, since we are getting repeated patterns from the bits for each step, wouldn't it be possible to compress those bits to make them take less space somehow? Or do they need to have all the information to store the value of infinity in the end? Like, some data is so unique it can't be compressed. Just thinking since I don't feel like testing.

Like, imagine you have a repeated pattern of 101101101101. That could be encoded into 100101 (4 times 101) where the bold part is the number of time to repeat the 101. Since that binary number takes less space than having all the 101's repeating, that will be lossless compression. It is more complex than that since you also need some kind of header bit to say the data is either compressed or not compressed, and sometimes, that data can be compressed as well by using the previous data, such as bits could be invented so they can be seen as new different data.

Just think of a scenario that one number out there is either infinity, or the closest you can ever get to infinity, and also compressed (kinda like finding the closest Prime number to infinity)! Well, as the number gets larger, so might the compression data itself, and eventually, it might even get too big so it wouldn't make sense to compress it.

@Splashee I'm trying to prove that the real numbers are a valid mathematical concept, because there is an algorithm (using infinite steps, but still) that generates them all. I'm using binary only for simplicity, you can do the same with decimal. And the representation I'm using is the same as the decimal representation of a number, but with binary instead. So 0.1 (binary) is the same as 0.5 (decimal), for example. And 0.11 (binary) is 0.75 (decimal). The 0. at the beginning is not the sign, it's the integer part which is equal to zero. (As I mentioned, we're only generating the numbers between 0 and 1, but expanding it to all the real numbers is easy). I think you misunderstood it as something like the binary representations of numbers in a computer, but it's not like that. I'm not going to write the steps in decimal, because it's 10 numbers in step 1, 100 in step 2, 1000 in step 3 and so on, so it would take too much space here.

As for 1.0, we don't really need it, because it's a known fact that 0.999... is equal to 1, so 0.111... in binary (the last number in our list after infinite steps) is also equal to 1. (I wrote that as 0.(9) and 0.(1) earlier, which may be confusing, because it's a notation used only in some countries. The digits in parentheses mean that they repeat infinitely many times, I think in the US notation you put a line above them.)

And I don't think run length encoding, or any other compression, would help us here, because we're dealing with all possible combinations of bits, so most of them are just like random series of bits that can't be compressed.

@PawelS

Yea I do understand that, as well as getting a little brushing up on the binary fractions (very much needed for my upcoming projects!)... However, 0.999... is equal to 1? In what mathematical world is that true? I know some people are lazy with rounding, like you know, it is almost this cold, or Pi is equal to exactly 3.1416.

Better check in my trustworthy C compiler:

Quote

    int IsEqual = (0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999L == 1.0L);

012D1CBE  mov         dword ptr [IsEqual],1

Quote

    int IsEqual = (0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999L != 1.0L);

012F1CBE  mov         dword ptr [IsEqual],0

No way....... I stand corrected
 

I decided to do a little C++ experiment showing the binary bits in mirrored order, and their corresponding decimal values:

step 0:
0.0     (Decimal 0.000000)
             
step 1:
0.1     (Decimal 0.500000)             

step 2:
0.01    (Decimal 0.250000)             
0.11    (Decimal 0.750000)

step 3:
0.001   (Decimal 0.125000)             
0.101   (Decimal 0.625000)
0.011   (Decimal 0.375000)
0.111   (Decimal 0.875000)

step 4:
0.0001  (Decimal 0.062500)             
0.1001  (Decimal 0.562500)
0.0101  (Decimal 0.312500)
0.1101  (Decimal 0.812500)
0.0011  (Decimal 0.187500)
0.1011  (Decimal 0.687500)
0.0111  (Decimal 0.437500)
0.1111  (Decimal 0.937500)

step 5:
0.00001 (Decimal 0.031250)             
0.10001 (Decimal 0.531250)
0.01001 (Decimal 0.281250)
0.11001 (Decimal 0.781250)
0.00101 (Decimal 0.156250)
0.10101 (Decimal 0.656250)
0.01101 (Decimal 0.406250)
0.11101 (Decimal 0.906250)
0.00011 (Decimal 0.093750)
0.10011 (Decimal 0.593750)
0.01011 (Decimal 0.343750)
0.11011 (Decimal 0.843750)
0.00111 (Decimal 0.218750)
0.10111 (Decimal 0.718750)
0.01111 (Decimal 0.468750)
0.11111 (Decimal 0.968750)

step 6:
0.000001        (Decimal 0.015625)             
0.100001        (Decimal 0.515625)
0.010001        (Decimal 0.265625)
0.110001        (Decimal 0.765625)
0.001001        (Decimal 0.140625)
0.101001        (Decimal 0.640625)
0.011001        (Decimal 0.390625)
0.111001        (Decimal 0.890625)
0.000101        (Decimal 0.078125)
0.100101        (Decimal 0.578125)
0.010101        (Decimal 0.328125)
0.110101        (Decimal 0.828125)
0.001101        (Decimal 0.203125)
0.101101        (Decimal 0.703125)
0.011101        (Decimal 0.453125)
0.111101        (Decimal 0.953125)
0.000011        (Decimal 0.046875)
0.100011        (Decimal 0.546875)
0.010011        (Decimal 0.296875)
0.110011        (Decimal 0.796875)
0.001011        (Decimal 0.171875)
0.101011        (Decimal 0.671875)
0.011011        (Decimal 0.421875)
0.111011        (Decimal 0.921875)
0.000111        (Decimal 0.109375)
0.100111        (Decimal 0.609375)
0.010111        (Decimal 0.359375)
0.110111        (Decimal 0.859375)
0.001111        (Decimal 0.234375)
0.101111        (Decimal 0.734375)
0.011111        (Decimal 0.484375)
0.111111        (Decimal 0.984375)

step 7:
0.0000001       (Decimal 0.007813)             
0.1000001       (Decimal 0.507813)
0.0100001       (Decimal 0.257813)
0.1100001       (Decimal 0.757813)
0.0010001       (Decimal 0.132813)
0.1010001       (Decimal 0.632813)
0.0110001       (Decimal 0.382813)
0.1110001       (Decimal 0.882813)
0.0001001       (Decimal 0.070313)
0.1001001       (Decimal 0.570313)
0.0101001       (Decimal 0.320313)
0.1101001       (Decimal 0.820313)
0.0011001       (Decimal 0.195313)
0.1011001       (Decimal 0.695313)
0.0111001       (Decimal 0.445313)
0.1111001       (Decimal 0.945313)
0.0000101       (Decimal 0.039063)
0.1000101       (Decimal 0.539063)
0.0100101       (Decimal 0.289063)
0.1100101       (Decimal 0.789063)
0.0010101       (Decimal 0.164063)
0.1010101       (Decimal 0.664063)
0.0110101       (Decimal 0.414063)
0.1110101       (Decimal 0.914063)
0.0001101       (Decimal 0.101563)
0.1001101       (Decimal 0.601563)
0.0101101       (Decimal 0.351563)
0.1101101       (Decimal 0.851563)
0.0011101       (Decimal 0.226563)
0.1011101       (Decimal 0.726563)
0.0111101       (Decimal 0.476563)
0.1111101       (Decimal 0.976563)
0.0000011       (Decimal 0.023438)
0.1000011       (Decimal 0.523438)
0.0100011       (Decimal 0.273438)
0.1100011       (Decimal 0.773438)
0.0010011       (Decimal 0.148438)
0.1010011       (Decimal 0.648438)
0.0110011       (Decimal 0.398438)
0.1110011       (Decimal 0.898438)
0.0001011       (Decimal 0.085938)
0.1001011       (Decimal 0.585938)
0.0101011       (Decimal 0.335938)
0.1101011       (Decimal 0.835938)
0.0011011       (Decimal 0.210938)
0.1011011       (Decimal 0.710938)
0.0111011       (Decimal 0.460938)
0.1111011       (Decimal 0.960938)
0.0000111       (Decimal 0.054688)
0.1000111       (Decimal 0.554688)
0.0100111       (Decimal 0.304688)
0.1100111       (Decimal 0.804688)
0.0010111       (Decimal 0.179688)
0.1010111       (Decimal 0.679688)
0.0110111       (Decimal 0.429688)
0.1110111       (Decimal 0.929688)
0.0001111       (Decimal 0.117188)
0.1001111       (Decimal 0.617188)
0.0101111       (Decimal 0.367188)
0.1101111       (Decimal 0.867188)
0.0011111       (Decimal 0.242188)
0.1011111       (Decimal 0.742188)
0.0111111       (Decimal 0.492188)
0.1111111       (Decimal 0.992188)

step 8:
0.00000001      (Decimal 0.003906)             
0.10000001      (Decimal 0.503906)
0.01000001      (Decimal 0.253906)
0.11000001      (Decimal 0.753906)
0.00100001      (Decimal 0.128906)
0.10100001      (Decimal 0.628906)
0.01100001      (Decimal 0.378906)
0.11100001      (Decimal 0.878906)
0.00010001      (Decimal 0.066406)
0.10010001      (Decimal 0.566406)
0.01010001      (Decimal 0.316406)
0.11010001      (Decimal 0.816406)
0.00110001      (Decimal 0.191406)
0.10110001      (Decimal 0.691406)
0.01110001      (Decimal 0.441406)
0.11110001      (Decimal 0.941406)
0.00001001      (Decimal 0.035156)
0.10001001      (Decimal 0.535156)
0.01001001      (Decimal 0.285156)
0.11001001      (Decimal 0.785156)
0.00101001      (Decimal 0.160156)
0.10101001      (Decimal 0.660156)
0.01101001      (Decimal 0.410156)
0.11101001      (Decimal 0.910156)
0.00011001      (Decimal 0.097656)
0.10011001      (Decimal 0.597656)
0.01011001      (Decimal 0.347656)
0.11011001      (Decimal 0.847656)
0.00111001      (Decimal 0.222656)
0.10111001      (Decimal 0.722656)
0.01111001      (Decimal 0.472656)
0.11111001      (Decimal 0.972656)
0.00000101      (Decimal 0.019531)
0.10000101      (Decimal 0.519531)
0.01000101      (Decimal 0.269531)
0.11000101      (Decimal 0.769531)
0.00100101      (Decimal 0.144531)
0.10100101      (Decimal 0.644531)
0.01100101      (Decimal 0.394531)
0.11100101      (Decimal 0.894531)
0.00010101      (Decimal 0.082031)
0.10010101      (Decimal 0.582031)
0.01010101      (Decimal 0.332031)
0.11010101      (Decimal 0.832031)
0.00110101      (Decimal 0.207031)
0.10110101      (Decimal 0.707031)
0.01110101      (Decimal 0.457031)
0.11110101      (Decimal 0.957031)
0.00001101      (Decimal 0.050781)
0.10001101      (Decimal 0.550781)
0.01001101      (Decimal 0.300781)
0.11001101      (Decimal 0.800781)
0.00101101      (Decimal 0.175781)
0.10101101      (Decimal 0.675781)
0.01101101      (Decimal 0.425781)
0.11101101      (Decimal 0.925781)
0.00011101      (Decimal 0.113281)
0.10011101      (Decimal 0.613281)
0.01011101      (Decimal 0.363281)
0.11011101      (Decimal 0.863281)
0.00111101      (Decimal 0.238281)
0.10111101      (Decimal 0.738281)
0.01111101      (Decimal 0.488281)
0.11111101      (Decimal 0.988281)
0.00000011      (Decimal 0.011719)
0.10000011      (Decimal 0.511719)
0.01000011      (Decimal 0.261719)
0.11000011      (Decimal 0.761719)
0.00100011      (Decimal 0.136719)
0.10100011      (Decimal 0.636719)
0.01100011      (Decimal 0.386719)
0.11100011      (Decimal 0.886719)
0.00010011      (Decimal 0.074219)
0.10010011      (Decimal 0.574219)
0.01010011      (Decimal 0.324219)
0.11010011      (Decimal 0.824219)
0.00110011      (Decimal 0.199219)
0.10110011      (Decimal 0.699219)
0.01110011      (Decimal 0.449219)
0.11110011      (Decimal 0.949219)
0.00001011      (Decimal 0.042969)
0.10001011      (Decimal 0.542969)
0.01001011      (Decimal 0.292969)
0.11001011      (Decimal 0.792969)
0.00101011      (Decimal 0.167969)
0.10101011      (Decimal 0.667969)
0.01101011      (Decimal 0.417969)
0.11101011      (Decimal 0.917969)
0.00011011      (Decimal 0.105469)
0.10011011      (Decimal 0.605469)
0.01011011      (Decimal 0.355469)
0.11011011      (Decimal 0.855469)
0.00111011      (Decimal 0.230469)
0.10111011      (Decimal 0.730469)
0.01111011      (Decimal 0.480469)
0.11111011      (Decimal 0.980469)
0.00000111      (Decimal 0.027344)
0.10000111      (Decimal 0.527344)
0.01000111      (Decimal 0.277344)
0.11000111      (Decimal 0.777344)
0.00100111      (Decimal 0.152344)
0.10100111      (Decimal 0.652344)
0.01100111      (Decimal 0.402344)
0.11100111      (Decimal 0.902344)
0.00010111      (Decimal 0.089844)
0.10010111      (Decimal 0.589844)
0.01010111      (Decimal 0.339844)
0.11010111      (Decimal 0.839844)
0.00110111      (Decimal 0.214844)
0.10110111      (Decimal 0.714844)
0.01110111      (Decimal 0.464844)
0.11110111      (Decimal 0.964844)
0.00001111      (Decimal 0.058594)
0.10001111      (Decimal 0.558594)
0.01001111      (Decimal 0.308594)
0.11001111      (Decimal 0.808594)
0.00101111      (Decimal 0.183594)
0.10101111      (Decimal 0.683594)
0.01101111      (Decimal 0.433594)
0.11101111      (Decimal 0.933594)
0.00011111      (Decimal 0.121094)
0.10011111      (Decimal 0.621094)
0.01011111      (Decimal 0.371094)
0.11011111      (Decimal 0.871094)
0.00111111      (Decimal 0.246094)
0.10111111      (Decimal 0.746094)
0.01111111      (Decimal 0.496094)
0.11111111      (Decimal 0.996094)

step 9:
0.000000001     (Decimal 0.001953)
0.100000001     (Decimal 0.501953)
0.010000001     (Decimal 0.251953)
0.110000001     (Decimal 0.751953)
0.001000001     (Decimal 0.126953)
0.101000001     (Decimal 0.626953)
0.011000001     (Decimal 0.376953)
0.111000001     (Decimal 0.876953)
0.000100001     (Decimal 0.064453)
0.100100001     (Decimal 0.564453)
0.010100001     (Decimal 0.314453)
0.110100001     (Decimal 0.814453)
0.001100001     (Decimal 0.189453)
0.101100001     (Decimal 0.689453)
0.011100001     (Decimal 0.439453)
0.111100001     (Decimal 0.939453)
0.000010001     (Decimal 0.033203)
0.100010001     (Decimal 0.533203)
0.010010001     (Decimal 0.283203)
0.110010001     (Decimal 0.783203)
0.001010001     (Decimal 0.158203)
0.101010001     (Decimal 0.658203)
0.011010001     (Decimal 0.408203)
0.111010001     (Decimal 0.908203)
0.000110001     (Decimal 0.095703)
0.100110001     (Decimal 0.595703)
0.010110001     (Decimal 0.345703)
0.110110001     (Decimal 0.845703)
0.001110001     (Decimal 0.220703)
0.101110001     (Decimal 0.720703)
0.011110001     (Decimal 0.470703)
0.111110001     (Decimal 0.970703)
0.000001001     (Decimal 0.017578)
0.100001001     (Decimal 0.517578)
0.010001001     (Decimal 0.267578)
0.110001001     (Decimal 0.767578)
0.001001001     (Decimal 0.142578)
0.101001001     (Decimal 0.642578)
0.011001001     (Decimal 0.392578)
0.111001001     (Decimal 0.892578)
0.000101001     (Decimal 0.080078)
0.100101001     (Decimal 0.580078)
0.010101001     (Decimal 0.330078)
0.110101001     (Decimal 0.830078)
0.001101001     (Decimal 0.205078)
0.101101001     (Decimal 0.705078)
0.011101001     (Decimal 0.455078)
0.111101001     (Decimal 0.955078)
0.000011001     (Decimal 0.048828)
0.100011001     (Decimal 0.548828)
0.010011001     (Decimal 0.298828)
0.110011001     (Decimal 0.798828)
0.001011001     (Decimal 0.173828)
0.101011001     (Decimal 0.673828)
0.011011001     (Decimal 0.423828)
0.111011001     (Decimal 0.923828)
0.000111001     (Decimal 0.111328)
0.100111001     (Decimal 0.611328)
0.010111001     (Decimal 0.361328)
0.110111001     (Decimal 0.861328)
0.001111001     (Decimal 0.236328)
0.101111001     (Decimal 0.736328)
0.011111001     (Decimal 0.486328)
0.111111001     (Decimal 0.986328)
0.000000101     (Decimal 0.009766)
0.100000101     (Decimal 0.509766)
0.010000101     (Decimal 0.259766)
0.110000101     (Decimal 0.759766)
0.001000101     (Decimal 0.134766)
0.101000101     (Decimal 0.634766)
0.011000101     (Decimal 0.384766)
0.111000101     (Decimal 0.884766)
0.000100101     (Decimal 0.072266)
0.100100101     (Decimal 0.572266)
0.010100101     (Decimal 0.322266)
0.110100101     (Decimal 0.822266)
0.001100101     (Decimal 0.197266)
0.101100101     (Decimal 0.697266)
0.011100101     (Decimal 0.447266)
0.111100101     (Decimal 0.947266)
0.000010101     (Decimal 0.041016)
0.100010101     (Decimal 0.541016)
0.010010101     (Decimal 0.291016)
0.110010101     (Decimal 0.791016)
0.001010101     (Decimal 0.166016)
0.101010101     (Decimal 0.666016)
0.011010101     (Decimal 0.416016)
0.111010101     (Decimal 0.916016)
0.000110101     (Decimal 0.103516)
0.100110101     (Decimal 0.603516)
0.010110101     (Decimal 0.353516)
0.110110101     (Decimal 0.853516)
0.001110101     (Decimal 0.228516)
0.101110101     (Decimal 0.728516)
0.011110101     (Decimal 0.478516)
0.111110101     (Decimal 0.978516)
0.000001101     (Decimal 0.025391)
0.100001101     (Decimal 0.525391)
0.010001101     (Decimal 0.275391)
0.110001101     (Decimal 0.775391)
0.001001101     (Decimal 0.150391)
0.101001101     (Decimal 0.650391)
0.011001101     (Decimal 0.400391)
0.111001101     (Decimal 0.900391)
0.000101101     (Decimal 0.087891)
0.100101101     (Decimal 0.587891)
0.010101101     (Decimal 0.337891)
0.110101101     (Decimal 0.837891)
0.001101101     (Decimal 0.212891)
0.101101101     (Decimal 0.712891)
0.011101101     (Decimal 0.462891)
0.111101101     (Decimal 0.962891)
0.000011101     (Decimal 0.056641)
0.100011101     (Decimal 0.556641)
0.010011101     (Decimal 0.306641)
0.110011101     (Decimal 0.806641)
0.001011101     (Decimal 0.181641)
0.101011101     (Decimal 0.681641)
0.011011101     (Decimal 0.431641)
0.111011101     (Decimal 0.931641)
0.000111101     (Decimal 0.119141)
0.100111101     (Decimal 0.619141)
0.010111101     (Decimal 0.369141)
0.110111101     (Decimal 0.869141)
0.001111101     (Decimal 0.244141)
0.101111101     (Decimal 0.744141)
0.011111101     (Decimal 0.494141)
0.111111101     (Decimal 0.994141)
0.000000011     (Decimal 0.005859)
0.100000011     (Decimal 0.505859)
0.010000011     (Decimal 0.255859)
0.110000011     (Decimal 0.755859)
0.001000011     (Decimal 0.130859)
0.101000011     (Decimal 0.630859)
0.011000011     (Decimal 0.380859)
0.111000011     (Decimal 0.880859)
0.000100011     (Decimal 0.068359)
0.100100011     (Decimal 0.568359)
0.010100011     (Decimal 0.318359)
0.110100011     (Decimal 0.818359)
0.001100011     (Decimal 0.193359)
0.101100011     (Decimal 0.693359)
0.011100011     (Decimal 0.443359)
0.111100011     (Decimal 0.943359)
0.000010011     (Decimal 0.037109)
0.100010011     (Decimal 0.537109)
0.010010011     (Decimal 0.287109)
0.110010011     (Decimal 0.787109)
0.001010011     (Decimal 0.162109)
0.101010011     (Decimal 0.662109)
0.011010011     (Decimal 0.412109)
0.111010011     (Decimal 0.912109)
0.000110011     (Decimal 0.099609)
0.100110011     (Decimal 0.599609)
0.010110011     (Decimal 0.349609)
0.110110011     (Decimal 0.849609)
0.001110011     (Decimal 0.224609)
0.101110011     (Decimal 0.724609)
0.011110011     (Decimal 0.474609)
0.111110011     (Decimal 0.974609)
0.000001011     (Decimal 0.021484)
0.100001011     (Decimal 0.521484)
0.010001011     (Decimal 0.271484)
0.110001011     (Decimal 0.771484)
0.001001011     (Decimal 0.146484)
0.101001011     (Decimal 0.646484)
0.011001011     (Decimal 0.396484)
0.111001011     (Decimal 0.896484)
0.000101011     (Decimal 0.083984)
0.100101011     (Decimal 0.583984)
0.010101011     (Decimal 0.333984)
0.110101011     (Decimal 0.833984)
0.001101011     (Decimal 0.208984)
0.101101011     (Decimal 0.708984)
0.011101011     (Decimal 0.458984)
0.111101011     (Decimal 0.958984)
0.000011011     (Decimal 0.052734)
0.100011011     (Decimal 0.552734)
0.010011011     (Decimal 0.302734)
0.110011011     (Decimal 0.802734)
0.001011011     (Decimal 0.177734)
0.101011011     (Decimal 0.677734)
0.011011011     (Decimal 0.427734)
0.111011011     (Decimal 0.927734)
0.000111011     (Decimal 0.115234)
0.100111011     (Decimal 0.615234)
0.010111011     (Decimal 0.365234)
0.110111011     (Decimal 0.865234)
0.001111011     (Decimal 0.240234)
0.101111011     (Decimal 0.740234)
0.011111011     (Decimal 0.490234)
0.111111011     (Decimal 0.990234)
0.000000111     (Decimal 0.013672)
0.100000111     (Decimal 0.513672)
0.010000111     (Decimal 0.263672)
0.110000111     (Decimal 0.763672)
0.001000111     (Decimal 0.138672)
0.101000111     (Decimal 0.638672)
0.011000111     (Decimal 0.388672)
0.111000111     (Decimal 0.888672)
0.000100111     (Decimal 0.076172)
0.100100111     (Decimal 0.576172)
0.010100111     (Decimal 0.326172)
0.110100111     (Decimal 0.826172)
0.001100111     (Decimal 0.201172)
0.101100111     (Decimal 0.701172)
0.011100111     (Decimal 0.451172)
0.111100111     (Decimal 0.951172)
0.000010111     (Decimal 0.044922)
0.100010111     (Decimal 0.544922)
0.010010111     (Decimal 0.294922)
0.110010111     (Decimal 0.794922)
0.001010111     (Decimal 0.169922)
0.101010111     (Decimal 0.669922)
0.011010111     (Decimal 0.419922)
0.111010111     (Decimal 0.919922)
0.000110111     (Decimal 0.107422)
0.100110111     (Decimal 0.607422)
0.010110111     (Decimal 0.357422)
0.110110111     (Decimal 0.857422)
0.001110111     (Decimal 0.232422)
0.101110111     (Decimal 0.732422)
0.011110111     (Decimal 0.482422)
0.111110111     (Decimal 0.982422)
0.000001111     (Decimal 0.029297)
0.100001111     (Decimal 0.529297)
0.010001111     (Decimal 0.279297)
0.110001111     (Decimal 0.779297)
0.001001111     (Decimal 0.154297)
0.101001111     (Decimal 0.654297)
0.011001111     (Decimal 0.404297)
0.111001111     (Decimal 0.904297)
0.000101111     (Decimal 0.091797)
0.100101111     (Decimal 0.591797)
0.010101111     (Decimal 0.341797)
0.110101111     (Decimal 0.841797)
0.001101111     (Decimal 0.216797)
0.101101111     (Decimal 0.716797)
0.011101111     (Decimal 0.466797)
0.111101111     (Decimal 0.966797)
0.000011111     (Decimal 0.060547)
0.100011111     (Decimal 0.560547)
0.010011111     (Decimal 0.310547)
0.110011111     (Decimal 0.810547)
0.001011111     (Decimal 0.185547)
0.101011111     (Decimal 0.685547)
0.011011111     (Decimal 0.435547)
0.111011111     (Decimal 0.935547)
0.000111111     (Decimal 0.123047)
0.100111111     (Decimal 0.623047)
0.010111111     (Decimal 0.373047)
0.110111111     (Decimal 0.873047)
0.001111111     (Decimal 0.248047)
0.101111111     (Decimal 0.748047)
0.011111111     (Decimal 0.498047)
0.111111111     (Decimal 0.998047)

step 10:
0.0000000001    (Decimal 0.000977)
0.1000000001    (Decimal 0.500977)

This bit-pattern gives a 1 to the very right hand side for every step just like I was trying manually above. Hopefully with my vague knowledge of math, I got the right decimal numbers

The proof of 0.999... =1 is not going to come from a program because they have a finite memory and have to round. The basic idea is if we assume 0.9 ... does not equal 1 then that would mean there is some number 0.0....x that is at the end of an infinite string of 0.9999. This number is bigger than 0 but smaller than any number imaginable? Such a number doesn't exist. That limit would go to 0.

14 hours ago, Thankful Brony 42 said:

The proof of 0.999... =1 is not going to come from a program because they have a finite memory and have to round. The basic idea is if we assume 0.9 ... does not equal 1 then that would mean there is some number 0.0....x that is at the end of an infinite string of 0.9999. This number is bigger than 0 but smaller than any number imaginable? Such a number doesn't exist. That limit would go to 0.

And the nature of floating point itself. As you could see in my test, the C compiler didn't even try to convert the value to any floating point, but simply just gave the variable a 1 for true, and a 0 for false (at compile time).
I have to do quite a lot of thinking to figure out how to write good algorithms for 3D graphics shaders since you can't just compare something to 1.0, or 0.0. That's why I immediately didn't like thinking about how 0.999... can be equal to 1. I mean, is 1.0 + 0.999... equal to 2.0? Or is it equal to 1.999...? How about 60 * 0.999...? Is it equal to 60? It just doesn't feel right to me.
One thing that I really dislike is that a pixel on the screen can have a digital value of 0 to 255 for red (just an example), but 3D uses normalized values, so the brightest read is 1.0. This causes a lot of frustration since graphics hardware sometimes try floating point with less precision to speed things up, and there is no clean way to get those colors (you just aren't allowed to reach the binary value, the hardware only takes floating point).

1 hour ago, Splashee said:

I mean, is 1.0 + 0.999... equal to 2.0? Or is it equal to 1.999...?

2.0 and 1.999... are the same, so both answers are true.

I still don't like this 0.999... = 1 thing @PawelS

I was hoping for a YouTube video explaining a good way. But all I see is the same problem as before, that one number equals another, that that's it:
 

To me, I still see the same difference between 1.0000...1 (that is probably impossible number to begin with) and 1.0 as 1.999... and 1.0. Though, it is hard to describe an infinitely long recurring string of 0's that ends with a 1 when it can't end in the first place..... Meh, this is making me go insane

Wait, why do we even have 0.999...? Is there like some equation that equals this? (like 1/3 = 0.333...) And why wouldn't it just be 1.0 then since they are the same? Seems more like a trick question

No.... I did 1/3 in calculator and added it 3 times, and pressed + and two = signs, got 1 instead of 0.999....... Okay, It is correct! Me give up! (You are a good teacher @PawelS)

4 minutes ago, Splashee said:

Wait, why do we even have 0.999...? Is there like some equation that equals this? (like 1/3 = 0.333...) And why wouldn't it just be 1.0 then since they are the same? Seems more like a trick question

Let's try to calculate 1/3 * 3 by hand, but using decimal fractions.

1/3 = 0.333...

So if we multiply it by 3, we should multiply every digit by 3 and in case of an overflow reduce it by 10 and increase the previous digit by 1.

But since there are only 3's (and a zero at the beginning), there is no overflow, we just multiply every 3 by 3 so they all become 9. So we get this:

0.333... * 3 = 0.999...

But we also know that 1/3 * 3 = 1, so 0.999... and 1 are the same thing.