1 = 0,99........

[BAndrew]

However weird this might sound 1 is equal to 0,99....(infinite 9s)

Proof (note that 0,9~ = 0,99......):

x = 0,9~
10x = 9,9~ (multiplied by 10)
10x - x = 9,9~ - 0,9~ (substracted x)
9x = 9
x = 1
1 = 0,9~


A more analytical proof is:





EDIT: Simple and Practical version by Unearthlybrutal

Simply version

1 / 3 = 0.33333... (infinite 3)
Then:
0.33333... x 3 = 0.9999999... (infinite 9)

(1 = 0.99999999...)

[Froge]

Pretty old news bro

Spoiler below!

But if you want a proof without all the maths, just consider the distance between each number. For example there are infinitely many smaller numbers between 0.9 and 1, but there isn't a single possible number between 0.9~ and 1.

[BAndrew]

Clever. But you forget to explain the point of this thread in all the fancy maths.

The goal of this discussion is to undersand why this is how it is. Not many people know that and if they see it their reaction is "This is impossible" or "No, it can't be. Something is wrong".

Pretty old news bro

Spoiler below!

But if you want a proof without all the maths, just consider the distance between each number. For example there are infinitely many smaller numbers between 0.9 and 1, but there isn't a single possible number between 0.9~ and 1.

I am aware of that . Great explanation for someone who is not good at maths.

[Bridge]

It's totally irrelevant though. No computer is capable of computing infinitesimals so 1 - 0.9~ on any calculator will always be 1 x 10^-∞. Personally it makes my brain hurt so I have a hard time accepting it.

EDIT: However note that even that is too much for modern computers, so replace ~ and infinity with whatever the amount max number of supported decimal places is.

[BAndrew]

It's totally irrelevant though. No computer is capable of computing infinitesimals so 1 - 0.9~ on any calculator will always be 1 x 10^-∞. Personally it makes my brain hurt so I have a hard time accepting it.

You calculate that with limits (needs advanced mathematics). A computer can calculate it with some mathematical methods. I cannot say the same for a calculator though. Well, like it or not, the proof is just above.

[Bridge]

It's totally irrelevant though. No computer is capable of computing infinitesimals so 1 - 0.9~ on any calculator will always be 1 x 10^-∞. Personally it makes my brain hurt so I have a hard time accepting it.

You calculate that with limits (needs advanced mathematics). A computer can calculate it with some mathematical methods. I cannot say the same for a calculator though. Well, like it or not, the proof is just above.

A calculator is a computer. I realize it is legit on paper but it is simply not practical. Almost all big calculations are handled by computers. You probably already know this, but there is no such thing as infinite memory. The number infinity has no calculable value and the value of 0.9~ requires an infinitesimal to precisely calculate and by definition computers need measurable data to calculate. Therefore neither of these operations are possible to truly calculate on computers. You always need to approximate, and with approximation comes a loss of data.

EDIT: I'm not saying I don't believe it, I just question its usefulness and whether or not the two numbers should be equal. When quantum computers become a reality we will most likely be able to precisely calculate the distance between 0.9~ and 1, because there is one, no matter how small, and the numbers will cease to be equal.

[BAndrew]

A calculator is a computer. I realize it is legit on paper but it is simply not practical. Almost all big calculations are handled by computers. You probably already know this, but there is no such thing as infinite memory. The number infinity has no calculable value and the value of 0.9~ requires an infinitesimal to precisely calculate and by definition computers need measurable data to calculate. Therefore neither of these operations are possible to truly calculate on computers. You always need to approximate, and with approximation comes a loss of data.

A calculator is a computer, but a computer is not a calculator. A modern computer cannot calculate the value "0,9~" if that is what you mean but it can calculate the value 1 which is equal to 0,9~. Also, a computer can calculate a sequence with infinite series.

There is no distance between 1 and 0,9~. If there is, it is infinitely small.

[Bridge]

Dude, the two are one and the same. Compute and calculate are synonyms, and computers were first invented to compute data. Calculators are just computer programs that take in an n amount of numbers and operators and return the calculated value of those numbers.

As I said, computers cannot work with infinite numbers. Regular integers on 32bit computers are 4 bytes large and the maximum value of a signed (both negative and positive) is 2,147,483,647. The absolute highest value I can get is an unsigned long long integer and that value is 18,446,744,073,709,551,615. Now I don't have a supercomputer but this is the absolute largest integer my compiler and computer support and it is far from infinite. I don't want to sound like a douchebag and pretend like I actually proved that 0.9~ is not equal to 1, but I think I proved that modern computers cannot work with infinite numbers. The number above is likely a 64bit integer which is the most that a 64bit computer architecture can handle.

All of these numbers took up physical memory while I was doing these calculations which should be sufficient proof that computers cannot work with infinite series. Again, there is no such thing as infinite memory. If you see a calculation in a computer program that works with an infinite series it is because somebody defined that infinite series with a finite number specified in some header somewhere. Infinitesimals are the same deal. They are too small to be measured, and if you want to do a calculation on a computer you need something measurable.

[Froge]

Well, Bridge, the question of usefulness shouldn't always be asked in mathematics. That's if you're a physicist - if you're a theoretical or pure mathematician, you'd see maths as art, a form of creative expression, rather than the tool it was created for.

[Bridge]

Well, Bridge, the question of usefulness shouldn't always be asked in mathematics. That's if you're a physicist - if you're a theoretical or pure mathematician, you'd see maths as art, a form of creative expression, rather than the tool it was created for.

I guess. I use math in programming to do stuff so I don't see too much artistic value in it. I find complex mathematical concepts interesting but the application of said concepts is much more interesting to me.