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0.999...= 1

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Re: 0.999...= 1

#231

Post by traderjane »

kangnkodos wrote: post #201
rrmm wrote: post #48

Earlier quoted context omitted.

Wildberger is great. His lectures that he teaches at UNSW (i think) are interesting, and he usually keeps a clear dividing line between std math and his own predilections. It threads the line between being a kook and legitimate published mathematician very finely. I actually have some sympathies with his contention that real numbers (limit points of infinite series) are somehow a different animal than rational number…

But on computers, you get things such as console.log(0.1 + 0.2) // 0.30000000000000004 A mathematician might say that this shows that you do not really have accurate floating point values and arithmetic in your computer, but instead something close to it.

That's a coincidence of the particular number system you use, and many programming languages have multiple number implementations.

Racket starts with arbitrary precision rationals.

Re: 0.999...= 1

#232

Post by zests »

undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

The real issue is that we don't define the real number system before we use it. The fact that 0.999... = 1 is a consequence of a formal definition of decimal numbers. We can create a new definition of decimal numbers that does not satisfy this equation and use it in place of our current one.

Let's imagine a new decimal number system with some vague notion of infinitesimal numbers. We lose some properties we enjoy in our current system but all of those properties still hold for numbers with no infinitesimal part. We can still use our every day numbers like nothing has changed yet we also have a notion to describe infinitesimal values. We can make statements like 1/3 is infinitesimally less than 0.333... and carry on like nothing else has changed.

Now let's sit someone down, start with the rational numbers, introduce Dedekind cuts to define the real numbers and prove that in the real number system that 0.999... is exactly equal to one. Let's also convince them that the real numbers are the unique complete ordered field and that each of these properties are indispensable. Then they will believe that 0.999... should be equal to 1.

Re: 0.999...= 1

#233

Post by sebringj »

The thing that helps me "understand" it is that the universe has finite sizes of things like the Planck length for example being a theoretical thing at the smallest distance I would imagine. Now imagine it going smaller than the Planck length (finite) in terms of the difference of .9 repeating and 1 since infinitely small differences can do that. Essentially there is no way to tell the difference between .9 repeating and 1 then from a practical or theoretical perspective of measurement. So not imagining infinity lets us at least imagine smaller than the smallest measurable thing.

Re: 0.999...= 1

#234

Post by zests »

tom-thistime wrote: post #156
SpicyLemonZest wrote: post #145

Earlier quoted context omitted.

Those number systems do exist, but I'm not sure it's right to say they work just as well for everyday purposes. They work only as long as you use them in a way that reduces to treating them as real numbers, either never computing an infintesimal in the first place or calculating 23 + 6ε and saying "oh that's basically just 23".

Sure. And it's true, 0.99... is equal to 1. All I'm saying is [SOPI below] it's all a little more technical than the junior high school proof. For example if 23+6\epsilon = 23, then how do I define 23 + 6\epsilon - 23? I can choose different approaches here, but "zero" is going to be pretty inconvenient when I go to do an integral. [SOPI] Statement of Personal Ignorance. I don't quite know what I'm talking about. If…

What are you integrating over?

Re: 0.999...= 1

#235

Post by ta1234567890 »

lukifer wrote: post #228
undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly p…

> As a static number, the concept of infinity makes no sense; but once reimagined as a process

Super insightful. That's the key right there.

The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.

Re: 0.999...= 1

#236

Post by scrimps »

tom-thistime wrote: post #169
Igelau wrote: post #164

Earlier quoted context omitted.

What's interesting is that people pretty quickly become comfortable with the idea that 1/3 = 0.333… So using that as a foothold, we can express 1/3 + 1/3 + 1/3 as 0.333… + 0.333… + 0.333… and it should be pretty easy to digest. At once we can see that in this little zone we've defined, 1 and 0.999… mean the same thing. Not a rigorous proof, and one or two people will probably bring up whataboutisms like "that's just…

This is a really good point. Maybe the problem is how we define equality. What's the test for when two numbers are equal? People accept that 1/3 = 0.333333... The same people don't always seem to accept that 3*0.33333... = 1. Well, how are we defining "equals"? If we can give that definition in black and white, I think that may help.

If 30.33333... = 1, then 30.33333... != 0.9999..., then 0.9999... != 1

Re: 0.999...= 1

#238

Post by TheRealPomax »

hilbertseries wrote: post #177
bognition wrote: post #168

Earlier quoted context omitted.

No .666666 is not equal to .6700000 0.666... is equal to 0.666...7

Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?

Can you explain what you mean with "real" in that sentence? Because in the context of maths, a real number is "a number in ℝ", which this absolutely qualifies for. Whereas in plain English the term doesn't really have a clear definition.

You might consider "real" numbers (in plain English) to mean physically measurable quantities, but there are plenty of numbers that we can write out because they're infinitely long, but trivially "made" (such as π, which just requires grabbing a compass and drawing a circle)

The bit you should be wondering about is why 0.666...6 and 0.666...7 are the same number: infinities cause digits written on paper (or a computer screen) to look like a kind of number that they're not. The two fractions (numbers in ℚ) 0.6666 and 0.66667 are 0.00001 apart, but the two reals (numbers in ℝ) 0.666...6 and 0.666...7 are 0.000...1 apart. That looks like a tiny tiny fraction, but it's not a fraction, it's an infinite number of zeroes, and thanks to that, this number, while it looks like a fraction, is just a silly way to write zero.

So thanks to infinities, the most-definitely-not-a-fraction number that we write as 0.666... is the same as the most-definitely-not-a-fraction number 0.666...(some numbers here). The difference between the two is zero.

Infinities are fun. And difficult. But also fun.

Re: 0.999...= 1

#239

Post by dsabanin »

tomtomtom1 wrote: post #180
johnhattan wrote: post #174

Earlier quoted context omitted.

I remember a conversation I had with my daughter in the car when she was starting out with algebra... Me: Is 9.999... the same as 10, or is it just really close to 10? Kid: Really close. It never gets all the way there. Me: Well then how close? What do you get when you subtract 9.999... from 10? Kid: (pause) An infinite number of zeroes. . .and then a one. . .wait, you can't do that. Me: Right. You just have an infin…

> An infinite number of zeroes. . .and then a one. . .wait, you can't do that. why not? why can't an infinitely small number exist?

Because if it has 1 at the end, then this will mark it’s end, thus making it finitely small.

Re: 0.999...= 1

#240

Post by adunk »

undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

One way to introduce the idea that a number represented as decimal digits can have multiple representations is to talk about the numbers 1 and 1.0 being exactly the same. And that 1.00 is the same as 1. Just like 0, 0.0, and 0.00 are the same number. Most people would agree at this point.

Then keep stretching the number of zeroes to 0.000... - which, again, is exactly the same as 0.

From there, it is not a huge stretch to be able to go from that 0.000... is another way to write 0, then 0.999... is another way to write 1.

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