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0.999...= 1

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Re: 0.999...= 1

#61

Post by baddox »

tomtomtom1 wrote: post #31
saagarjha wrote: post #13

Earlier quoted context omitted.

Arithmetic breaks, as multiplication is no longer the inverse of division. (For example, 1/3 * 3 = 0.999… would no longer work.)

why? 1/3 * 3 could still be equal to one. but 1/3 != 0.33333... that is, 1/3 is not representable in base 10. Which makes way more sense. I wonder if taking 0.9999.. != 1, that is 0.0000...1 exists would allow us to reslove, the fact that some possible events have probability 0?

Limits would break, which might break or at least cause problems with calculus and other branches of math.

Re: 0.999...= 1

#62

Post by vfinn »

Sorry for my naivety, but why one couldn't prove by induction that adding 9s never close the gap, or let's say, that by definition the operation is such, that it never closes the gap. If you can always halve the pie, then you can continue eating forever. To me it would be much easier to accept that (1/3)*3 is not 1.

Re: 0.999...= 1

#63

Post by ssivark »

knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

> The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise.

Sure, but the point of "elementary school" arithmetic is not "elementary" arithmetic, as a mathematician would define it :-)

The goal is to teach people to reason by matching patterns. Deductive/Inductive reasoning can slowly proceed from that, as they try to frame their intuition for patterns into increasingly more general abstractions.

Re: 0.999...= 1

#64

Post by atq2119 »

tomtomtom1 wrote: post #31
saagarjha wrote: post #13

Earlier quoted context omitted.

Arithmetic breaks, as multiplication is no longer the inverse of division. (For example, 1/3 * 3 = 0.999… would no longer work.)

why? 1/3 * 3 could still be equal to one. but 1/3 != 0.33333... that is, 1/3 is not representable in base 10. Which makes way more sense. I wonder if taking 0.9999.. != 1, that is 0.0000...1 exists would allow us to reslove, the fact that some possible events have probability 0?

The crux of the matter is that you have to define what things like "0.3333..." mean in the first place. Any reasonable definition of it as a representation of a real number is going to lead to it being equal to 1/3.

If you want to redefine it explicitly as not a real number, you can do that, and maybe even get to some amusing math that way, but you're no longer talking the same language as the rest of the world.

Re: 0.999...= 1

#66

Post by saagarjha »

champagneben wrote: post #51
hinkley wrote: post #40

Earlier quoted context omitted.

Yeah that’s way more complicated than it needs to be and I’m tempted to replace that whole section with: x = 0.9999... 2x = 1.9999... 2x - x = 1 x = 1

How do you jump from x = 0.999... to 2x = 1.999..

It’s less intuitive, but still relatively straightforward from performing the “long” multiplication. You do need to convince people that the “8 at the end“ is irrelevant, however.

Re: 0.999...= 1

#67

Post by agumonkey »

knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

And I thought mathematicians would reject normal arithmetic operation over the domain of `N...` elements. They're always so ultra rigorous to classify what is or is not, what is defined, the domain .. but then they let an infinite sequence be treated as any finite number.

Re: 0.999...= 1

#68

Post by tomtomtom1 »

knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

if we say that infinitesimals exist. that 1/3 != 0.33.. and 1 != 0.9999... and the probability of possible events is never 0.

what are the properties that we would lose?

Re: 0.999...= 1

#69

Post by saagarjha »

emerongi wrote: post #57
ginko wrote: post #9

I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious: 1/3 = 0.333.. 3 * 1/3 = 3 * 0.333.. 3/3 = 0.999.. 1 = 0.999..

This also happens to be the test for whether your calculator is any good.

My high-quality scientific calculator makes this a bit uninteresting:

  1/3            ⅓
  Ans * 3        1

Re: 0.999...= 1

#70

Post by saagarjha »

vfinn wrote: post #62

Sorry for my naivety, but why one couldn't prove by induction that adding 9s never close the gap, or let's say, that by definition the operation is such, that it never closes the gap. If you can always halve the pie, then you can continue eating forever. To me it would be much easier to accept that (1/3)*3 is not 1.

Induction can be non-intuitive to people who are troubled by this problem.
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