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0.999...= 1

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Re: 0.999...= 1

#91

Post by polotics »

ball_of_lint wrote: post #74
baddox wrote: post #49

Earlier quoted context omitted.

But you absolutely can evaluate that limit as N goes to infinity and correctly conclude that 1/3 does equal 0.333 repeating.

Repeating decimals may be introduced in a mathematics education long before any other infinite series or the methods used to tame them, such ass limits.

I see the introduction of a conceptually incomplete notion as one of the big hurdle of school-level math: without limits, children are asked to just adopt the axiom that there is such a thing as a concept of infinity with any form of practical usefulness, and this riles.

It's only with limits and proper formalism that I was reconciled with maths that frankly were just tending towards approaching an equality with bullshit.

Re: 0.999...= 1

#92

Post by himinlomax »

iso1631 wrote: post #12
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..

Another secondary school 'proof' x = 0.9999..... 10x = 9.9999..... (10x -x) = 9x = (9.9999.... - 0.9999....) = 9 x = 9/9 = 1

I like this one better:

    0.9        = 1 - 0.1
    0.99       = 1 - 0.01
    0.999      = 1 - 0.001
    0.9999     = 1 - 0.0001
    0.99999... = 1 - 0.00000... with a 1 at the end of the infinite series of 0

Re: 0.999...= 1

#93

Post by rs23296008n1 »

AlEinstein wrote: post #14
rs23296008n1 wrote: post #6

I'm actually curious what impact it would have on various proofs if 0.999... wasn't accepted as 1. What gets broken? What consequences do we hit?

You can multiply 1/3 by 3 and not get 1.

I have a pizza. I divide it into three parts.

You'd be asserting that if I eat the three parts I have not eaten the whole pizza.

I'm unconvinced.

Re: 0.999...= 1

#94

Post by CogitoCogito »

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…

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards.

I've never considered them right at all. By saying something like

0.9... x 10 = 9.9...

and then saying that

9.9... - 0.9... = 9

you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you're basically just defining it to be 1.

I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Unless you're making use of limits, completeness, or something equivalent I don't see how a proof could possibly make any sense.

Re: 0.999...= 1

#95

Post by mcphage »

CogitoCogito wrote: post #94
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…

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…

> I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning.

Peano arithmetic only covers nonnegative whole numbers, so one will never exist.

Re: 0.999...= 1

#96

Post by CogitoCogito »

mcphage wrote: post #95
CogitoCogito wrote: post #94

Earlier quoted context omitted.

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…

> I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Peano arithmetic only covers nonnegative whole numbers, so one will never exist.

> Peano arithmetic only covers nonnegative whole numbers, so one will never exist.

Thank you for the pedantism. How about I replace "Peano arithmetic" with the "operations of multiplication/addition/division/etc. expressible upon the rational numbers"?

Re: 0.999...= 1

#97

Post by mFixman »

tomtomtom1 wrote: post #68
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?

If we say that infinitesimals exist, it still happens that 1 = 0.999…. It just happens that 0.999 ≠ 1 - 𝛚.

0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.

Re: 0.999...= 1

#98

Post by _v7gu »

tomtomtom1 wrote: post #68
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?

Nonstandard analysis exists (with infinitesimal and infinite numbers) , but 1/3 and 9/9 is the same there. The problem is that the numbers 0.333... and 0.999... don't really exist.

Re: 0.999...= 1

#99

Post by jayd16 »

himinlomax wrote: post #92
iso1631 wrote: post #12

Earlier quoted context omitted.

Another secondary school 'proof' x = 0.9999..... 10x = 9.9999..... (10x -x) = 9x = (9.9999.... - 0.9999....) = 9 x = 9/9 = 1

I like this one better: 0.9 = 1 - 0.1 0.99 = 1 - 0.01 0.999 = 1 - 0.001 0.9999 = 1 - 0.0001 0.99999... = 1 - 0.00000... with a 1 at the end of the infinite series of 0

>[...] at the end of the infinite series of [...]

...uhh

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