Daughter's math homework (1 Viewer)

[Hooper31]

1/10 .1 1/100 is .011/infinity is .00000...11/infinity plus .99999... = 1therefore 1/infinity = 0therefore 0 * infinity = 1therefore 1/0 = infinity

Infinity isn't a number you can divide by. It's not a number at all, but rather a symbol that represents a concept (limit, or lack thereof).

 

[Hooper31]

at some point the convention of -5^2 was accepted that it is -(5^2)?as opposed to (-5)^2?...when did that become the convention out of curiosity?

When? Why would you ever assume that it just happened? Isn't it more likely that is has always been, but perhaps you are only now learning it?

 

[tommyboy]

at some point the convention of -5^2 was accepted that it is -(5^2)?as opposed to (-5)^2?...when did that become the convention out of curiosity?

When? Why would you ever assume that it just happened? Isn't it more likely that is has always been, but perhaps you are only now learning it?

guess so. i read it as negative 5 squared first time i saw it. so did a lot of people. we are all morons apparently.by the way the reason a negative number multiplied times a negative number is a positive is because you are losing a negative which is a positive result. its that simple

 

[Short Corner]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

Can you 'prove' 1 = 2?

Only if you let me divide by 0.

 

[dparker713]

So some here are honestly telling me that 5^2 - 5^2= 50?

yes, they typed that above. it's great, isn't it?

Actually, what I would say is the 5^2 + -5^2 = 50. I would agree that 5^2 - 5^2 = 0. Spaces matter. Or would you ever write the integer negative 5 as "- 5"?

 

[gonzobill5]

-5^2 is read "the opposite of 5 squared"

If you can communicate well there is much less confusion.

Sure, but "the opposite of 5 squared" can be read as, "the opposite of five, squared" or "the opposite of five-squared".It's like that thing with the girl spinning and you can't tell which direction but once you see it one way you can't see how someone could see it the other way. http://www.moillusions.com/2007/06/spinnin...l-illusion.html

Not unless you add a comma or a pause. That would be important to articulate too.

 

[Bull Dozier]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

 

[Maurile Tremblay]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

Dead serious.ETA: Well, except for the part about needing calculators.

 

[Short Corner]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....

Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....

Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

Let A = .999.....10A = 9.999...

1A = .9999... (subtract)

9A = 9

A = 1

 

[Bronco Billy]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's a valid proof, but could have left the 9*.111111... step out as it adds nothing to that proof. 0.9 repeating does equal 1. It's an extremely important concept in mathematics since it lends proof to limits.

 

[Ignoratio Elenchi]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's not meant to be funny, just one of those strange things in math that seem like they shouldn't be true, but are.

 

[Ignoratio Elenchi]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's a valid proof, but could have left the 9*.111111... step out as it adds nothing to that proof.

Without that part, he'd just be saying1/9 = 0.111....9 * 1/9 = 1Therefore, 0.999... = 1.That conclusion doesn't follow unless you recognize/specify that 0.999... = 9 * 0.111...

 

[Bronco Billy]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's a valid proof, but could have left the 9*.111111... step out as it adds nothing to that proof.

Without that part, he'd just be saying1/9 = 0.111....9 * 1/9 = 1Therefore, 0.999... = 1.That conclusion doesn't follow unless you recognize/specify that 0.999... = 9 * 0.111...

Okay, I'll concede that.

 

[dgreen]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's not meant to be funny, just one of those strange things in math that seem like they shouldn't be true, but are.

I think we've all had bad math teachers during our formal education. We've had good ones, too, of course. But, when you have a bad teacher, some simple things are just never really explained well. They may be able to tell you what the answer is, but they may have difficulty explaining why that's the answer.I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

 

[Short Corner]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's not meant to be funny, just one of those strange things in math that seem like they shouldn't be true, but are.

I think we've all had bad math teachers during our formal education. We've had good ones, too, of course. But, when you have a bad teacher, some simple things are just never really explained well. They may be able to tell you what the answer is, but they may have difficulty explaining why that's the answer.I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

Except for 0^0 and inf^0

 

[Bronco Billy]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

Unfortunately, not all teachers have expertise in their subject matter. It's really unfortunate, but most especially with something that is necessarily incremental as math. When I taught, it was uncanny how often I could track stuggling students roots down one one particular teacher whom they had in common and had dropped the ball, putting those kids behind the others.Great teachers are unique in that not only do they have expertise in a subject, but they can comminicate that expertise at a common level - making the difficult sound/seem simple, as it were.

 

[Ignoratio Elenchi]

Uh oh.

 

[dgreen]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

Except for 0^0 and inf^0

Explain, please.

 

[Bull Dozier]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

Dead serious.ETA: Well, except for the part about needing calculators.

Nevermind. I need to stay out of math thread prior to 10 am.

 

[Ignoratio Elenchi]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

Except for 0^0 and inf^0

Explain, please.

00 is indeterminate, although in most contexts (e.g. if we're not concerned with things like continuity) it makes sense to treat 00 = 1. Infinity isn't a number, so it doesn't really make sense to say something like infinity0 as if it's a number being raised to a power. In the context of limits, infinity0 is also indeterminate.

 

[Hooper31]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

Except for 0^0 and inf^0

Explain, please.

00 is indeterminate, although in most contexts (e.g. if we're not concerned with things like continuity) it makes sense to treat 00 = 1. Infinity isn't a number, so it doesn't really make sense to say something like infinity0 as if it's a number being raised to a power. In the context of limits, infinity0 is also indeterminate.

 

[Hooper31]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's not meant to be funny, just one of those strange things in math that seem like they shouldn't be true, but are.

I find that people have an easier time understanding this proof if you uses thirds instead.1/3 = .333...2/3 = .666...Add them together and you get:1/3 + 2/3 = .999...1 = .999...

 

[Ignoratio Elenchi]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....

Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....

Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

I assume this is some sort of math humor?

It's not meant to be funny, just one of those strange things in math that seem like they shouldn't be true, but are.

I find that people have an easier time understanding this proof if you uses thirds instead.1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

 

[Chaka]

We need to get an onslaught going with the top 10 posters in this thread.

 

[Hooper31]

I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

Belief. That word really made me think about this for a bit. Do I "believe" this? Is it comparable to having faith? And how does that concept compare to religious faith?

 

[Maurile Tremblay]

I find that people have an easier time understanding this proof if you uses thirds instead.

1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

If someone knows how to do long division, and they start in on solving 1/3 using long division by hand, they will quickly realize that the answer is 0.333333... that just keeps going on forever. The pattern is never going to change.

 

[rolyaTy]

I find that people have an easier time understanding this proof if you uses thirds instead.

1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

If someone knows how to do long division, and they start in on solving 1/3 using long division by hand, they will quickly realize that the answer is 0.333333... that just keeps going on forever. The pattern is never going to change.

So where does the extra .00000000...1 come from when you add .333333... to .66666666... and get 1?

 

[Maurile Tremblay]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

I had learned that x^0 = 1 without ever questioning why. Then some guy came to talk at a school assembly about his long years in solitary confinement, and the kinds of things he'd do to pass the time. As an example, he mentioned that he'd figured out why x^0 = 1, and he said that not a lot of people knew why. He didn't give us the answer, but after thinking about it, I came up with my own explanation.x^n represents a fraction consisting of only x's or 1's in both the numerator and the denominator, where there are n more x's in the numerator than in the denominator. For example, x^2 = (x*x*x)/x, or (x*x*x*x)/(x*x), or simply (x*x)/1. (There are two more x's in the numerator than in the denominator in each case.)By the same token, x^(-2) = x/(x*x*x), or (x*x)/(x*x*x*x), or simply 1/(x*x). (There are two fewer x's in the numerator than in the denominator in each case.)For the same reason, x^0 = (x*x)/(x*x), or (x*x*x)/(x*x*x), or simply x/x or 1/1. (There are zero more x's in the numerator than in the denominator in each case.)

 

[dgreen]

I find that people have an easier time understanding this proof if you uses thirds instead.

1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

If someone knows how to do long division, and they start in on solving 1/3 using long division by hand, they will quickly realize that the answer is 0.333333... that just keeps going on forever. The pattern is never going to change.

So where does the extra .00000000...1 come from when you add .333333... to .66666666... and get 1?

It comes from realizing that fractions are sometimes a better way than decimal places to represent an exact value.

 

[rolyaTy]

It comes from realizing that fractions are sometimes a better way than decimal places to represent an exact value.

Had an interesting question that a friend and I talked about in high school that's similar to this.If you have a point on a number line, and you have another point on the number line, and you kept moving the 1 point towards the second point by moving it half the distance between the two points, the two points would never overlap, according to common sense, but with numbers like decimals, there's a magical bump that takes place when you get infinitely close to the other point that just automatically makes them overlap.The decimal stuff reminds me of that question.

 

[dgreen]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

I had learned that x^0 = 1 without ever questioning why. Then some guy came to talk at a school assembly about his long years in solitary confinement, and the kinds of things he'd do to pass the time. As an example, he mentioned that he'd figured out why x^0 = 1, and he said that not a lot of people knew why. He didn't give us the answer, but after thinking about it, I came up with my own explanation.x^n represents a fraction consisting of only x's or 1's in both the numerator and the denominator, where there are n more x's in the numerator than in the denominator. For example, x^2 = (x*x*x)/x, or (x*x*x*x)/(x*x), or simply (x*x)/1. (There are two more x's in the numerator than in the denominator in each case.)By the same token, x^(-2) = x/(x*x*x), or (x*x)/(x*x*x*x), or simply 1/(x*x). (There are two fewer x's in the numerator than in the denominator in each case.)For the same reason, x^0 = (x*x)/(x*x), or (x*x*x)/(x*x*x), or simply x/x or 1/1. (There are zero more x's in the numerator than in the denominator in each case.)

Yep, thinking about how you subtract exponents when you are dividing them leads you to realize that x^n/x^n = x^0 or x/x or 1.But, it's just strange to see something like 42^0 = 1.

 

[videoguy505]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

I remember wondering why too. Teacher charted it, it made sense... like, for 5, reducing the exponent by 1 is dividing by 5.5^3 = 1255^2 = 255^1 = 55^0 = 15^-1 = 0.25^-2 = .04

 

[rolyaTy]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

I had learned that x^0 = 1 without ever questioning why. Then some guy came to talk at a school assembly about his long years in solitary confinement, and the kinds of things he'd do to pass the time. As an example, he mentioned that he'd figured out why x^0 = 1, and he said that not a lot of people knew why. He didn't give us the answer, but after thinking about it, I came up with my own explanation.x^n represents a fraction consisting of only x's or 1's in both the numerator and the denominator, where there are n more x's in the numerator than in the denominator. For example, x^2 = (x*x*x)/x, or (x*x*x*x)/(x*x), or simply (x*x)/1. (There are two more x's in the numerator than in the denominator in each case.)By the same token, x^(-2) = x/(x*x*x), or (x*x)/(x*x*x*x), or simply 1/(x*x). (There are two fewer x's in the numerator than in the denominator in each case.)For the same reason, x^0 = (x*x)/(x*x), or (x*x*x)/(x*x*x), or simply x/x or 1/1. (There are zero more x's in the numerator than in the denominator in each case.)

Yep, thinking about how you subtract exponents when you are dividing them leads you to realize that x^n/x^n = x^0 or x/x or 1.But, it's just strange to see something like 42^0 = 1.

I always thought it was this:X^(a-b)=X^(a)/X^(b)if (a-b)=0, then a=b, then X^(a)/X^(b), by substitution, equals, X^(a)/X^(a)=1

 

[dgreen]

I remember once asking why X^0 is always 1. It was just presented as fact that X^1 always = X (which made sense) and X^0 always = 1. The first response I got was, "What else could it be?" That answer didn't exactly help me understand. Once it was explained, it was so simple to understand. But, unfortunately, I had a teacher once who just couldn't explain it.

I had learned that x^0 = 1 without ever questioning why. Then some guy came to talk at a school assembly about his long years in solitary confinement, and the kinds of things he'd do to pass the time. As an example, he mentioned that he'd figured out why x^0 = 1, and he said that not a lot of people knew why. He didn't give us the answer, but after thinking about it, I came up with my own explanation.x^n represents a fraction consisting of only x's or 1's in both the numerator and the denominator, where there are n more x's in the numerator than in the denominator. For example, x^2 = (x*x*x)/x, or (x*x*x*x)/(x*x), or simply (x*x)/1. (There are two more x's in the numerator than in the denominator in each case.)By the same token, x^(-2) = x/(x*x*x), or (x*x)/(x*x*x*x), or simply 1/(x*x). (There are two fewer x's in the numerator than in the denominator in each case.)For the same reason, x^0 = (x*x)/(x*x), or (x*x*x)/(x*x*x), or simply x/x or 1/1. (There are zero more x's in the numerator than in the denominator in each case.)

Yep, thinking about how you subtract exponents when you are dividing them leads you to realize that x^n/x^n = x^0 or x/x or 1.But, it's just strange to see something like 42^0 = 1.

I always thought it was this:X^(a-b)=X^(a)/X^(b)if (a-b)=0, then a=b, then X^(a)/X^(b), by substitution, equals, X^(a)/X^(a)=1

Yeah, that's the same thing we said.

 

[Ignoratio Elenchi]

I find that people have an easier time understanding this proof if you uses thirds instead.

1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

If someone knows how to do long division, and they start in on solving 1/3 using long division by hand, they will quickly realize that the answer is 0.333333... that just keeps going on forever. The pattern is never going to change.

Sure, but .3 is strictly less than 1/3, and .33 < 1/3, and .333 < 1/3, and on and on forever. If someone isn't convinced that 0.999... = 1, I'd think it's also natural for them to think that the infinitely repeating decimal 0.333... doesn't ever equal 1/3, it just keeps getting closer and closer. It's the same difficult to grasp concept, so it doesn't make sense that they'd balk at the idea that 0.999... = 1 but then be convinced by a proof that relies on the premises that 0.333... = 1/3 and 0.666... = 2/3.

 

[Ignoratio Elenchi]

I find that people have an easier time understanding this proof if you uses thirds instead.

1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

If someone knows how to do long division, and they start in on solving 1/3 using long division by hand, they will quickly realize that the answer is 0.333333... that just keeps going on forever. The pattern is never going to change.

So where does the extra .00000000...1 come from when you add .333333... to .66666666... and get 1?

There is no extra .000...1. When you add 0.333... and 0.666..., you get 0.999..., which is equal to 1. That's the thing with these "proofs" - if you don't already have some understanding of the nature of an infinitely repeating decimal (and if you did, you'd already know that 0.999... = 1) then I'm not sure how you can be easily convinced by the 1/3 + 2/3 trick. It should be problematic to believe that 0.333... = 1/3 if you have trouble believing that 0.999... = 1.Similarly, I'd always seen this:

x = 0.999...

10x = 9.999...

10x - x = 9.999... - 0.999...

9x = 9

x = 1

and I've heard the objection that there's got to be a zero at the end of the decimal expansion of 10x in the second line, that there will be one less 9 at the end, so when you perform the subtraction in the third line you don't get 9, you get something like 8.999...1.

The fact that 0.999... = 1 is actually proven in a more rigorous fashion than these methods.

 

[dgreen]

The fact that 0.999... = 1 is actually proven in a more rigorous fashion than these methods.

The problem for some people could be with the word "equal".

 

[Ignoratio Elenchi]

The fact that 0.999... = 1 is actually proven in a more rigorous fashion than these methods.

The problem for some people could be with the word "equal".

I'm not sure I follow. They are equal. 0.999... and 1 are just two different ways of writing the same exact number.

 

[Short Corner]

I find that people have an easier time understanding this proof if you uses thirds instead.

1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

If someone knows how to do long division, and they start in on solving 1/3 using long division by hand, they will quickly realize that the answer is 0.333333... that just keeps going on forever. The pattern is never going to change.

So where does the extra .00000000...1 come from when you add .333333... to .66666666... and get 1?

There is no extra .000...1. When you add 0.333... and 0.666..., you get 0.999..., which is equal to 1. That's the thing with these "proofs" - if you don't already have some understanding of the nature of an infinitely repeating decimal (and if you did, you'd already know that 0.999... = 1) then I'm not sure how you can be easily convinced by the 1/3 + 2/3 trick. It should be problematic to believe that 0.333... = 1/3 if you have trouble believing that 0.999... = 1.Similarly, I'd always seen this:

x = 0.999...

10x = 9.999...

10x - x = 9.999... - 0.999...

9x = 9

x = 1

and I've heard the objection that there's got to be a zero at the end of the decimal expansion of 10x in the second line, that there will be one less 9 at the end, so when you perform the subtraction in the third line you don't get 9, you get something like 8.999...1.

The fact that 0.999... = 1 is actually proven in a more rigorous fashion than these methods.

can also be proven by summation of an infinite geometric series where a = .9*10^-na(1) = .9, r = .1

s(inf) = a(1)/(1-r) = .9/(1-.1) = .9/.9 = 1

 

[Ilov80s]

The fact that 0.999... = 1 is actually proven in a more rigorous fashion than these methods.

The problem for some people could be with the word "equal".

Equal means the same. Problem solved.

 

[Psychopav]

1/10 .1 1/100 is .011/infinity is .00000...11/infinity plus .99999... = 1therefore 1/infinity = 0therefore 0 * infinity = 1therefore 1/0 = infinity

A strange game. The only winning move is not to play. How about a nice game of onslaught?

 

[Psychopav]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

Can you 'prove' 1 = 2?

Only if you let me divide by 0.

1. a=b. (assumed)a^2=b^2 (square both sides)a^2 - b^2 = 0. (Subtract b^2 from both sides)(a+b)(a-b) = 0 (factor)a+b = 0 (divide both sides by a-b)a = -b (subtract b from both sides)a = -a (substitution)Thus rendering this entire discussion, along with all of mathematics, moot.2. 10*( 0.999...) - 1*(0.999...) = 9*(0.999...). (Factoring and subtracting 10-1)10*0.999... = 9.999... (Multiplying)(9.999...) - (0.999...) = 9 (Subtracting)SO,9*(0.999...) = 9 (substitution)Therefore, 0.999... must be the multiplicative identity, or 1.

 

[shuke]

According to Google, -5^2 = 25.

Looks like they changed their mind five years later.

You've been checking this every day, haven't you?

 

[tommyboy]

would like to get Shick! back in here to set us all straight again

 

[jon_mx]

I find that people have an easier time understanding this proof if you uses thirds instead.

1/3 = .333...

2/3 = .666...

Add them together and you get:

1/3 + 2/3 = .999...

1 = .999...

That's a good one. I think the hard part for most people is really understanding the idea of an infinitely repeating decimal, though. Someone who doesn't initially believe that 0.999... = 1 might very well be convinced by the above proof, but they probably shouldn't be - because for the same reason they doubt 0.999... = 1, it seems like they should also doubt that 0.333... = 1/3 and 0.666... = 2/3. Either way it appears to be a decimal that keeps getting closer and closer to but never actually reaches its rational representation.

If someone knows how to do long division, and they start in on solving 1/3 using long division by hand, they will quickly realize that the answer is 0.333333... that just keeps going on forever. The pattern is never going to change.

So where does the extra .00000000...1 come from when you add .333333... to .66666666... and get 1?

There is no extra .000...1. When you add 0.333... and 0.666..., you get 0.999..., which is equal to 1. That's the thing with these "proofs" - if you don't already have some understanding of the nature of an infinitely repeating decimal (and if you did, you'd already know that 0.999... = 1) then I'm not sure how you can be easily convinced by the 1/3 + 2/3 trick. It should be problematic to believe that 0.333... = 1/3 if you have trouble believing that 0.999... = 1.Similarly, I'd always seen this:

x = 0.999...

10x = 9.999...

10x - x = 9.999... - 0.999...

9x = 9

x = 1

and I've heard the objection that there's got to be a zero at the end of the decimal expansion of 10x in the second line, that there will be one less 9 at the end, so when you perform the subtraction in the third line you don't get 9, you get something like 8.999...1.

The fact that 0.999... = 1 is actually proven in a more rigorous fashion than these methods.

Most people don't quite get what infinity really means.

 

[chet]

According to Google, -5^2 = 25.

Looks like they changed their mind five years later.

You've been checking this every day, haven't you?

Still -25 in my google.

 

[Hooper31]

would like to get Shick! back in here to set us all straight again

Nobody likes math.

 

[chet]

would like to get Shick! back in here to set us all straight again

Nobody likes math.

I like Shick!

 

[tommyboy]

According to Google, -5^2 = 25.

Looks like they changed their mind five years later.

You've been checking this every day, haven't you?

Still -25 in my google.

me too. funny

 

[tommyboy]

yahoo says its 25