Why can't 10 be divided by 3 perfectly?

[chaseter]

I think you're mixing this up in your head. 1/3 is not an approximation of .333333 to infinity, it's the actual representation of that in fractal form. The actual length of an inch is an arbitrary human choice. If you were the person to decide what an inch was, you could take a single stick, and say "an inch will be this length times three" and then a third of an inch would be the exact length of the stick. There is nothing mathematically limiting here about using fractions, and technically no need to approximate.

1/3 is the simplified form of 0.333333 to infinity. They are the same thing. 1/3 = 0.333333333 to infinity.

 

[Hound55]

Whether it's mechanical or not it woud never "know" where to make the initial mrking.

For the same reason that no calculator or human being could accurately show a third in decimals.

It would simply make it "close enough".

Technically if a calculator or computer program were created that would take as long as necessary (over infinity if required) to provide an answer to any problem it would never give you a response to 1 divided by 3 because it would continue dividing on an infinitesimal scale.

 

[Calvin]

it seemed like you were implying that it was an approximation of that. Yes, it's a simpler way of writing it, but it is exact, and it is mathematically possible to measure a third of an inch.

 

[Calvin]

Whether it's mechanical or not it woud never "know" where to make the initial mrking.

For the same reason that no calculator or human being could accurately show a third in decimals.

It would simply make it "close enough".

Technically if a calculator or computer program were created that would take as long as necessary (over infinity if required) to provide an answer to any problem it would never give you a response to 1 divided by 3 because it would continue dividing on an infinitesimal scale.

Here's an easy way to designate where you mark your units. Let's create a unit of measurement, let's call it a "hobbes.". Now we're going to take a singular and unique object or length, say the physical length of a specific hair that I pull from my head. I can decide that the length of my hair times 3 will be the length of a hobbes. Now I have an extremely easy and exact way to designate thirds of my hobbes unit, with no approximation needed. The same thing can basically be done with an inch. Stop thinking of everything in terms of decimals in bases of ten. That's what's confusing you. You're thinking of it in a limiting format.

 

[Hound55]

The markings on your ruler ARE an approximation. Because it's not possible to accurately mark a third of an inch.

 

[Calvin]

The markings on your ruler ARE an approximation. Because it's not possible to accurately mark a third of an inch.

A computer has the means to mark a third of an inch, if you're just arguing that the mechanical process itself won't be perfect in the creation of the physical rulers, no duh, that's a useless observation and not in the spirit of the original mathematical exercise that was put forth.

 

[Hound55]

Here's an easy way to designate where you mark your units. Let's create a unit of measurement, let's call it a "hobbes.". Now we're going to take a singular and unique object or length, say the physical length of a specific hair that I pull from my head. I can decide that the length of my hair times 3 will be the length of a hobbes. Now I have an extremely easy and exact way to designate thirds of my hobbes unit, with no approximation needed. The same thing can basically be done with an inch. Stop thinking of everything in terms of decimals in bases of ten. That's what's confusing you. You're thinking of it in a limiting format.

It's not confusing me at all...

When I came in here, my first comment was going to be "because we use a base ten counting system".

That said, you still don't have a ruler that accurately and exactly measures thirds of an inch because it's not possible to accurately create one.

 

[Victarion]

Also, aldol condensation.

 

[Calvin]

It's not confusing me at all...

When I came in here, my first comment was going to be "because we use a base ten counting system".

That said, you still don't have a ruler that accurately and exactly measures thirds of an inch because it's not possible to accurately create one.

But my point is there's nothing mathematically limiting about creating an accurate one, and saying we're mechanically limited from creating a perfect ruler for it physically is really just a cop out of an observation.

 

[Hound55]

It's not really "a cop out of an observation" when that sentence there (the result of you having to deal with my semantic BS) pretty much most accurately hits at the crux of the issue that the topic was created for...

The difference between the mathematical and the mechanical/physical when dealing with thirds in our base ten perspective of the world...

 

[Calvin]

It's not really "a cop out of an observation" when that sentence there (the result of you having to deal with my semantic BS) pretty much most accurately hits at the crux of the issue that the topic was created for...

The difference between the mathematical and the mechanical/physical when dealing with thirds in our base ten perspective of the world...

But our world doesn't exist in a base ten perspective in any physical way, and especially not in modern mathematics (and by modern, you can probably go back more than a century), and it certainly doesn't seem to be the crux of the issue to me at all, it seems like a silly way to go about it, as it doesn't involve any conceptual thinking at all. That'd be like getting a word problem, and then claiming there's no exact answer because mechanically humans won't be able to create a physically perfect ruler. You could say that about the ruler even if the question was asking how to measure one whole inch. It doesn't take any thinking at all to come up with that answer, and it's not in the spirit of figuring out any sort of legitimate math question.

 

[Hound55]

If you have a 10 inch ruler and you cut it into 3 equal parts, what would each part equal? Whatever the answer is, that number would STOP. It has to stop, because it's a physical object which can be measured.

Now, take that answer and apply it to a calculator. If you divide 10 by 3, you get 3.333333333 for infinity. How is that possible? Why can something be divided perfectly in a physical way, but mathematically last for eternity?

If you have a 10 mile strip of road, you can split up that road into 3 separate roads. But on a calculator, these roads last forever.

But our world doesn't exist in a base ten perspective in any physical way, and especially not in modern mathematics (and by modern, you can probably go back more than a century), and it certainly doesn't seem to be the crux of the issue to me at all, it seems like a silly way to go about it, as it doesn't involve any conceptual thinking at all. That'd be like getting a word problem, and then claiming there's no exact answer because mechanically humans won't be able to create a physically perfect ruler. You could say that about the ruler even if the question was asking how to measure one whole inch. It doesn't take any thinking at all to come up with that answer, and it's not in the spirit of figuring out any sort of legitimate math question.

I disagree... I think it addresses the initial question in red perfectly...

And the best answer for the question in green is either 3 and 1/3 of an inch or 3.33 recurring inches.

 

[HighFivingMF]

That's like... Whooooa, man!

 

[John Locke]

Don't forget to carry the one.

Yes I am that guy. You need one of those guys. It's like a rule look it up.

 

[Wolfwood]

Oh man. Seriously? This thread hurted my brain.

 

[Optimus_Prime_]

If you have a 10 inch ruler and you cut it into 3 equal parts, what would each part equal? Whatever the answer is, that number would STOP. It has to stop, because it's a physical object which can be measured.

Now, take that answer and apply it to a calculator. If you divide 10 by 3, you get 3.333333333 for infinity. How is that possible? Why can something be divided perfectly in a physical way, but mathematically last for eternity?

If you have a 10 mile strip of road, you can split up that road into 3 separate roads. But on a calculator, these roads last forever.

In the case of physical objects, rather than mathematical paradoxes, look up the term Planck length. It can most certainly be divided evenly (if technology permitted of course). Also, as far as physical lengths go, no number truly terminates, they all have a bunch of zeros after them, which as far as the measurement is concerned would matter if you wanted an objected divided ABSOLUTELY perfectly.

 

[Calvin]

I disagree... I think it addresses the initial question in red perfectly...

And the best answer for the question in green is either 3 and 1/3 of an inch or 3.33 recurring inches.

The part in red just seems to have a misunderstanding of how math works, and I did answer the green part with 10/3 inches.

Here's something to try, and I'm not sure if this will work because I'm not sure how exacting photoshop software is (and I don't have it anymore). Create a 12 inch ruler or rectangle in photoshop, cropped so that the available space is exactly 12 inches across, and line it up with their ruler lines, then make a mark at 4 inches, and at 8 inches. Next, scale the whole image so that it is ten inches wide, and if photoshop is really exact, it should scale all the lines accordingly, and the marks you made will split the now ten inch ruler into three equal parts.

 

[NHawk19]

1/3 being a non-terminating number has nothing to do with an equal cut. Just because the number has an infinite number of 3's behind the decimal point doesnt mean the number doesn't exist. The fact is that is is possible cut something at exactly the number 1/3 which is equal to a non terminating .3333. That's why the number is expressed more commonly as 1/3 and not .3333333333333 (repeating)

The problem is at some point any calculator will need to round-off the answer.

 

[Sentinel X]

Jesus christ, does nobody understand fractions? You could cut a ten inch ruler into three equal parts that are 10/3 in. each. You don't have to do it in decimals, that's arbitrary, and it certainly isn't "impossible."

I certainly have rulers that can designate a third of an inch. Measure ten of those at a time, cut, and call it a day.

Thats not really the question though and besides 10/3 (whether written as a fraction or decimal) is still an irrational number

 

[Teelie]

This is why I hate math. No matter how much you try to explain or understand it, it will never, ever be simple. In reality, it's 3.33 of an inch. You can nitpick the decimals that at 3.333 or 3.332 it's no longer equal but at that level of physical existence who cares? Mathmatically on paper it's wrong, but on paper is not reality.

 

[terry78]

Somebody is going to come in here and just scream "nerd fight!" at some point.

 

[SpideyVille]

I love math because its one of the few things that I feel like I'm very good at since it comes easy to me, but for some reason, I can never explain or teach it to someone.

 

[Teelie]

I'm better at English and nit-picking it but I'm far from an English professional or grammar nazi. Unlike some of the math-heads I've met I can see people with terrible, terrible writing all the time but never feel the need to correct it like they would incorrect math. That and I know I'm not grammatically perfect either but English is a bastard language anyways.

 

[Calvin]

Thats not really the question though and besides 10/3 (whether written as a fraction or decimal) is still an irrational number

Wrongo, my friend. First of all, an irrational number is still a real number, so it's beside the point. Second, your fact is incorrect to begin with, as an irrational number is a real number that cannot be expressed as a fraction using integers (with the denominator being a non-zero integer). 10/3 is not an irrational number. It is real and rational.

 

[Mike_D202]

Someone texted me earlier and said "My buddy said that 5+5+5=550 if you just make a slight change, what is it?"

I had no clue.