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My son has just shamed me considerably as I couldn't answer his simple question: if ten divided by three is 3.333r, (which doesn't make ten if you times it by three again), how is it possible to have three equal parts of ten in a physical sense? I know there is probably a simple asnwer to this, but I couldn't fathom it and didn't want to fob him off. Please, clever people, help me explain it in simple language!  

Quote from: "weedoch"My son has just shamed me considerably as I couldn't answer his simple question: if ten divided by three is 3.333r, (which doesn't make ten if you times it by three again), how is it possible to have three equal parts of ten in a physical sense? I know there is probably a simple asnwer to this, but I couldn't fathom it and didn't want to fob him off. Please, clever people, help me explain it in simple language!  

The only way I could figure it is to cut a round cake or pie (as the round number 10) into three equal sections.
Other than using a geometric shape, that one's beyond me.

I'll take your convention of using an 'r' to mean 'repeating'

it's because 3.333333r  times 3 is equal to 9.999999r  correct?
Well it is a fact that 9.99999r  is equal to 10.  So that's what you need to show your son.

There are a number of ways to prove this.  1 way is to just enter it into a calculator and it will show up as 10 lol.
Ok a better method is this:

The usual proof is done with a 1 instead of 10.  But it should be the same, I'll show you how it's done with 1.  This will prove that 0.99999r is equal to 1:

x = 0.999999r
10x = 9.999999r
so,
10x - x =   9.999999r  -  0.999999r
which is equal to 9
so,
10x - x = 9
9x = 9           <----edited, sorry for the mistake hope I caught it before you read this, originally wrote "9x=x"
x = 1

look at the beginning, x was defined as 0.99999r  but at the end we get x=1, so they are the same thing.
There are other ways as well but some are more complex and I also forgot them.
Anyway I hope that's good enough.

to do it with 10 instead of 1 just change them to x=9.9999r  10x = 99.999999r   and you'll see that x=10 in that case.

Also, dealing with decimals is considered kind of an approximation which is why you run into trouble when you have an infinite amount of 9's.  The exact answer to 10 divided by three is actually "10/3" (ten-thirds, simple right?) so then when you multiply by 3 it's obviously what you would expect  (10/3)*3  =  10.

Hope that helps clarify.

Cheers Perillux, that helps a lot. My 5 year old was playing with the calculator, and trying to explain the difference between fractions and decimals was frankly out of my league. I don't think he'll fully understand the answer, but at least I do now... I think!

You have bright kid there!

Here's a more simple answer maybe (or maybe not).   In our physical reality, you can't divide a pie (peach, banna, apple, etc.) into 3 equal sections.

But let's say your pie is composed of 1 million atoms of pure gold so we don't have to be confused by the molecular structure of the thing.  We have some pretty spectacular cutting equipment and can make super precise cuts.  There is no way to make 3 equal slices of gold pie at the atomic level.  You can only get a 33,3333 atom slice, another 33,3333 atom slice and 33,3334 atom slice.  Splitting that last atom would probably be a very bad thing, with nuclear fission and all. So no satisfaction here.


Now lets say you have a pie composed of 1 million protons (yum yum!).  You can only get  a 33,3333 proton slice, another 33,3333 proton slice and a 33,3334 proton slice.  But in quark theory, protons are composed of three quarks.  And so if you go down a level of the atomic structure into the quark level, you can get three equal parts of proton pie, with one extra quark per slice.  Yippeee!!!

Quote from: "LARA"Now lets say you have a pie composed of 1 million protons (yum yum!). You can only get a 33,3333 proton slice, another 33,3333 proton slice and a 33,3334 proton slice. But in quark theory, protons are composed of three quarks. And so if you go down a level of the atomic structure into the quark level, you can get three equal parts of proton pie, with one extra quark per slice.  Yippeee!!!

That's IT!  :D

3.33... times 3 does equal 10.

If the pie analogy is confounding you, it's because you're confusing .33... with infinity, or something which goes on forever in a physical sense. The 1/3 pie slice really is .33... of the pie.

Think of the circumference of a circle. This involves a different kind of pie: namely, the number pi. This number, too, goes on forever: 3.14... it never stops. But look at the circle's circumference. It is finite. It is a specific length. In fact, you can do the same thing to the diameter: if you have an apple with a circumference of 5, its diameter is 1.59154943... and so on. Is the apple infinite? No. But depending on how precise you want to measure it, the description of its diameter could go on forever.

This thread has done nothing but proven that math is just not my friend.. the numbers conflict with reality sometimes.

It makes as much sense as being told that there is a 40% chance of rain today and then it raining.. Wouldn't the fact that its raining mean that there's a 100% chance of it raining?

Quote from: "Jolly Sapper"This thread has done nothing but proven that math is just not my friend.. the numbers conflict with reality sometimes.

It makes as much sense as being told that there is a 40% chance of rain today and then it raining.. Wouldn't the fact that its raining mean that there's a 100% chance of it raining?

Yes, the moment it rains the odds jump to 100%. For the same reason, if it doesn't rain, when the day is over the odds fall to 0%. Probability only applies before the incident, because it's about the unknown. The answer is actually 100% or 0%, but we don't have enough information to say, so based on past weather we have to approximate. What "40%" means is that, in the past when conditions were like they are today, it rained the next day 40% of the time.

As the science of meteorology improves (and as the historical weather record grows), you'll see less and less predictions around the 50% range and more predictions around the 10% or 90% ranges. More understanding of weather will lead to nearer certainty.

It's just like flipping a coin. Before you flip it, there's a 50% chance of getting heads. But once you flip it, and get heads, you learn that there was actually a 100% chance of getting heads for that flip.

HA! thou hast been fooled by the sheer simplicity of the question. It is philosophical, not numerical - and the philosophy is quite weak. Zeno would blush.

First off - ten, three, and 1/3 are symbolic abstractions, not physical things. I may decide to divide a thing - any thing - into ten equal parts, or three equal parts, arbitrarily. All that matters is the degree of precision.

Just because the number 10 appears a "finite" representation as opposed to 3.333... does not mean one is less physically "real" than the other - it is an illusion of a number system based upon an even number - 10.

So - take one of the ten parts, divide it into three parts, and add each of these to each of the three sets of three parts remaining from ten, and each set contains - 3.333... parts.

Not so much as seventh-grade algebra required.  

Quote from: "Promethium147"So - take one of the ten parts, divide it into three parts, and add each of these to each of the three sets of three parts remaining from ten, and each set contains - 3.333... parts.

 

(1/10)/3... Then what? ((1/10)/3)+(1/10) ..?

I just thought of another way to show this.  Do you know what limits, and infinite series are?  I won't describe it, it might be complicated.  If anyone wants me to, then I will but if not then I'll just try and explain it qualitatively.

ok, so, consider:
10 - 9 = 1
10 - 9.9 = 0.1
10 - 9.99 = 0.01
10 - 9.999 = 0.001
10 - 9.9999 = 0.0001
10 - 9.99999 = 0.00001

As you can see the difference  between 10 and the decimal number gets smaller and smaller.
in other words:
as the series (9.999...) gets closer and closer to the infinite series (9.999r) then the difference from 10 approaches zero.  Therefore 9.999r = 10

GEEZ, Asmodean - I never suspected this would rise to the level of a Great Debate on THIS board.

I've been awake about 70 hrs., but here I go again.

You've got one Thing, divided into ten equal parts, but - you want three equal parts. Problem is - ten is an even number, three is odd.

You want to divide the original Thing into three equal parts by arranging these ten parts, but - the best you can do without further subdivision of each 1/10th part is - arrange nine of the 1/10th parts into three sets of three parts each. Each of these three sets now constitutes 3/10 of the Thing ( 0.3 ).

But one unused 1/10th part remains.

Divide the remaining part into three equal parts (that's (1/10)/3) and each of the three new SUBparts constitutes 0.0333... of the original Thing.

Your gonna hafta divide something by three at some point - in order to divide something by three. We must face this fact, and reconcile ourselves to it.

If you now add one of the SUBparts (0.0333... ) to each of the three sets of three parts (0.3), each of the three sets equals 0.333... of the Thing, viola.

Check your work; 3 x 0.333... = 0.999... = 1 Thing.

-------------------------------------

But really, you should just rejoin the Thing into one part, and divide the Thing into three equal parts - I was just doing it long way 'round, because that was the source of this ConFuSiOn.

Just let me get some shuteye, and I'll be right back with some Nonlinear Finite Element Analysis for Dummies.  

i heard this idea the other day and had to ignore it.

take a piece of string and cut it once. now you have two strings

cut again and you get three.

do this an infinite number of times and you get a number more than infinity.  

By the way, Lara - you can divide the Golden Pie into three equal pieces down to any desired level of precision - such as atoms, beyond which we no longer have Gold.

All that is required is a Golden Pie containing a number of atoms divisible by three. The MOST you could possibly need to add (or subtract)to obtain this Pie of Pies is two atoms - and no one should mind, really.

OH DARN IT, so hard to walk away, jcm...

Of course, Infinity is not a number.

Divide Infinity by any arbitrarily large number, and you get an arbitrarily large number of Infinities.

Infinity is not a Number - in fact, it is not a Thing.  

Quote from: "Promethium147"OH DARN IT, so hard to walk away, jcm...

Of course, Infinity is not a number.

Divide Infinity by any arbitrarily large number, and you get an arbitrarily large number of Infinities.

Infinity is not a Number - in fact, it is not a Thing.  

oh well, i thought it was cool

Quotetake a piece of string and cut it once. now you have two strings

cut again and you get three.

do this an infinite number of times and you get a number more than infinity. :)
However, if the function was reversed x / x^(2) then in that case the answer to (infinity)/(infinity) is zero, because in this new case the bottom increases faster than the top.

I'm sorry if that's confusing.  It's something you learn in calculus.  Probably more so in calculus 2.

Quote from: "Perillux"ANY number even 10^(5 billion)  divided by infinity equals zero.

Are you sure it would 0 (zero) and not "undefined", kinda like dividing a number by zero (since zero is a place holder and not an actual number) is undefined?

Yes you are correct Jolly, thank you for pointing that out.

Sorry, I should have worded it like this ("the limit of the fraction is bla bla bla...")  The actual value of the fraction is undefined like you said.

However, in this situation (at least the way I was using it) I'm actually treating infinity as a limit.  so the limit as x goes to infinity of (any constant) / x  is equal to zero.  But without treating it as a limit, then ya it would be undefined.
This is because when calculating a limit you never get to the exact value.  Your just basically looking at what it appears to approach.  So if you keep dividing the constant by an increasing number you will find the answer gets smaller and smaller, and it "approaches" zero.  If the increasing number goes on forever(infinite) then we can say that the limit is zero.

Saying it's undefined doesn't help anyone to visualize what's happening to the graph of the function because it can have any value (undefined).  But using limits allow you to see the overall trend of the graph.