# Is this possible!

1. Nov 30, 2011

### bmcgann

Sorry for the very vague title but my brother and I had a long chat the other day and we could not come to a conclusion.

Our question was, can every object be split into 3 equal parts?

Scenario 1: If you split a 60cm ruler into 3 20cm parts you would say that yes the object is in 3 completely equal parts.

Scenario 2: Now when you try to split a 100cm ruler into 3 parts it doesn't seem possible, no matter how accurate you measure it, one part will always measure longer than the other 2 parts.

Now the question is can every physical property be split into 3 equal parts or is maths misleading us.

We imagined changing the scale of the maths involved, i.e. calling the 100cm ruler 60(whatever units), now the ruler is still the same size but maths now tells us that it can be split into 3 equal parts, but how can this be?

I know that I am talking ridiculously small distances and accurate measuring beyond imagination, but the picture in my head that all the pieces line up against the (whatever scale) and that one of the pieces is always bigger against the cm scale is really confusing! Its the same piece

So what are your opinions, are we missing something simple?

2. Nov 30, 2011

### Cicnar

You are able to split 60 cm equally because factoring 60 gives you 60=2x2x3x5, and dividing by 3 cancels out, leaving multiple of 2x2x5=20 as your answer, that is, pretty, whole number. If you try the same with 100 you have 100=2x2x5x5 dived by 3. Because your prime factorization of 100 clearly has no factor of 3 in it, the division of it by number 3 will not produce whole number as quotient. So youll end up with a repeating decimal, going forever, that is, 33.333.... In a number system where base has a factor of 3 too, that would not be the case.

3. Nov 30, 2011

### AlephZero

You can divide a line into N equal segments without measuring anything "exactly". All you need is a way to "copy" the same length several times. Here's one way to do it.

http://www.mathopenref.com/constdividesegment.html

4. Nov 30, 2011

### DaveC426913

The fact that one stick has markings on it that you can assign numbers to and that you can then divide those numbers without a remainder has no effect on whether either physical object can be divided into smaller pieces.

You could take either one of those rulers and erase the markings and redraw them so that the metre long stick has markings divisible by 3 and the 60cm one does not - and it will not make a whit of difference.

5. Nov 30, 2011

### micromass

I agree with Dave. In fact, I don't even think you are able to split the stick of 60cm in three! You could argue: I split the stick at 20cm and 40cm. But how do you measure 20cm? You can never do it exactly! You always have a degree of uncertainty.

In fact, if space were quantized, then you would never be able to split it exactly!!

Also, (as Dave noticed): what if I had another measuring scale where the stick of 60cm now measures 100 units. Could you then split the stick?