Question

irrational numbers - question : my teacher said the following:
"The Pythagoreans believed that magnitudes could always be measured using whole numbers, which would imply that lengths are not infinitely divisible." I dont understand the implication
my teacher also said
If everything is measurable by whole numbers, and you can divide infinitely, surely you can find the contradiction there,
my teacher is a bit of a wingspan

Answer 1

sounds like it

Answer 2

my teacher also said :
If quantities could always be measured using whole numbers, eventually there is something with measure 1, which cannot be divided

Answer 3

i think the point is that if you have a "unit" of length say, like feet or meters or inches, then any length can be measured using that unit. for example to say that a rope is 1.4 feet long means 5 ropes = 7 feet (because 1.4=7/5).

turns out you cannot so that for every length. for example if you have a square with side 1 then the diagonal of the square is
\[\sqrt{2}\] units long, and since that number in not a fraction (is irrational) then you cannot say "some number of diagonals = some number of feet"

Answer 4

The implication is that fractions and decimals are unnecessary for measuring things. For example, the length of an object would always be expressible as an integer amount of units. However, this obviously isn't true with a finite number of units of measurement. Essentially what your teacher is saying is that if everything is measurable in integer units of length, then if an object is of length 1, it can't possibly be any shorter (because to express its length you would have to use a fraction or decimal). But since obviously things are (for practical purposes) infinitely divisible, there is no length at which an object cannot be shorter.

Answer 5

Strictly speaking there is a length at which things can't be shorter, but that's something for physicists to argue about...

Answer 6

@blacksteel i thinks that may be is correct, but the pythagoreans knew about ratios. they just thought you could measure things by comparing them, not necessarily that something could have measure less than one. to say a rope is
\[\frac{2}{3}\] of a foot means 3 ropes = 2 feet.

Answer 7

so "measurable by whole numbers" in this context means that you can compare some whole number of what you are measuring to some whole number of the unit you are measuring it by

Answer 8

which is of course false

Answer 9

I agree, and getting into irrational numbers is the RIGHT way to disprove the Pythagoreans, but my response in in regards to what the teacher said.

Answer 10

but i never assumed there was a common measure to all pairs of lengths

Answer 11

the greeks claimed, given any two lengths, there exists a third length that fits evenly into the two lengths (measures them or divides them evenly)

Answer 12

Quoting Cantor:
"my teacher also said : If quantities could always be measured using whole numbers, eventually there is something with measure 1, which cannot be divided"

I was basing my response off of this, but you are quite correct that this isn't a strong argument against the Pythagorean theories.

Answer 13

so for each pair of lengths, we can choose different measures .

Answer 14

i never assumed there was a common measure to all the lengths

Answer 15

what does this mean?
" If quantities could always be measured using whole numbers, eventually there is something with measure 1, which cannot be divided"

how does this follow

Answer 16

Your teacher is saying that essentially there would have to be some sort of base unit for length. In other words, if everything can be expressed as whole number ratios of other things, eventually you'll have to have some object that's used as a base length for measurement. This is what I tried to say above, but in fact, this is a weak argument against Pythagorean theory. The proper way to disprove the theory is to show that there are numbers that are incommensurable with 1. (An early proof showing this to be true for the square root of 2 is what actually disproved the Pythagorean theory of measurement).

Answer 17

(Two numbers being incommensurable means that one number cannot be expressed as a ratio of the other.)

Answer 18

but every two new pairs of lengths, there is a different common measure

Answer 19

That's why proving incommensurability is the key to disproving that theory. If two numbers are incommensurable, they cannot be expressed as ratios of one another, so there is no measurement that divides both evenly. For example, there is no number that evenly divides both 1 and the square root of 2. Thus, it is impossible to construct a ratio a/b using whole numbers such that a/b * 1 = sqrt(2)

Answer 20

no what im saying is that it doesnt make sense to say
if quantities could always be measured, we would get finally a measure 1.
each two quantities has a new measure

Answer 21

ok let me go back.
the greeks believed, before they discovered irrationals,
that given any two lengths there was a third length that went into each of them evenly. ie, all pair of lengths was conmeasurable

Answer 22

Not exactly. Consider, for example, two boards of lengths 6 in and 9 in. The common measure is 3 in; both boards' lengths can be expressed in terms of this common measure. However, the units of the measure are still inches; there will still be objects of length 1 inch. However, you can of course introduce new, smaller units of length as necessary, which is why, as I said before, this is a weak argument.

Answer 23

you can also use 1 inch, 1/2 inch ,

Answer 24

there always exists some other length that goes into it. there is no bound on how small it can be .

Answer 25

youre saying that we want a uniform measure? that measures all things in terms of this unit? like 1 inch or 1 mm or .000001 mm