Pythagoras Theorem: Exploring Real vs Math Worlds

[xMonty]

Hi,

I have a really stupid question

suppose the base and perpendicular are both length 1 (whatever units)
then the hypotenuse's length comes out to be SquareRoot of 2

but square root of 2 is 1.41421356.... (not a fixed number)

so that means if i physically measure the hypotenuse upto the accuracy of 3 decimal places i would get 1.414

but the actual length is bigger then that, its bigger by .00021356... which is not much

but the point is "Since the square root of 2 is not an exact number" with whatever arbitrary precision i choose to measure the hypotenuse its length will always be somewhat bigger than what i just measured

Whats going on.. is there s disconnect between the real and mathematics world

Thoughts please

 

[fourier jr]

math is exact, humans aren't. i think that's all it is

 

[robert Ihnot]

The Pythagoreans dealt with this problem 2000 years ago: the discovery that not all numbers are the ratio of two integers--particularly the case you mention. They referred to these as "unutterables."

It's not just lower bounds, we can get upper bounds too. We consider the the Pellian equation Y^2-2X^2 = plus or minus 1. Take (1-sqr2)^n and get a series of terms (1-sqr2)^2 = 3-2sqr2, (1-sqr2)^3 = 7-5sqr2, etc.

We then arrive at a series of lower and upper bounds on the square root of 2. 1/1<sqr2; 3/2>sqr2; 7/5<sqr2, 17/12 >sqr2; 41/29<sqr2...etc.

Those are the best approximations possible considering the size of the integers.

 

[Tac-Tics]

but square root of 2 is 1.41421356.... (not a fixed number)
[...]
Whats going on.. is there s disconnect between the real and mathematics world

First, the square root of 2 IS a fixed number. It has an exact value. The problem is you are using an underpowered definition of what a "number" is. Numbers are more than their decimal places. Rational numbers, the kind you learned to add and multiply in elementary school, are represented by fractions of integers. You don't learn much about irrational numbers in school. Most people know that pi has a non-repeating decimal expansion, but beyond that, nothing.

The details are complicated (and interesting), but aren't something you learn unless you're a mathematician (the class is usually called Real Analysis). But almost everyone can get by in life with rational numbers. Rational numbers can always get "close enough" to any irrational number. And close enough goes a long way in life.

There's a ton of disconnect between math and the real world. Math is a formalization of our ideas about the world and our understanding of the world is pretty darn weak.

 

[Mentallic]

But you know what. You wouldn't have been able to get the side lengths of 1 to the exact length either - or in another case, if you take some length and define it to be 1, you would still have a rounding error on the other length that you try to make equal 1. It's an error that we just account for.
It's not the fact that the number is irrational that we can't measure it, it's just our precision that can't measure it perfectly accurately.

If we could measure it perfectly, you'll find that as you magnify the ruler's reading, you'll keep following an infinite string of decimals, sqrt(2).

 

[Mensanator]

You're describing truncating, not rounding.

 

[HallsofIvy]

Hi,

I have a really stupid question

suppose the base and perpendicular are both length 1 (whatever units)
then the hypotenuse's length comes out to be SquareRoot of 2

but square root of 2 is 1.41421356.... (not a fixed number)

square root of 2 certainly IS a "fixed number". It just is not a terminating decimal nor is it a rational number.

so that means if i physically measure the hypotenuse upto the accuracy of 3 decimal places i would get 1.414

but the actual length is bigger then that, its bigger by .00021356... which is not much

but the point is "Since the square root of 2 is not an exact number"

Absolutely not true. square root of 2 is as "exact" a number as 0 or 1 or 1/3. It is the measurement that is not exact. And no one expects measurement to be exact. That's why you have to talk about "accuracy".

with whatever arbitrary precision i choose to measure the hypotenuse its length will always be somewhat bigger than what i just measured

No, that's not true either. For example, if you measure to an accuracy of 7 decimal places you would get 1.4142136 and the correct result is somewhat less than that. But it is "bigger" or "smaller" by less than your accuracy.

Whats going on.. is there s disconnect between the real and mathematics world

Thoughts please

All measurement is approximate, just as you said when you talked about "accuracy". Nothing new about that. As any scientist knows, it is the measurement that is the problem.

 

[xMonty]

Thanks a lot for clarifying but how come square root of 2 is fixed?

 

[HallsofIvy]

It is a specific number! It doesn't change! That's what 'fixed' means. What did you think it meant?

 

[blitz.km]

Root(2) is a "Fixed Number"
Just as we say that 1/3 is a fixed number.. we can represent it on the number line..(not using a ruler.. but a compass or divider)
the same way we can represent root(2)..maybe we can.. don't know about root(2).
Does anyone know if we can do so?

 

[robert Ihnot]

Root(2) is a "Fixed Number"
Just as we say that 1/3 is a fixed number.. we can represent it on the number line..(not using a ruler.. but a compass or divider)
the same way we can represent root(2)..maybe we can.. don't know about root(2).
Does anyone know if we can do so?

OF COURSE YOU CAN! This was easily discovered in the days of Pythagoras. Simply draw a one unit X, and a one unit Y, and then CONNECT THE DIAGONIAL.

 

[marcusmath]

OF COURSE YOU CAN! This was easily discovered in the days of Pythagoras. Simply draw a one unit X, and a one unit Y, and then CONNECT THE DIAGONIAL.

Ha, I think I hear a face plant.

 

[robert Ihnot]

Well, if that is too harsh a way to put it, I think we can just start with Xmonte, who started this problem by saying:

Hi,

I have a really stupid question

suppose the base and perpendicular are both length 1 (whatever units)
then the hypotenuse's length comes out to be SquareRoot of 2

and HallsofIvy repeated the same statements..

 

[blitz.km]

OF COURSE YOU CAN! This was easily discovered in the days of Pythagoras. Simply draw a one unit X, and a one unit Y, and then CONNECT THE DIAGONIAL.

Thanks.

 

[HallsofIvy]

Root(2) is a "Fixed Number"
Just as we say that 1/3 is a fixed number.. we can represent it on the number line..(not using a ruler.. but a compass or divider)
the same way we can represent root(2)..maybe we can.. don't know about root(2).
Does anyone know if we can do so?

Any number that is "Algebraic of order a power of 2" can be "constructed" using straight edge and compass (or compass alone assuming that marking 2 points "gives" the line through those two points). Any number that is not algebraic of order a power of 2 cannot.

Of course, \sqrt{2} satisfies x^2- 2= 0 and so is algebraic of order 2.