hypotenuse of a right angle triangle?

Question

If you draw a right angle triangle with 1m width at the base and a height of 1m with a machine that ensures that the measurements are precise. Would it be possible to use another machine with 100% accuracy to measure the hypotenuse?

These machines are just theoretical ; )

edit: the hypotenuse cannot be found by calculating it as it is an irrational number

Answer 1

There is no machine to measure sqrt(2) with 100% accuracy. All machines will have some measurement error. Also, this is an irrational number and never ends.

Answer 2

If, by 100% accurate, you mean ALL the digits in the answer, then I would say it is impossible.

An irrational number has an infinite number of essentially non-repeating decimal digits - your "machine" will, I presume only be able to calculate a finite number of digits in a given time. There will still be an infinite number of digits to go to achieve that desired "100% accurate" label, so the machine will always require an infinite amout of time to complete the task. i.e. it can't.

However, how about the following reasoning . . is it flawed ?

Your machine takes one second to work out the first decimal digit, half a second to work out the second one, 1/4 second to do the next, 1/8 the next, and so on.

1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . . = 2, so it will have completed the task once two seconds have elapsed . . . and will have have incinerated itself in the process.

. . . Yes, perhaps time is quantised, so the infinitessimally small units of time required to do the above calculation may not actually exist !!

Answer 3

Depends on what type of measurements your hypothetical machines can give you. If they are allowed to give you infinite expansions or represent numbers as the result of certain operations (ie sqrt(2)), then yes. Otherwise no. It has become a problem of what type of output the machine can give you. These methods where used to approximate values for things like pi a long time ago. Take a look at "A History of Pi" by Beckmann.

Answer 4

If you build the base and height of a triangle with 90 deg (i.e. Right Triangle) then you don't need to measure the hypotenuse. It can be calculated

Hypotenuse = Square Root of (base sqaured time height squared)

(pythagoras theorem)

Answer 5

The problem here is that this measurement will be defined as continuous not a discreet measurement.
Continuous measurements always have variations because of the measuring device.
Therefore the 100% accuracy doesnot apply here.

Answer 6

Why would you need a machine? The Pythagorean theorem tells you the length of the hypotenuse.

Then again, I don't see why you couldn't use a machine to measure it.

Answer 7

uh.. yeah theoretically... but if you had these perfect theoretical machines drawing the base and height perfectly you woldnt need to measuse anything u would know the exact measuse exactly from pythagorean

Answer 8

you r just messing with Pythagoras Theorem. Use formula instead of machine to measure hyp.

Answer 9

Theoretically, yes. :)

If you know of any practical machines with 100% accuracy, let me know - I'll invest!

Answer 10

yes