How do you measure an irrational number?

Question

I am trying to get a brick layer to build a wall for my small bar-b-q pit. It is a right triangle with each of the two sides 1 foot long. The hypotenuse is therfore the square root of two feet. The bricklayer is confused with hytpotneuse wall that is an irrational length. So I told him to shorten one of the sides just enough so all 3 sides are rational numbers that he can measure. I want the one foot wall shortend as little as possible. What is the length of the shortened side?

I went back to the bricklayer and told him to just round the hypotenuese to something like like many suggest. But this irrational number really confuses him. He is a bricklayer, not a mathematican. His comment to me was "This is a real, rational wall. Why can't you give me a real, rational length?" His contract states the measurements must fit pythagoreanm's equation or he won't gurantee there won't be gaps. So I need to give him three rational numbers with one leg as 1 foot, one leg close to 1 foot as possible (but still rational) and makes the hypotenuese some rational number.

Answer 1

Why don't you try
3/4, 1 and 5/4
There could be a better one, but this is the first one I found.
(3/4)^2 + 1^2
= 9/16 + 1
= 9/16 + 16/16
= (9+16)/16
= 25/16
= (5/4)^2

Answer 2

This sounds like a problem from a book (or some sort of class problem) and not a real problem. You can actually get as close to 1 foot as you want. If you tell me how accurate you want it, I can do that. You can actually get it so close to 1, that the bricklayer wouldn't be able to measure the difference between 1 and the number.

If this is an actual problem that you are having. Round.

If you need an answer, for hw or something, I need to know how close you want it. Because "as close as you can get" doesn't work; there is no such value.

Although 3/4 of a foot (or 8") is the most popular answer, making the wall 20/21 of a foot or 11 3/7" is a better answer (and reasonably measured). This will give the hypotenuse a length of 29/21 ft or 16 4/7"

Like I said, you can get more and more accurate.

Answer 3

This is a practical matter not necessarily solved with math.. Have the builder build the 2 one foot walls first. Do not measure the hypotenuese wall but instead just fill the space between the ends of these two walls.

Answer 4

this problem is less rational than the hypotenuse in question

what kind of a barbecue pit is a triangle one foot on two sides?
how would you get a brick layer to come to your house for that tiny project (those guys are hard to find these days)?

and finally, where would you get a measuring tape with infinitely small measuring capability

the problem only arises if you measure the one foot sides as 1.00000000000000000 with infinite 0's feet.

if you measure it with any actual number of significant digits (3, or 1,000,000 or whatever) then the you can calculate your hypotenuse to the same number of significant digits and measure it with the same tape and voila, no problems at all

these infinite rulers that mathmeticians have cause all kind of problems, but only in a theoretical world

Answer 5

The great thing about analog measuring tools, such as a yardstick or a meterstick, is that they can measure irrational lengths very nicely. It all depends on how good your eyes are. Measure the hypotenuse, make a mark on the stick, cut bricks to the proper length.

But, no matter how irrational your distances are, the manufacturing tolerances on bricks will ensure that it won't matter. That is why mortar was invented.

Answer 6

Just round off square root of two from 1 foot 4.97 inches to 1 foot 5inches. Give him this measurement, and I'm sure the slightly larger hypotenuse will look better than throwing off the symmetry of the triangle. Any discrepancies should be fixable by narrowing or widening a joint slightly.

Answer 7

ingenious question.

Basics.
1.
One Leg is equal to 1
2.
The other leg should be as close to 1 as possible. Define as a/b
3.
The sum of square of (a/b)^2 and 1 needs to be rational

Restating 3. We get

(a^2 +b^2) need to be a perfect square. So A, B,C need to be a pythagorean triple.

Using what we know about pythagorean triples:
3,4,5
5,12,13
7,24,25

The ratio that is largest with a/b is 3/4

So 3/4, 1, 5/4 is the correct answer.

Answer 8

As other people have said - round.
Anything you would use to measure with has accuracy limits anyway. Most tape measures, for example, wont measure anything below 1/16th of an inch. You don't need to have any of your measurements accurate to more than 3 decimal places.

Answer 9

the brick layer does not NEED to measure the third wall... he just needs to measure the two 1 foot walls and make sure they are square to each other... and build the third wall to fill in between them.. in a straight line.. i'm sure he has something longer than 1.414 feet that is straight.

Answer 10

The square root of 2 isn't an irrational number, it is 1.414. An irrational number is one that doesn't exist (except mathematically), and that is the square root of -1, for example.