Based on my answer to the question "How is it possible to draw or make a 1x1 right triangle if the square root of 2 is irrational?" which was posted January 2, 2021 on Quora.
I think what you’re asking is how we can create a physical object (or a drawing of one) if one or more of its dimensions is irrational. In other words, if a number ends in a decimal that never terminates or repeats (such as the square root of two), how can it actually exist?
The simplest explanation is that we don’t need a name for something in order for it to exist. My daughter and I were making waffles for breakfast one weekend morning. She’s inherited my love for language, and she was learning some Spanish in her kindergarten class, so we were naming the ingredients in Spanish. We started with harina and leche. Then we cracked and added some huevos. She asked while she was mixing them, “Daddy, how do you say waffles in Spanish?” I never learned that in my high school Spanish, so I bluffed: “You know, there isn’t a word for waffles in Spanish. That’s why they’re illegal in Spain.” “Really‽” “Sure. How can they let you have something if there’s no name for it?” It took only a few moments for her five-year-old brain to process that, and apparently to realize it didn’t make sense: “You’re teasing me, Daddy, aren’t you? We don’t need names for everything!”
And so it is with numbers, too. We’ve chosen a number system that makes it easy to represent numbers most of us use frequently, like 6, or 3½, or 98.6. But one of the perhaps less desirable results is that it takes more work to represent other types of numbers. But those numbers are just as valid, and their value is just as well defined, independent of the way we choose to represent them.
Suppose we draw an isosceles right triangle and define one of the legs to be exactly 1 unit long. The other leg will also have length 1, give or take some errors in drawing and measurement, and its hypotenuse will be about 1.4 units long.
Of course, since this is a real drawing in our imperfect world, the second leg of the triangle won’t be exactly one unit long, and the “right angle” won’t be exactly 90 degrees. So let’s draw and measure more carefully. Maybe this time we’ll measure the hypotenuse to be 1.42 units long. After some more refinements and maybe an electron microscope for measurement, we’ll get an even better estimate of, say, 1.414 units.
As we improve our drawing and measurement, we’ll find the measured length of the hypotenuse gets closer and closer to some number. Mathematicians say that number is the limit as we approach infinite precision. It doesn’t matter that we’ll never achieve that precision. (The best we can do today is within a few trillionths of a meter, or maybe it's quadrillionths or quintillionths by the time you read this; what's a half-dozen orders of magnitude between friends?) We can imagine a perfect shape and calculate the exact length. We just can’t represent it exactly as a decimal, so we need to be more resourceful. In this case we name it based on one of its fundamental properties, namely that if we multiply it by itself, we get exactly 2.
There are other numbers that can’t be described with a simple algebraic expression. For example, if we repeat the draw-and-measure experiment with a circle, we find the ratio of the circumference to the diameter always approaches a specific limit, and this limit is the same for every circle. We can represent it as accurately as we like by adding together more of an ever-decreasing series of numbers, but unless we resort to summing an infinite series we’ll never get there exactly. So instead, we give it a name: pi, and we define π to be that limit, which is exact.
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