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this post was submitted on 31 Jan 2024
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No Stupid Questions
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Thanks for the answer. I'm confident you're correct but I'm having a hard time wrapping my head around drawing a line with an irrational length. If we did draw a right angled triangle with two sides equal to 1cm and we measured the hypotenuse physically with a ruler, how would we measure a never ending number? How would we able to keep measuring as the numbers after the decimal point keep going forever but the physical line itself is finite?
You're talking about maths, maths is theoretical. Measuring is physics.
In the real world you eventually would have to measure the atoms of the ink on your paper, and it would get really complicated. Basically .. you can't exactly meassure how long it is because physics gets in the way (There is an entire BBC documentary called "How Long is a Piece of String" it's quite interesting).
Thanks for the answer and for suggesting the documentary!(excited to have my head hurt even more after watching it😂)
Another way of thinking about it:
Numbers offer a sense of scale. As numbers go further left from the decimal, they get bigger and bigger. Likewise, as they go right from the decimal, they get smaller and smaller.
If I’m looking with just my eyes, I can see big things without issue, but as things get smaller and smaller, it becomes more and more difficult. Eventually, I can’t see the next smallest thing at all.
But we know that smaller thing is there— I can use a magnifying glass and see things slightly smaller than I can unaided. With a microscope, I can see smaller still.
So I can see the entirety of a leaf, know where it begins and ends, even though I can’t, unaided, see the details of all its cells. Likewise, you can see the entirety of the line you drew, it’s just that you lack precise enough tools to measure it with perfect accuracy.