No, they don’t have to be rational. It’s counter-intuitive but you can accurately draw a line with an irrational length, even though you can’t ever finish writing that length down.
The simplest example is a right-angled triangle with two side equal to 1. The hypotenuse is of length root 2, also an irrational number but you can still draw it.
Yup. This is corollary to the other post talking about diameter. If you make a perfect circle with your perfect meter of perfect string, suddenly you can no longer perfectly express the diameter in SI units, but rather it's estimated at 31.8309886... cm. Nothing is wrong with the string in either scenario.
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?
It’s not that it can be measured forever, it’s just that it refuses to match up with any line on the ruler.
For a line of length pi: it’s somewhere between 3 or 4, so you get a ruler and figure out it’s 3.1ish, so you get a better ruler and you get 3.14ish. get the best ruler in existence and you get 3.14159265…ish
…and when you go deep enough you suddenly lose the line in a jumble of vibrating particles or even wose quantum foam, realising the length of the line no longer makes sense as a concept and that there are limits to precision measurements in the physical world.
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).
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.
In the real world, you’re measuring with significant figures.
You draw a 1 cm line with a ruler. But it’s not really 1 cm. It’s 0.9998 cm, or 1.0001, or whatever. The accuracy will get better if you have a better ruler: if it goes down to mm you’ll be more accurate than if you only measure in cm, and even better if you have a nm ruler and magnification to see where the lines are.
When you go to measure the hypotenuse, the math answer for a unit 1 side triangle is 1.414213562373095… . However, your ruler can’t measure that far. It might measure 1.4 cm, or 1.41, or maybe even 1.414, but you’d need a ruler with infinite resolution to get the math answer.
Let’s say your ruler can measure millimeters. You’d measure your sides as 1.00 cm, 1.00 cm, and 1.41 cm (the last digit is the visual estimate beyond the mm scoring.) Because that’s the best your ruler can measure in the real world.
And this comes up in some fields like surveying. The tools are relatively precise, but not enough to be completely accurate in closing a loop of measurements. Because of the known error, there is a hierarchy of things to measure from as continual measurements can lead to small errors becoming large.
Irrational numbers can be rounded to whatever degree of accuracy you demand (or your measuring instrument allows). They’re not infinite, it just requires an infinite number of decimal places to write down the exact number. They’re known to be within two definite values, one rounded down and one rounded up at however many decimal places you calculate.
Pi is irrational because it represents an imperfect ratio. Curves are different than straight lines, so when you try to relate them to one another, some things don’t match up. It happens any time straight lines and curved lines interact, I think. Pi and e both show up in the weirdest friggin places.
It has nothing to do with the curve being weird. You’re working with two length, and a length of a curve is no different from a length of a radius. You can have a circle where the circumference is exactly 1. In that case it’s the radius that’s bringing in the irrationality.
Why must the circumference and diameter of a circle be related in such a way by two integers precisely?
IE: Why are you so confident in “proving” that these two values are related to integers? Especially if you’re a modern mathematician who knows about irrational numbers (aka: can never be represented by a ratio of two integers) or imaginary numbers (which truly appear in electricity: phasors and the like. Just because the name is “imaginary” doesn’t mean that they’re not real!!!)
I don’t know that the common proof by contradiction is even remotely straightforward for this community. Niven’s proof relies on way more shit than you’d expect someone asking the question this way to know. I’m honestly not sure there is a simple proof because even Lambert’s relies on continued fractions.
I’m kind of dissatisfied with the answers here. As soon as you talk about actually drawing a line in the real world, the distinction between rational and irrational numbers stops making sense. In other words, the distinction between rational and irrational numbers is a concept that describes numbers to an accuracy that is impossible to achieve in real life. So you cannot draw a line with a clearly irrational length, but neither can you draw a line with a clearly rational length. You can only define theoretical mathematical constructs which can then be classified as rational or irrational, if applicable.
More mathematically phrased: in real life, your line to which you assign the length L will always have an inaccuracy of size x>0. But for any real L, the interval (L-x;L+x) contains both an infinite number of rational and an infinite number of irrational numbers. Note that this is independent of how small the value of x is. This is why I said that the accuracy, at which the concept of rational and irrational numbers make sense, is impossible to achieve in real life.
So I think your confusion stems from mixing the lengths we assign to objects in the real world with the lengths we can accurately compute for mathematical objects that we have created in our minds using axioms and definitions.
What’s interesting is that no matter how big or how small your circle is, pi is a constant ratio of the diameter to the perimeter (or circumference) of your circle. If you were to cut a string to the length of your circle’s diameter, it WILL take 3.14 lengths of string to wrap around the circle (or π times). That’s where that number comes from.
Because of this ratio, there will never be a situation in which both the diameter and circumference are both rational numbers at the same time. Either your Diameter is a rational number or your circumference. For example:
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