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.
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.
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.
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.
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.
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.
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).
Is that basically the coastline paradox?
Yes!
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.
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.
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.
Millimeters are 1/1000 of a meter, or 1/10 of a centimeter (which is 1/100 of a meter).
Whoops, fixed.
It’s not fixed. Millimeters aren’t 1/100 of a centimeter.
It is fixed. Your ruler shows 1.0, and then you estimate 1 digit past to 1.00 +/- 0.01.
You’re not making any estimation within 1/10 like that. 1/2 is as close as you can reasonably get.
Ok, well I didn’t come up with the system so please write to the heads of science to get it changed.
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.
I used to think that “1 + 1 = 3 for high enough values of 1” was a joke until I realised it’s actually true when it comes to real-world measurements.
I’d like to point out that rational numbers can easily be written in finite length, just not in decimal format.
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.
This is a great life lesson. Even though it’s irrational, you can still do it!