Physical representation of irrational numbers

[PhysDrew]

My question relates to a specific example, namely the square root of two. If one forms a right isosceles triangle with the hypotenuse equal to 2 (be it metres, centimetres or whatever) then the other two sides must equal the square root of 2. But the square root of 2 is an irrational number. If one was to make this triangle physically, wouldn't it be impossible, as the number (and hence the length of the sides) continues on forever? Is this a reflection of the incompleteness of mathematics? Or is the physical representation of any number (or length) impossible to achieve precisely, and hence the sides aren't exactly equal to 2 (and hence 2^(1/2))?

EDIT: Or is Euclidean geometry fundamentally incorrect?

 

[tiny-tim]

Hi PhysDrew!

… Is this a reflection of the incompleteness of mathematics?

No, it's a reflection of the incompleteness of the rational numbers.

Mathematics completes the rational numbers by creating the algebraic numbers (solutions to a polynomial = 0) or the real numbers (including non-algebraic numbers like π and e).

If you want to model the real world, you need to choose the right maths for it …

if you don't, the maths isn't incomplete, you are! ​

 

[DoctorBinary]

If one was to make this triangle physically, wouldn't it be impossible, as the number (and hence the length of the sides) continues on forever?

The length is finite -- just our place value representation of it is infinite.

 

[CRGreathouse]

The length is finite -- just our place value representation of it is infinite.

Ditto. This 'problem' happens with lengths as 'simple' as 1/3, and for some reason those attract less antipathy, from the time of Pythagoras on... I don't really know why. I guess I distinguish between a number and its representation and most people don't?

 

[dkotschessaa]

So could we say the number of decimal places in the real world just represents how precisely you are measuring something? An object that's 1 foot long might really be 1.00001 foot if you can measure it that far, or perhaps 1.000012 or 1.0000123 etc. as you get more and more precise. The only limits would be those of our measuring equipment.

So I suppose if you are going to create something the length of sqrt2 you'd just have to decide how precise you want to be about it.

Just rephrasing the above responses in more rough language so I can understand it. :)

-DaveK

 

[HallsofIvy]

Yes, that's reasonable. But recognize that line segment representations of numbers do not reflect the properties of the numbers themselves.

 

[DoctorBinary]

So I suppose if you are going to create something the length of sqrt2 you'd just have to decide how precise you want to be about it.

I think our inability to draw something of exactly length sqrt(2) is a separate issue. In any case, whatever we do manage to draw, might it not have an irrational length anyhow?

 

[CRGreathouse]

In any case, whatever we do manage to draw, might it not have an irrational length anyhow?

It's not clear how you could test whether the length of a physical object was irrational or not, even in principle.

 

[dkotschessaa]

I think our inability to draw something of exactly length sqrt(2) is a separate issue. In any case, whatever we do manage to draw, might it not have an irrational length anyhow?

Yeah, that's kind of my point. In real life, anything we create is likely to have an irrational length, as we keep measuring with more and more precision. I actually have no idea if this is true or how to prove this. Isn't the "exactness" of something relative to whatever measuring device you are using?

-DaveKA

 

[DoctorBinary]

It's not clear how you could test whether the length of a physical object was irrational or not, even in principle.

I didn't say you could measure it -- just that it might be irrational. (But I'm getting into "philosophical" issues beyond my expertise I'm afraid.)

 

[dkotschessaa]

Maybe we can say (and perhaps I'm just repeating what somebody else has said in another way) that "irrational" is a mathematical concept and not a physical one.

 

[PhysDrew]

Maybe we can say (and perhaps I'm just repeating what somebody else has said in another way) that "irrational" is a mathematical concept and not a physical one.

Yeah, I remember reading a discussion on the nature of mathematics itself, and the overwhelming consensus seemed to be that the mathematical laws and relationships are "out there" in nature, and we simply uncover them. But it seems to me that mathematics is the construct of the human mind to model the world which we observe. We then create laws, proofs and theorems based on these observations to create new mathematical derivations that we can apply to the unobservable, based on logic and deductive reasoning. I just don't know which is correct. I just get the strange feeling that somethings not right! I think I have to study a bit more

 

[CRGreathouse]

Maybe we can say (and perhaps I'm just repeating what somebody else has said in another way) that "irrational" is a mathematical concept and not a physical one.

That's certainly not what I'm saying, just that to know whether a number is rational or irrational requires exact measurement which in all or most cases seems to be impossible in the 'real world'.

 

[PhysDrew]

That's certainly not what I'm saying, just that to know whether a number is rational or irrational requires exact measurement which in all or most cases seems to be impossible in the 'real world'.

Just being able to measure it or not being able to measure it doesn't diminish the fact that the length continues on forever. So where does the extra material go? Does it loop around into another dimension? Does nature recognise the paradox of an irrational number length and truncates it? It's interesting to think about anyway..

 

[dkotschessaa]

Just being able to measure it or not being able to measure it doesn't diminish the fact that the length continues on forever. So where does the extra material go? Does it loop around into another dimension? Does nature recognise the paradox of an irrational number length and truncates it? It's interesting to think about anyway..

There's no extra - just smaller and smaller divisions, chopping something up more and more. There is a Zeno's paradox - go half way to distance X (x/2), then go half that distance (x/4) then half that distance etc. etc. You can continue for infinity but you will never get to X.

-DaveKA

 

[CRGreathouse]

length continues on forever

?

 

[chiro]

My question relates to a specific example, namely the square root of two. If one forms a right isosceles triangle with the hypotenuse equal to 2 (be it metres, centimetres or whatever) then the other two sides must equal the square root of 2. But the square root of 2 is an irrational number. If one was to make this triangle physically, wouldn't it be impossible, as the number (and hence the length of the sides) continues on forever? Is this a reflection of the incompleteness of mathematics? Or is the physical representation of any number (or length) impossible to achieve precisely, and hence the sides aren't exactly equal to 2 (and hence 2^(1/2))?

EDIT: Or is Euclidean geometry fundamentally incorrect?

It really depends on what reference or base system you are defining.

Typically we use base 10 number system (probably because we have 10 fingers), but you can use any base you want including irrational bases.

Also a number in one base can be rational and the same number in another base can be irrational. Think about representing 1/3 in base 10 and 1/3 in base two. In the first it is a rational number however in the second it is not.

There's no reason why an irrational number can not be used as a base (ie something like pi, or e, or ln(2) etc). The thing is though that historically we used integers and then generalized to real numbers and not the other way around, so typically our thinking and introduction to numbers is with integers which might explain why so many people in history struggled like Pythagoras did with irrational numbers.

 

[HallsofIvy]

Just being able to measure it or not being able to measure it doesn't diminish the fact that the length continues on forever. So where does the extra material go? Does it loop around into another dimension? Does nature recognise the paradox of an irrational number length and truncates it? It's interesting to think about anyway..

It might be interesting to think about if I had a clue what you were saying! What "length" are you saying 'continues on forever'? Are you back to claiming that since \sqrt{2} cannot be written, in base 10, as a terminating decimal, that, in some sense, a line segment of length \sqrt{2} must be infinite?

Does the fact that everyone responding here has said that is wrong bother you at all? Whether a number is rational or irrational, can be written as a terminating decimal or not, says nothing about a line segment of that length "going on forever"- in fact just saying that is has a length means it does not. How we represent a number has nothing to do with "how large" it is or how long segments of that length are.

 

[DoctorBinary]

Also a number in one base can be rational and the same number in another base can be irrational.

Being rational or irrational is a property of a number, not its base.

Think about representing 1/3 in base 10 and 1/3 in base two. In the first it is a rational number however in the second it is not.

1/3 is rational in any base. In base 10, it is 1/3 = 0.(3). In base 2, it is 1/11 = 0.(01). (I used parentheses to indicate repeating digits.)

 

[PhysDrew]

Thank you to all that posted.
Admittedly I do feel very embarassed about my evidently total lack of understanding in this topic, and on number theory in general. However I am determined to raise my knowledge of this subject. I often blurt things out that pop into my head straight away, when I should have simply thought about it more.
Anyway I have since learned from my mistakes, thanks again.

 

[ramsey2879]

Thank you to all that posted.
Admittedly I do feel very embarassed about my evidently total lack of understanding in this topic, and on number theory in general. However I am determined to raise my knowledge of this subject. I often blurt things out that pop into my head straight away, when I should have simply thought about it more.
Anyway I have since learned from my mistakes, thanks again.

PhysDrew, don't be embarrassed about in making a mistake in your question. As you said, you learned from it and are determine to increase your knowledge of the subject. I hope that you will continue to post your questions as you continue to learn. Only a fool doesn't ask questions about something he doesn't understand since only fools are too proud to admit or show their lack of understanding in order to do something to gain the understanding they know they lack.
As you now know, finite rational numbers such as 1/3 (in fact any rational number with a prime factor in the denominator other than a factor in the decimal base) will have an infinite decimal representation. That the representation is infinite does not mean that the number it self is infinite. If there is anything in the above posts that you still don't understand don't hesitate to ask another question, if you still can't find an answer that explains it to you after some effort.

 

[dkotschessaa]

Thank you to all that posted.
Admittedly I do feel very embarassed about my evidently total lack of understanding in this topic, and on number theory in general. However I am determined to raise my knowledge of this subject. I often blurt things out that pop into my head straight away, when I should have simply thought about it more.
Anyway I have since learned from my mistakes, thanks again.

I would think that even the learned here should at least understand where the confusion comes from. The key is whether the rest of us are willing to listen, I guess. I'm still not 100% clear.

Allow me to re-confuse us all over again :)

My boss asks me to measure a thingamajig, and so using a standard ruler I find that it is .5 inches. I call my boss with this information and he tells me this isn't precise enough. So I measure more precisely and I get .54 inches. Again I am told I need to be more accurate, so I begin to break out more precise instruments and measure.

My TI-84 evaluates that the statement .542 > .54 is true. As is .5421 > .542, and so on.

.542 = .54 + .002 So I've added a value .002, getting a larger number. My calculator seems to be telling me that my thingamajig is getting larger, though of course I know this not to be the case. I am simply being more precise.

What if I have two objects, A and B - both of which measure .5 inches with a standard ruler.

A B
.5 .5 - objects are the same size
.54 .52 - at this point A is larger - B has gotten "larger" but "shrunk" relative to A.
.542 .524332

I decided in the third case to measure B more precisely than A, to see if I could make it catch up, but no matter how many decimal places I measure to I can never make B equal to or larger than A.

If anybody can tell me what the heck I've just demonstrated here I'd love to know.

-DaveKA

 

[Quisquis]

I would think that even the learned here should at least understand where the confusion comes from. The key is whether the rest of us are willing to listen, I guess. I'm still not 100% clear.

Allow me to re-confuse us all over again :)

My boss asks me to measure a thingamajig, and so using a standard ruler I find that it is .5 inches. I call my boss with this information and he tells me this isn't precise enough. So I measure more precisely and I get .54 inches. Again I am told I need to be more accurate, so I begin to break out more precise instruments and measure.

My TI-84 evaluates that the statement .542 > .54 is true. As is .5421 > .542, and so on.

.542 = .54 + .002 So I've added a value .002, getting a larger number. My calculator seems to be telling me that my thingamajig is getting larger, though of course I know this not to be the case. I am simply being more precise.

What if I have two objects, A and B - both of which measure .5 inches with a standard ruler.

A B
.5 .5 - objects are the same size
.54 .52 - at this point A is larger - B has gotten "larger" but "shrunk" relative to A.
.542 .524332

I decided in the third case to measure B more precisely than A, to see if I could make it catch up, but no matter how many decimal places I measure to I can never make B equal to or larger than A.

If anybody can tell me what the heck I've just demonstrated here I'd love to know.

-DaveKA

You're demonstrating the idea of a limit. As you measure more and more accurately, you approach the actual value of the measurement.

Think about it on larger scales. I have a car that only measures in 100 mile increments. Either I've traveled 100 miles or zero or 200 miles etc..., but not anything in between. Now I take a trip and I see I've gone 700 miles. If someone with a more accurate odometer measures the same trip at 740 miles was it longer for him? If another person measures 738, was it shorter for him? No, everybody traveled the same, finite, distance. Their measuring was just more accurate.

As for your real \sqrt{2}, \sqrt{2}, 2 triangle, it is only those lengths to the accuracy we can measure them to; in fact, I think it's not just that we can't measure it, so we don't know, but instead is that it's literally impossible for it to actually be the length it's measured to be OR is supposed to be.

 

[dkotschessaa]

Thanks Quisquis.

Since this thread started, I got (for Christmas) a copy of Timothy Gowers' book "Mathematics: a Very short Introduction," and he actually touches on this topic in Chapter 4: Limits and Infinity.

I highly recommend this book to anybody trying to understand, as the intro says "the most fundamental differences, which are primarily philosophical, between advanced mathematics and what we learn at school, so that one emerges with a clearer understanding of such paradoxical sounding concepts as "infinity", "curved space," and "imaginary numbers."

And the book is very short - 140 pages that will fit in your pocket and can be read in a weekend.

-DaveKA

 

[ramsey2879]

One should note that technically it is no harder to construct a length equal to the \sqrt{2} times a unit length than it is to divide a unit length in half. In either case, in base 3, the decimal representation is infinite although the steps of construction are presumed to accurately result in the length they are supposed to construct.

 

[Hippasos]

Hi PhysDrew!


No, it's a reflection of the incompleteness of the rational numbers.

Mathematics completes the rational numbers by creating the algebraic numbers (solutions to a polynomial = 0) or the real numbers (including non-algebraic numbers like π and e).

If you want to model the real world, you need to choose the right maths for it …

if you don't, the maths isn't incomplete, you are! ​

I'd say it's a reflection of the incompleteness of a MEASUREMENT.