what is so strange about triangle with length of sqrt(2) and circumference of circle?

Discussion in 'Math' started by PG1995, Sep 26, 2011.

  1. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hi :)

    Two of the concepts which at a time really confused me are sqrt(2) and the other related one is circumference of a circle.

    If we have a triangle with a base and perpendicular of unit length, then it's hypotenuse would be sqrt(2) long. Obviously, such a length sandwiched between two other definite and measurable lengths should also be definite and measurable. Here, I use the words "definite" and "measurable from mathematics' point of view. If humans can't measure any thing definitely or absolutely, it's their limitations, not mathematics'. As sqrt(2) is irrational number and has never-ending trail of digits after the decimal point, so as a result one might argue that if something never ends then how can it be used to represent something definite (in this case hypotenuse of the triangle)? In calculus we find limits for functions using mathematical methods and we says it's the limit which the function tries to reach without ever actually reaching there. Likewise, the length of the hypotenuse which equals the value of sqrt(2) is the 'pictorial' limit and in this case human methods are the reason which prevent us from really calculating the value of sqrt(2).

    The same argument goes for circumference of a circle which is 2*pi*r. Pi is an irrational number just like sqrt(2) with unending trail of digits after the decimal point. We have the 'pictorial' limit before us but we don't have the instrument to reach it.

    I understand my reasoning is, perhaps, convoluted and a little erroneous. But I'm sure you can see beyond what I wrote and can extract the correct bits of information, and can fill the gaps between them to make the reasoning understandable for yourself. What is your opinion on length sqrt(2) and circumference? Please let me know. Thank you.

    Regards
    PG
     
  2. Georacer

    Moderator

    Nov 25, 2009
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    The length of the hypotenuse of a unit square is very defined. The length sqrt(2) is real (it belongs to the real set). It is just irrational, that means that we can't write where it is on the ruler, but we can definitely notch its place with the aid of a compass.
    Geometry has little problem with sqrt(2).

    In the same manner we can measure the circumference of a circle by rolling it along a line and marking the start and the end point.

    The fact that we can't write all of pi's digits doesn't make it undefined. We can define it as closely as we want, for example we know that it's larger that 3.13 and smaller that 3.15.

    What I want to say is that irrational numbers are part of this world (hence real), but we have trouble writing their value.
    Is that a deficiency of mathematics? Not really, because mathematics are made to depict this world.

    If you need to build a metal rod with length of sqrt(2) you will be more challenged by the technology limitations, rather than the mathematical ones.
    If you want to exactly depict the length of pi in centimeters you will run out of molecules of paper and lead long before you run out of digits.
    We can know the magnitude of the irrational numbers for all our practical purposes.

    For the purists that want to know the length of pi exactly, it is proven that pi has no end. Take it or leave it.
    It's like asking for the exact length of the universe. Has that number any meaning, given all those theories about relativity and space-time continuum?

    My 2c.
     
  3. BillO

    Well-Known Member

    Nov 24, 2008
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    Well, there is nothing that irrational about irrational numbers.

    let me explain.

    Irrational numbers are those that cannot be expressed as a ratio, like 13/23 can be. However, they both, expressed as a decimal, have a string of digits that must go on forever. It's just that 13/23 repeats a sequence of 22 digits, but it must be repeated forever to express the number properly. There really is no difference between a quantity like 'e', 'pi' or 'sqrt(2)' and any other number.

    In fact, when we write a number like 2, if its precision is absolute, we really mean 2.0000000000000000000....

    So, sqrt(2) is just a measurable and definite as is 13/23 or 2.

    In fact, the great preponderance of numbers in the set R are irrational numbers. They are infinitely more common than rational numbers.
     
  4. Georacer

    Moderator

    Nov 25, 2009
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    On a side note, if I 'm not wrong, there is a crucial difference between sqrt(2) and pi, in respect of geometry:
    We can construct a linear section with a length of sqrt(2) with a ruler and a compass (by drawing a unit square), but we can't make one with a length of pi.

    This is the cause that the circle can't be "rectangularized". Pi is a transcendal number and therefore cannot be constructed.
    Read more here: http://en.wikipedia.org/wiki/Squaring_the_circle
     
  5. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    @GeoRacer: Could you elaborate a bit what you mean by "cannot be constructed"? Thanks.
     
  6. BillO

    Well-Known Member

    Nov 24, 2008
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    He means the number is not an algebraic constructable number. It can not be constructed with a straightedge and a compass given some unit measure.

    It should be noted that there are other reasons a number can be transcendental other than not being constructable. However, pi is not constructable.


    Edit: Again, it should be noted that most numbers are transcendental, for one reason or another. The reason we are not familiar with them is, we choose to work with easy numbers to study math and physics. Also, pi is only transcendental in the chosen number system. We could define the fundamental mathematical unit as the ratio between the diameter and circumference of a circle. Then pi would not be such a burden. Other things would be though.
     
    Last edited: Sep 26, 2011
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  7. t_n_k

    AAC Fanatic!

    Mar 6, 2009
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    Is there any real difference between trying to make a metal rod exactly 1 inch in length or a another rod of the same material exactly √2 inch in length? Presumably there are always issues of precision and accuracy in the fabrication of a real physical object. I know that scientists have been making near perfect metal spheres 'exactly' 1kg in mass [as the potentially new form of a mass standard] - I believe they are "shaving off" layers of dimensions approaching the metal atomic radii.

    http://www.newscientist.com/article/dn14229-roundest-objects-in-the-world-created.html
     
    Last edited: Sep 27, 2011
  8. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hello everyone

    The guys here are very helpful and most importantly they aren't arrogant freaks! :)

    Thanks. I didn't know that. So, Pi cannot be constructed using math tools such as compass, straight edge.

    Hi TNK

    Good point. From real world's point of view there isn't much difference between the two because making a rod exactly as long as 2 inches is as hard as making one of sqrt(2) length. Still, numbers like Pi and sqrt(2) are hard to conceptualize because they seem be never ending numbers which suggests as if they don't have fixed 'length' yet they have definite lengths as I said in first post above. Thank you.

    Best wishes
    PG
     
  9. Georacer

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    Nov 25, 2009
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    From what we discussed in this thread it seems that given a compass and a straightedge making the 1inch and the sqrt(2)inch rod is the same. However, you cannot construct a pi-inch rod.

    The first two differ in that you can't construct a sqrt(2)-inch rod with the use of a scaled ruler.
     
  10. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hi Geo

    I understand what you say. Say, we use a pencil to make 1 inch and sqrt(2) inch rods. The pencil's mark would have a definite thickness. So, where does the end for, say 1 inch rod, lie? You can't tell. No matter how thin you try to make the pencil's mark its thickness would never become zero (on the other hand, why would we want it to become zero? How would we draw then?!). More interestingly, you can't even exactly pinpoint the beginning of such a rod because the markings on the ruler used would also have a definite thickness. When I wrote my previous post I had this in mind. I hope I make sense. Please correct me if I'm wrong. Thank you.

    Regards
    PG
     
  11. Georacer

    Moderator

    Nov 25, 2009
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    In our theoretical line of thoughts we don't take mark widths into account. We assume that we can duplicate lengths exactly, without number marks being a nuisance.

    Using a straightedge and a compass, create an arbitrary unitary linear segment. That is your unit. Using your compass you can duplicate this exact unit as many times as you want.
    Then using euclidean geometry, create a square with a side of one unit. Using your compass, create a linear segment as long as its diameter. You now have a linear segment with length of sqrt(2)!
    You couldn't have constructed that linear segment using the notches of a ruler, because you can't locate the number sqrt(2) on a ruler.
    However, no matter what you do, you can't construct a linear segment with length of pi. Bummer.
    That is the difference between sqrt(2) and pi in respect of euclidean geometry.
     
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  12. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hi

    I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied? If my question is not clear, then please let me know. I would try to rephrase it. Thanks a lot.
     
    Last edited: Oct 5, 2011
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