Irrational Lengths

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yalialp06

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We often say that the length of the hypotenuse is square root of 2 or the square root of 5. How can a length be irrational?
 
If you have a right triangle and the legs are both 1, then the hypotenuse is sqrt(2). Just ask Pythagoras. Why do you think that a length can't be sqrt(2)??
 
Check out this video I made a while ago. It shows that the square root of any positive integer is constructible. Click here
 
yalialp06 said:
We often say that the length of the hypotenuse is square root of 2 or the square root of 5. How can a length be irrational?
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Why do you think that length cannot be irrational?

The circumference of a circle - which is a "length" - is also irrational.
 
"Irrational" as a word might lead one to think that the number is preposterous or nonsensical, but it has an explicit and well-defined scientific meaning when used in a mathematical context. Irrationality simply means that the number cannot be expressed as a fraction of integers. It is, however, still an ordinary number otherwise.
 
I know that the hypotenuse is sqt of 2 in this triangle. However, the fact that I can't measure it is throwing me off.
 
Subhotosh Khan said:
Why do you think that length cannot be irrational?

The circumference of a circle - which is a "length" - is also irrational.
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I know it can be irrational. I just don't know how to explain how it can be irrational. Meaning, how can there be a length that I can't measure.
 
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yalialp06 said:
I know it can be irrational. I just don't know how to explain how it can be irrational. Meaning, how can there be a length that I can't measure.
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This length which is finite is represented by a number that is infinite!
 
Come on you know that sqrt(2) is NOT infinite. Just because as a decimal number it goes on forever you can't conclude that sqrt(2) is infinite. After all you can write 1 as 1.000000.... and 1 is finite. sqrt(2) < 2 so how can it be infinite?

I can measure the sqrt(2)!! And I can do it with absolute precision! Again, what is the hypotenuse of a right triangle whose legs are both 1? Just put a straight edge along that hypotenuse, mark off the beginning and end of that hypotenuse and you have the length of sqrt(2).

Did you watch the video that I posted above? Please watch it.
 
Just for the record can you please explain how you can measure say 1 inch? Answering this question will greatly help you understand why you can measure sqrt(2), sqrt(3), sqrt(4), ....
 
There is no such thing as a physically observable number that can be proven to be irrational. The irrational numbers exist in a Platonic world of ideas where such things as perfect triangles, dimensionless points, lines of infinite length, and errorless measurements exist. Clearly, that is not the world of physical reality. It is surprising how useful such idealized concepts are when applied to the real world.
 
Jomo. Your explanations are very helpful. Especially when you explained that the sqt of 2 is less than 2, therefore it cannot be infinite. I also liked how you explained that it being represented by a decimal that goes on forever does not mean it's infinite. Thank you for that. I also watched your video. I understand the construction somewhat, but I have to think about it more deeply. I also added a link to a document that I was reading as part of my exploration. It is below. The document explains the construction in your video. Thank you again, and I will continue to explore these ideas.
 

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Another way that I think it can be explained is with 1/3 for example. 1/3 when written as a decimal, repeats forever. That doesn't mean that 1/3 is an infinite number. It just means that 1/3 doesn't fit our number system of base 10, therefore the decimal repeats. These are just some ideas going on in my mind. Does anyone agree or want to add anything?
 
from a circle radius 1 we can obtain the irrational lengths sqrt(2), sqrt(3), sqrt(5), sqrt(6) etc...
kv-vs-siemens2.jpg
 
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