[Mathematics] Sum of angles of triangle [Updated]
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d@nish wrote:
that cannot be drawn
Can you draw a straight line of length 1 meter? If you accept your pencil/pen/whatever has a certain width and you are satisfied that lengths and widths should not be more accurate than said width, then you can draw it perfectly. Same for circles, and hence also for SQRT(3) and many more irrational numbers. :)
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Luc Pattyn wrote:
surface of a sphere is a two-dimensional object
don't think that's true since it moves through length, width AND height.
in your Cartesian mind, yes. If you were the captain of a ship on one of earth's oceans, you would see longitude and latitude (hence two dimensions), and no third one, at least as long as you stay afloat. :)
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no, it just means you can't accurately measure the sqrt(3) side.
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Luc Pattyn wrote:
Can you draw a straight line of length 1 meter?
Depends on the accuracy rate we agree upon. :)
OK, you choose the accuracy, we verify I can do 1 meter; next I will do SQRT(3) meter with the same accuracy, as outlined earlier. :)
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No. You not being able to do something does not prove or disprove something else. :)
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OK, you choose the accuracy, we verify I can do 1 meter; next I will do SQRT(3) meter with the same accuracy, as outlined earlier. :)
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Well since the square-root of 3 is a non-finite number, no you couldn't draw the line. Hence the figure drawn would not be a triangle at all since the two lines would never meet and the figure would not be closed. Ergo, the "point" were one side "doesn't meet" with the 3^1/2 side has no angle.
ahmed zahmed wrote:
the square-root of 3 is a non-finite number
It is finite. I think the word you are looking for is "irrational". Another irrational number is 3.14159...
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in your Cartesian mind, yes. If you were the captain of a ship on one of earth's oceans, you would see longitude and latitude (hence two dimensions), and no third one, at least as long as you stay afloat. :)
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actually you do see a third one: the horizon is curved. (and not because there are waves)
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ahmed zahmed wrote:
the square-root of 3 is a non-finite number
It is finite. I think the word you are looking for is "irrational". Another irrational number is 3.14159...
no, I said the correct word: non-finite (or infinite). The number of digits is not countable, ergo it is infinitely long and yes, not rational, i.e., not complete comprehensible.
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I see it more like this: if X states Y is possible and I know Y is not possible, X has to be wrong.
well, I can draw a line of length 1 or of length SQRT(3) equally well as you can not prove I can't draw them. :)
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If we go for 100% accuracy, latter is not possible. If we go for anything less than that, latter wont be correct.
if you want 100% accuracy, then stick to math, and don't draw anything. :)
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if you want 100% accuracy, then stick to math, and don't draw anything. :)
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That's what I am saying with 100% accuracy, one cannot draw the 30,60,90 triangle. Never mind. I am off to sleep now. Good day. :)
Nor can you draw a 1 kilometer straight line with 1 micron accuracy. Good night to you. :)
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Surface of sphere "can be" considered 2-d if we are considering a part of its surface where sphere is of astronomical radius or we consider extremely small part of the surface. Otherwise I guess it has to be 3-d.
No. the surface of any 3d object is 2d. It's not a Euclidean plane, but that's not the definition of 2d; it just means you need 2 (no more, no less) variables to define any point on the surface; eg latitude and longitude.
3x12=36 2x12=24 1x12=12 0x12=18
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no, it just means you can't accurately measure the sqrt(3) side.
You can accurately measure sqrt(3) side as well as you measure the 1 side. The difficult is to measure the sqrt(3) side by means of 1 side. :)
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no, I said the correct word: non-finite (or infinite). The number of digits is not countable, ergo it is infinitely long and yes, not rational, i.e., not complete comprehensible.
You're wrong: it is finite. The number of digits representing the number is an infinite one. According to your definition, even
1/3is an infinite number.If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
This is going on my arrogant assumptions. You may have a superb reason why I'm completely wrong. -- Iain Clarke
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
Geometry simply doesn't care you can't draw exactly a sqrt(2) line (and you can't draw exactly a 1 line as well). :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
This is going on my arrogant assumptions. You may have a superb reason why I'm completely wrong. -- Iain Clarke
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actually you do see a third one: the horizon is curved. (and not because there are waves)
You cannot perceive that: you're a bidimensional captain. :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
This is going on my arrogant assumptions. You may have a superb reason why I'm completely wrong. -- Iain Clarke
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Rob Graham wrote:
On a 2D Plane surface, a 30,60,90 triangle can easily be drawn accurately, but you may not be able to precisely measure the length of the side that is a multiple of the square root of 3.
Which means sum of angles is not 180 degree. Right?
d@nish wrote:
Which means sum of angles is not 180 degree. Right?
Absolutely NOT! you can accurately measure and construct the angles. The siomplest construction was mentioned by another poster (perhaps you failed to comprehend): construct an equilateral triangle using any convenient side length you wish. Locate the center of one side, then draw a line to the opposite vertex. You now have two perfect 30,60,90 triangles, whose short side and hypotenuse are lengths you know very accurately, but whose long side is some multiple of the square root of 3.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree. Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
The sum of angles of a triangle is 180 degrees if and only if the edges occur within a surface having zero curvature; that is to say, if and only if the triangle is drawn on a flat surface. If the triangle is drawn on a positively curving surface -- such as the outside of a ball -- the sum of its angles will be greater than 180 degrees. If the triangle is drawn on a negatively curving surface -- such as the surface of a hyperboloid -- the sum of its angles will be less than 180 degrees. This topic is covered in depth in the mathematical field of topology. Cosmological evidence is mounting that the universe as a whole has a negative curvature; that is one of the explanations given for the increasing rate of the universe's expansion. That would mean that a triangle defined by any three points in the universe would, by definition, have angles that added up to less than 180 degrees.