Making a Triangle with >180 Degrees

[bobross419]

I remember from some math class I took that there is a principle (could be the wrong term) that says that an infinitely repeating decimal like .999999999... can be considered to be the whole number 1. If I recall correctly this also applies on the other side: 1.0000...001 = 1

Couldn't you technically use this principle to create a triangle with 3 angles: 90, 45, 45.0000...001

Based on the principle that 45.0000...001 equals wouldn't this allow the criteria for a triangle to be met?

http://www.mathopenref.com/triangle.html
A closed figure with three sides which are straight line segments linked end-to-end.
A 3-sided polygon.


http://www.geom.uiuc.edu/~wano...nitionofatriangle.html
A triangle is a closed figure with three sides

I am by no means a mathematician, so if this is totally retarded please let me know.

Also, if anyone knows the name of the principle that I am referring to please let me know.

 

[CoachB]

If I recall my H.S. Geometry correctly....

You can have a triangle with 90, 44.999999, and 45.0000001 as everything MUST add up to 180 degrees. This is NOT because a triangle must total 180 but because, when two lines intersect any two adjacent angles must total 180 degrees and opposite angles MUST be equal.
In essence, enlarging angle #1 to 45.00001 creates angle #2 with a 44.999999 degree measure. No way around it.

I can see it in my head, but putting it in words is a bit difficult :>)

 

[bobross419]

That makes sense CoachB, but what I'm saying is instead of the 44.99999... to offset the 45.0000...001 you just have 90, 45, and 45.000...001 because the 45.0000...001 equals 45 for all practical purposes (Yes the Engineer joke thread spawned this idea).

 

[Born2bwire]

45.0000...001 is not 45 due to the fact that it is a terminating decimal. 44.999... = 45 due to the fact that it is an infinite series. So lines at angles of 90, 45, and 45.000...001 are not a valid set of angles for a triangle.

 

[bobross419]

Originally posted by: Born2bwire
45.0000...001 is not 45 due to the fact that it is a terminating decimal. 44.999... = 45 due to the fact that it is an infinite series. So lines at angles of 90, 45, and 45.000...001 are not a valid set of angles for a triangle.

Alright, now lets look at the 45.00000...001 angle this way. It could be the opposite site of a 134.999999... angle on a 180 degree line. Wouldn't this allow for the 45.000...001 side of the angle to be considered non-terminating since it is the opposite side of a non-terminating angle?

 

[Special K]

45.0000...0001 is not a number. By using the "...", you are implying an infinite sequence of 0's. You can't then terminate an infinite sequence of 0's with a 1.

 

[CycloWizard]

No, it can't be done. Opening the last angle by 1e-15° causes the 90° angle to no longer be 90°. Try it and you'll see right away. The only way what you propose works is if the triangle has curved sides.

 

[bobross419]

Originally posted by: Special K
45.0000...0001 is not a number. By using the "...", you are implying an infinite sequence of 0's. You can't then terminate an infinite sequence of 0's with a 1.

Then what would the other side of a 134.9999.... angle be?

I'm not trying to be difficult, I'm just trying to understand.

 

[dinkumthinkum]

If you want a triangle with > 180 degrees summed from the angles then try your hand at Spherical Geometry.

 

[CP5670]

Originally posted by: bobross419

Originally posted by: Special K
45.0000...0001 is not a number. By using the "...", you are implying an infinite sequence of 0's. You can't then terminate an infinite sequence of 0's with a 1.

Then what would the other side of a 134.9999.... angle be?

I'm not trying to be difficult, I'm just trying to understand.

If those 9s go on forever, that number is exactly equal to 135 (there are a couple of different ways to see this, using decimal shifts or limits), so the other side would just be 45.

As far as the triangles go, the only way you could have the angles not add up to 180 degrees is in a non-Euclidean space.

 

[CycloWizard]

Originally posted by: bobross419
Then what would the other side of a 134.9999.... angle be?

I'm not trying to be difficult, I'm just trying to understand.

Like I said, the easiest way for you to understand why this doesn't make sense is to actually try to draw it. You will quickly see that to increase one angle of a triangle, at least one other angle must change. I don't know a simpler way to say it than that. If you have two angles that are exactly 45° and 90°, the other angle is completely determined by nature of the intersection of the other two segments.

 

[TridenT]

Originally posted by: dinkumthinkum
If you want a triangle with > 180 degrees summed from the angles then try your hand at Spherical Geometry.

Damn youuuu!!! That was going to be my answer.

 

[TuxDave]

The whole notation of 45.0000...0001 doesn't make sense. The "..." implies infinite number and never ending train of zeros. It doesn't mean "a million or a billion or a gazillion zeros". It means an INFINITE number of zeros.

So how can the number terminate with a 1 after an infinite number of zeros?

 

[TuxDave]

Originally posted by: bobross419

Originally posted by: Special K
45.0000...0001 is not a number. By using the "...", you are implying an infinite sequence of 0's. You can't then terminate an infinite sequence of 0's with a 1.

Then what would the other side of a 134.9999.... angle be?

I'm not trying to be difficult, I'm just trying to understand.

Here's a brain burster for you.

2 - 0.999... = 0.999...

 

[Cogman]

Originally posted by: TuxDave

Originally posted by: bobross419

Originally posted by: Special K
45.0000...0001 is not a number. By using the "...", you are implying an infinite sequence of 0's. You can't then terminate an infinite sequence of 0's with a 1.

Then what would the other side of a 134.9999.... angle be?

I'm not trying to be difficult, I'm just trying to understand.

Here's a brain burster for you.

2 - 0.999... = 0.999...

oh I love it. People just don't innately understand what exactly infinity implies.

Like Tux Dave said, there is no difference between 1, and .999...

 

[BrownTown]

BAN

 

[PowerEngineer]

As the OP concedes, 0.999... is equal to 1.0. OP also seems to understand that 1.00...001 (yes, by this I mean an infinite number of zeros in fornt of the one) is also equal to one (which might also be expressed as 2.0 - 0.999...).

So why is it then that OP seems to think that 45.00...001 (or 46.0 - 0.999...) is not equal to 45.0? It's just as true as 44.999... = 45.0.

So, it seems to me that the original question about the possibility of constructing a triangle with angles of 90, 45, and 45.00...001 degrees is really equivalent to asking if you can contruct a triangle with angles 0f 90, 45, and 45 degrees...

I'd say that's possible! :laugh:

 

[randay]

the problem here is understanding "=".

 

[bobross419]

Originally posted by: PowerEngineer

As the OP concedes, 0.999... is equal to 1.0. OP also seems to understand that 1.00...001 (yes, by this I mean an infinite number of zeros in fornt of the one) is also equal to one (which might also be expressed as 2.0 - 0.999...).

So why is it then that OP seems to think that 45.00...001 (or 46.0 - 0.999...) is not equal to 45.0? It's just as true as 44.999... = 45.0.

So, it seems to me that the original question about the possibility of constructing a triangle with angles of 90, 45, and 45.00...001 degrees is really equivalent to asking if you can contruct a triangle with angles 0f 90, 45, and 45 degrees...

I'd say that's possible! :laugh:

So you're saying that this doesn't work because I can't say that 44.99999.... equals one then turn around and say it doesn't?

That makes sense to me. And sorry to everyone, but as I explained I am not a mathematician and wanted some clarification on this question. Thanks for all the answers

 

[silverpig]

Use curved space.

 

[iCyborg]

Originally posted by: bobross419
So you're saying that this doesn't work because I can't say that 44.99999.... equals one then turn around and say it doesn't?

Well, you can't say 44.999... equals one because it equals 45

44.999... is a shorthand for a decimal representation of a certain number that happens to have two representations. 45.000...001 is ill-defined and meaningless. Look up decimal notation and you'll see why.

 

[DrPizza]

Let's say for a moment that the notation 45.000...001 actually did mean something. Then, the compliment would be: 44.999...999 Notice, it's now a TERMINATING decimal, not an infinitely repeating decimal. If the OP wants the .999... part to repeat FOREVER, then the .000... must also repeat FOREVER. Hence, there is no trailing 1.

And, to expand on what others have suggested: spherical geometry, grab a globe, start at some point on the equator. Travel due East (or West) 1/4 of the way around the globe. Stop, turn due north, go to the north pole. Turn 90 degrees and turn south toward your starting point. You now have a triangle (in spherical geometry) with 3 right angles. This is why the definition of a triangle in Euclidean geometry includes the phrase "in a plane."

 

[bobross419]

Cool. Thanks for helping me understand this folks

 

[bobdole369]

"The only way what you propose works is if the triangle has curved sides."

Or exists in more than 2 dimensions - such as on the surface of a sphere, or in 4-D or greater space.

 

[silverpig]

Originally posted by: bobdole369
"The only way what you propose works is if the triangle has curved sides."

Or exists in more than 2 dimensions - such as on the surface of a sphere, or in 4-D or greater space.

A triangle, in no matter what space, is a 2 dimensional object.

The surface of a sphere is also 2 dimensional.

You can also have 4-D flat space.