Really basic stuff that was never proven in class

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Re: Really basic stuff that was never proven

Postby D.B. » Wed Sep 22, 2010 10:28 am UTC

voidPtr wrote:One thng I don't ever remember being proved in grade-school maths is that the angles in every triangle add up to 180 degrees.

Actually, I remember a lot more of what wasn't proved than what actually was proved, some of it for good reason, some not.


Actually, I rememebr this one being shown specifically to me when I was in school. School grade "proof" follows:

Imagine walking round the edges of a triangle. Suppose you start at a vertex of the triangle, and walk along one of the edges until you reach the next vertex. You now rotate, through the smallest angle possible (i.e. the exterior angle), so that you are facing in the direction of the next edge. Let's call the amount you rotate to do this, in degrees, [imath]r_1[/imath]. You carry on your journey, repeating this process at the next vertex (rotation [imath]r_2[/imath]), and finally at the third vertex (rotation [imath]r_3[/imath]) so that you've gone all the way round and are now facing the same direction as whe you began.

As you've turned in a full circle in the course of talking your walk, we know
[math]r_1 + r_2 + r_3 = 360[/math].

Now consider the interior angles at each vertex of the triangle, calling them [imath]a_1, a_2, a_3[/imath]. As each one, when added to the exteroir angles (the ones you rotated through) must add up to 180 degrees (as together they must make a straight line), we now know

[math]a_1 = 180 - r_1[/math]
[math]a_2 = 180 - r_2[/math]
[math]a_3 = 180 - r_3[/math]

And so we can get the sum of the interior angles like so

[math]a_1 + a_2 + a_3 = 180 - r_1 +180 - r_2 + 180 - r_3 = 540 - (r_1 +r_2 + r_3) = 540 - 360 = 180[/math]

so the interior angles of any triangle must add up to 180.


I've not thought about that for years. With hindsight I'm impressed at my maths teacher for going through it with us, as I'm doubtful it would have been on the syllabus.

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Re: Really basic stuff that was never proven

Postby SlyReaper » Wed Sep 22, 2010 12:03 pm UTC

The way I always envision it was by copying the triangle twice, rotating one of them 180 degrees, then tesselating them like this:

tri.JPG
tri.JPG (6.5 KiB) Viewed 3783 times


If you look at where the blue, red and green angles intersect, you see they make a straight line. Thus, red + blue + green = 180 degrees. And those are each one of the angles of the same triangle.
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Re: Really basic stuff that was never proven

Postby gmalivuk » Wed Sep 22, 2010 1:56 pm UTC

The advantage of D.B.'s proof over that one, and what makes it particularly impressive for gradeschool, is that you can similarly use it to prove the angle formulas for all other polygons.
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Re: Really basic stuff that was never proven

Postby Spambot5546 » Wed Sep 22, 2010 1:57 pm UTC

Alternatively, look at the largest angle of a given triangle and try to increase it. As it approaches 180o the other angles approach 0 to the point that at 180o it is no longer a triangle but a straight line.
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Re: Really basic stuff that was never proven

Postby jestingrabbit » Wed Sep 22, 2010 2:12 pm UTC

gmalivuk wrote:The advantage of D.B.'s proof over that one, and what makes it particularly impressive for gradeschool, is that you can similarly use it to prove the angle formulas for all other polygons.


You can pretty easily do that with the result for triangles, and to formalise it you need the results about lines crossing parallel lines.

Spambot5546 wrote:Alternatively, look at the largest angle of a given triangle and try to increase it. As it approaches 180o the other angles approach 0 to the point that at 180o it is no longer a triangle but a straight line.


That doesn't work. You've shown that the angles in a degenerate triangle sum to a straight line, you haven't shown that the sum is the same for all triangles.
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Re: Really basic stuff that was never proven

Postby mike-l » Wed Sep 22, 2010 2:12 pm UTC

Spambot5546 wrote:Alternatively, look at the largest angle of a given triangle and try to increase it. As it approaches 180o the other angles approach 0 to the point that at 180o it is no longer a triangle but a straight line.


How do you know the sum of the angles is constant though?
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Re: Really basic stuff that was never proven in class

Postby Spambot5546 » Wed Sep 22, 2010 2:29 pm UTC

I guess i was assuming you were looking for proof that the sum of the angles is 180, not that there is a constant sum of the angles at all.
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Re: Really basic stuff that was never proven in class

Postby SlyReaper » Wed Sep 22, 2010 2:37 pm UTC

Spambot5546 wrote:I guess i was assuming you were looking for proof that the sum of the angles is 180, not that there is a constant sum of the angles at all.

But you proved neither. You proved that the sum of angles in one particular case was 180 degrees. You have to prove it for all possible combinations of angles. From your starting point, the obvious way to attack it would be to show that the sum of angles is constant.
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Re: Really basic stuff that was never proven in class

Postby Tyl. » Wed Sep 22, 2010 3:02 pm UTC

No no, assuming we have proven that there is a constant sum of angles, he did prove (well, yea..) that constant must be 180.

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Re: Really basic stuff that was never proven in class

Postby SlyReaper » Wed Sep 22, 2010 3:07 pm UTC

Tyl. wrote:No no, assuming we have proven that there is a constant sum of angles, he did prove (well, yea..) that constant must be 180.

Which is a bad assumption because he didn't show the sum was constant. Once he does, his proof will be complete.
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Re: Really basic stuff that was never proven in class

Postby Eebster the Great » Wed Sep 22, 2010 6:50 pm UTC

Actually, I remember proving a LOT in my geometry class. That was the one class where proofs were very important. We pretty much proved everything we learned that year from a few postulates.

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Re: Really basic stuff that was never proven in class

Postby Tirian » Wed Sep 22, 2010 8:06 pm UTC

I agree. The only thing in plane geometry that wasn't proven were the axioms. Pick up a copy of Euclid's Elements; if you don't recognize everything there, it's because you weren't paying attention in class. :P

The standard proof is essentially SlyReaper's without quite so many points. Imagine the triangle in the middle to be the "official" one, then construct the unique line that passes through the lowest point and is parallel to the highest line (as is drawn in the diagram). The two red and the two blue points are both internal angles of parallel lines cut by a transversal, so they are equal, and therefore the angle sum of the triangle is equal to the angle sum of a straight line.

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Re: Really basic stuff that was never proven in class

Postby gmalivuk » Wed Sep 22, 2010 8:16 pm UTC

Tirian wrote:Pick up a copy of Euclid's Elements; if you don't recognize everything there, it's because you weren't paying attention in class.
Or because you never learned Greek...
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Re: Really basic stuff that was never proven in class

Postby Dason » Wed Sep 22, 2010 8:19 pm UTC

Eebster the Great wrote:Actually, I remember proving a LOT in my geometry class. That was the one class where proofs were very important. We pretty much proved everything we learned that year from a few postulates.

Well I wish I could say the same but sadly the only proofs we dealt with were pretty lame. It wasn't even until undergrad that I did something resembling a real proof.
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Re: Really basic stuff that was never proven in class

Postby voidPtr » Thu Sep 23, 2010 12:33 am UTC

Dason wrote:
Eebster the Great wrote:Actually, I remember proving a LOT in my geometry class. That was the one class where proofs were very important. We pretty much proved everything we learned that year from a few postulates.

Well I wish I could say the same but sadly the only proofs we dealt with were pretty lame. It wasn't even until undergrad that I did something resembling a real proof.


Yeah, not all school and not all teachers within those schools are equal, and I can't say I had strong middle-school math teachers. Thankfully, that changed in high school, and my Grade 11/12 pre-cal teacher especially was really good.

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Re: Really basic stuff that was never proven in class

Postby Eebster the Great » Thu Sep 23, 2010 3:26 am UTC

You know, I never proved the intermediate value theorem in any class, and that's a pretty important one.

For that matter, I never proved the fundamental theorem of algebra either.

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Re: Really basic stuff that was never proven in class

Postby Tirian » Thu Sep 23, 2010 4:30 am UTC

Eebster the Great wrote:You know, I never proved the intermediate value theorem in any class, and that's a pretty important one.

For that matter, I never proved the fundamental theorem of algebra either.


100 Great Problems of Elementary Mathematics is teh awesome.

I suppose IVT is proved in any calculus textbook, and is only two lines long in topology.

Speaking of topology, this thread needs to mention the Jordan Curve Theorem. Obviously true, and assumed all over the place, but a bitch to prove.

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Re: Really basic stuff that was never proven in class

Postby phlip » Thu Sep 23, 2010 8:05 am UTC

For IVT:
Spoiler:
So we have f:[a,b]->R is continuous, f(a) is negative, and f(b) is positive, or vice-versa.

First, find f((a+b)/2), the value at the midpoint of a and b... either it's 0, in which case we're done, or it's not, in which case one side or the other of the range is now a range of half size, which itself is positive at one end and negative at the other. Repeat this process, the basic binary-search root-finding algorithm, and take the limit as we go to infinity - call c the point it converges to.

Now, we know we have arbitrarily small intervals around c for which f is positive at one end and negative at the other. So think about the limit of f at c... if this limit exists, it's L, such that for any epsilon around L, there's a delta around c, such that the image of the latter range is within the former. But any delta around c must include some of those arbitrarily-small intervals that are positive at one end and negative at the other... so the image of that range must contain both positive and negative numbers. So the epsilon range around L must contain 0. So L can't be non-zero, as there'd be an epsilon around L that didn't include 0, and thus didn't include either the positive numbers or the negative numbers (depending on the sign of L). So if the limit of f at c exists, it must be 0.

Which is where the continuity comes in... if f is continuous at c, then the limit must exist, and must equal f(c), by definition of continuity. So f(c) = 0.

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Re: Really basic stuff that was never proven in class

Postby Eebster the Great » Thu Sep 23, 2010 7:19 pm UTC

phlip wrote:For IVT:
Spoiler:
So we have f:[a,b]->R is continuous, f(a) is negative, and f(b) is positive, or vice-versa.

First, find f((a+b)/2), the value at the midpoint of a and b... either it's 0, in which case we're done, or it's not, in which case one side or the other of the range is now a range of half size, which itself is positive at one end and negative at the other. Repeat this process, the basic binary-search root-finding algorithm, and take the limit as we go to infinity - call c the point it converges to.

Now, we know we have arbitrarily small intervals around c for which f is positive at one end and negative at the other. So think about the limit of f at c... if this limit exists, it's L, such that for any epsilon around L, there's a delta around c, such that the image of the latter range is within the former. But any delta around c must include some of those arbitrarily-small intervals that are positive at one end and negative at the other... so the image of that range must contain both positive and negative numbers. So the epsilon range around L must contain 0. So L can't be non-zero, as there'd be an epsilon around L that didn't include 0, and thus didn't include either the positive numbers or the negative numbers (depending on the sign of L). So if the limit of f at c exists, it must be 0.

Which is where the continuity comes in... if f is continuous at c, then the limit must exist, and must equal f(c), by definition of continuity. So f(c) = 0.

Thanks, I have since read the proof, just not in class. Wikipedia's proof is also pretty simple.

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Re: Really basic stuff that was never proven in class

Postby pizzazz » Sun Sep 26, 2010 5:14 am UTC

No one ever proved for me that the integral from a to b of a function is equal to its antiderivative evaluated at a minus the antiderivative at b.

Of course, we also never proved the fundamental theorem of calculus, is that included (we should be doing that this year).

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Re: Really basic stuff that was never proven in class

Postby Mike_Bson » Sun Sep 26, 2010 5:27 am UTC

I've never gotten a decent, easy proof that there are infinite Gaussian Primes.

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Re: Really basic stuff that was never proven in class

Postby Eebster the Great » Sun Sep 26, 2010 7:35 am UTC

pizzazz wrote:No one ever proved for me that the integral from a to b of a function is equal to its antiderivative evaluated at a minus the antiderivative at b.

Of course, we also never proved the fundamental theorem of calculus, is that included (we should be doing that this year).

You got the order wrong, but yes, that is the fundamental theorem of calculus. Well, it is part two of it, anyway.

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Re: Really basic stuff that was never proven in class

Postby supremum » Sun Sep 26, 2010 11:11 am UTC

Mike_Bson wrote:I've never gotten a decent, easy proof that there are infinite Gaussian Primes.

I'm assuming we're not allowed to take for granted that there are infinitely many primes that are congruent to 3 (mod 4)?

Edit: Actually, scratch that, it's easy to show even without that fact. it's not super hard to show that if z=a+bi, and a^2 + b^2 is prime, then z itself has to be prime. So, in particular, every prime p in the integers that's congruent to 1 (mod 4) factors into exactly two gaussian primes of the form (a + bi) (a-bi) where a^2 + b^2 = p, and every prime in the integers congruent to 3 (mod 4) has no such factorization. Clearly if a gaussian prime [imath]\pi[/imath] is a factor of p, then [imath]\pi[/imath] isn't a factor of any other prime.

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Re: Really basic stuff that was never proven in class

Postby Doraki » Mon Sep 27, 2010 2:23 pm UTC

If p is a prime divisor of n²+1, then p is a gaussian prime.
If there is only finitely many gaussian primes p1...pn, look at the prime divisors of (p1...pn)²+1, and you have a contradiction.

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Re: Really basic stuff that was never proven in class

Postby supremum » Mon Sep 27, 2010 2:54 pm UTC

Doraki wrote:If p is a prime divisor of n²+1, then p is a gaussian prime.

No, that's not true at all:
(2 + i)(2-i) = 5
(4-i)(4+i) = 17

In general, if n^2 +1 is a prime, then it is congruent to 1 mod 4 and therefore factors over the gaussian integers.

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Re: Really basic stuff that was never proven in class

Postby antonfire » Mon Sep 27, 2010 4:15 pm UTC

What's wrong with just reusing the proof from the integers?
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Re: Really basic stuff that was never proven in class

Postby Eastwinn » Mon Sep 27, 2010 5:20 pm UTC

Back in 6th grade my "Honors Math" teacher was a big Number Theory freak. Prime numbers were his favorite topic to discuss with us. He told us that there were infinitely many. Being in 6th grade, none of us questioned him, but thinking back I don't think he ever proved it to us. That's pretty strange because Euclid's proof is super easy. Likewise, he showed us how an infinite series could converge using the most simple example, 1/2+1/4+1/8+1/16...=1 , and he told us that the harmonic series, which he did not name, didn't converge. It kept getting bigger and bigger, according to him, but he never showed us why.

Ignoring those slight oversights, he did prove a lot of things to us. It's thanks to him that I began to appreciate mathematics.
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Re: Really basic stuff that was never proven in class

Postby supremum » Mon Sep 27, 2010 6:00 pm UTC

Eastwinn wrote:Back in 6th grade my "Honors Math" teacher was a big Number Theory freak. Prime numbers were his favorite topic to discuss with us. He told us that there were infinitely many. Being in 6th grade, none of us questioned him, but thinking back I don't think he ever proved it to us. That's pretty strange because Euclid's proof is super easy. Likewise, he showed us how an infinite series could converge using the most simple example, 1/2+1/4+1/8+1/16...=1 , and he told us that the harmonic series, which he did not name, didn't converge. It kept getting bigger and bigger, according to him, but he never showed us why.

Ignoring those slight oversights, he did prove a lot of things to us. It's thanks to him that I began to appreciate mathematics.

I wish that more elementary/middle school math teachers made an effort to show cool little fragments of math like this to their students. It sounds like he must have been a great teacher.

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Re: Really basic stuff that was never proven in class

Postby achan1058 » Mon Sep 27, 2010 7:20 pm UTC

Eastwinn wrote:Back in 6th grade my "Honors Math" teacher was a big Number Theory freak. Prime numbers were his favorite topic to discuss with us. He told us that there were infinitely many. Being in 6th grade, none of us questioned him, but thinking back I don't think he ever proved it to us. That's pretty strange because Euclid's proof is super easy. Likewise, he showed us how an infinite series could converge using the most simple example, 1/2+1/4+1/8+1/16...=1 , and he told us that the harmonic series, which he did not name, didn't converge. It kept getting bigger and bigger, according to him, but he never showed us why.
The proof for the harmonic series isn't hard either, if you don't be too rigorous about it. Namely,
1+1/2+1/3+...
>=1+1/2+1/2+1/4+1/4+1/4+1/4+...
=1+1+1+...

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Re: Really basic stuff that was never proven in class

Postby Eastwinn » Mon Sep 27, 2010 7:24 pm UTC

^ Yeah, I've seen that before. It's not rigorous but I'm sure we'd all agree that it's suitable to show to a 6th grader.

I did have a great math teacher back then. I did visit him one day and I told him how much I appreciated what he taught me.
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Re: Really basic stuff that was never proven in class

Postby Tirian » Mon Sep 27, 2010 7:54 pm UTC

achan1058 wrote:
Eastwinn wrote:Back in 6th grade my "Honors Math" teacher was a big Number Theory freak. Prime numbers were his favorite topic to discuss with us. He told us that there were infinitely many. Being in 6th grade, none of us questioned him, but thinking back I don't think he ever proved it to us. That's pretty strange because Euclid's proof is super easy. Likewise, he showed us how an infinite series could converge using the most simple example, 1/2+1/4+1/8+1/16...=1 , and he told us that the harmonic series, which he did not name, didn't converge. It kept getting bigger and bigger, according to him, but he never showed us why.
The proof for the harmonic series isn't hard either, if you don't be too rigorous about it. Namely,
1+1/2+1/3+...
>=1+1/2+1/2+1/4+1/4+1/4+1/4+...
=1+1+1+...


That's not rigorous to the point of being not true. 1/3 is not greater than or equal to 1/2. It is true that 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... > 1 + 1/2 + 1/4 + 1/4 + 1/8 + ... = 1 + 1/2 + 1/2 + 1/2 + ...

And it's not hard to make that point rigorously. Let H(n) be the sum of the first n terms of the harmonic sequence. You can easily and firmly show by induction that H(2^n) >= 1 + n/2 for all natural n by making this argument without an ellipsis. Therefore, H is an isotone sequence that eventually exceeds any integer, so it is not convergent.

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Re: Really basic stuff that was never proven in class

Postby Eebster the Great » Mon Sep 27, 2010 8:40 pm UTC

Next step: teach [imath]1 + 2 + 3 + \dots = -\frac{1}{12}[/imath] to seventh graders.

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Re: Really basic stuff that was never proven in class

Postby achan1058 » Mon Sep 27, 2010 9:31 pm UTC

Tirian wrote:
achan1058 wrote:The proof for the harmonic series isn't hard either, if you don't be too rigorous about it. Namely,
1+1/2+1/3+...
>=1+1/2+1/2+1/4+1/4+1/4+1/4+...
=1+1+1+...


That's not rigorous to the point of being not true. 1/3 is not greater than or equal to 1/2. It is true that 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... > 1 + 1/2 + 1/4 + 1/4 + 1/8 + ... = 1 + 1/2 + 1/2 + 1/2 + ...
Somehow my brain just farted on this.

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Re: Really basic stuff that was never proven in class

Postby Talith » Mon Sep 27, 2010 9:47 pm UTC

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8...

the 3rd and 4th in the sequence are greater than or equal to 1/4,
the 5th, 6th, 7th and 8th in the sequence are greater than or equal to 1/8,
in general there are 2^(n-1) terms in the nth subsequence that are greater than or equal to 1/(2^n) so we have a sequence* 1 + 1/2 + 2(1/4) + 4(1/8) + 8(1/16) + ... + 2^(n-1) [1/(2^n)] + ... = 1 + 1/2 + 1/2 + 1/2 + .... which is an obvious divergent sequence and happens to be less than or equal to the harmonic sequence term-wise.

* you should read a(x) as meaning the x term appears a times in the sequence in order.

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Re: Really basic stuff that was never proven in class

Postby Eastwinn » Tue Sep 28, 2010 12:44 am UTC

Google reveals tons of proofs for the harmonic series's divergence. A lot of them are really clever!
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Re: Really basic stuff that was never proven in class

Postby Tirian » Tue Sep 28, 2010 1:07 am UTC

Eebster the Great wrote:Next step: teach [imath]1 + 2 + 3 + \dots = -\frac{1}{12}[/imath] to seventh graders.


That would be an interesting day. I'd bet that 1 + 2 + 4 + 8 + ... = -1 gets demonstrated on a regular basis, though.

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Re: Really basic stuff that was never proven in class

Postby Eebster the Great » Tue Sep 28, 2010 2:32 am UTC

Tirian wrote:
Eebster the Great wrote:Next step: teach [imath]1 + 2 + 3 + \dots = -\frac{1}{12}[/imath] to seventh graders.


That would be an interesting day. I'd bet that 1 + 2 + 4 + 8 + ... = -1 gets demonstrated on a regular basis, though.

That one is admittedly easier to show very informally, but I still would never expect to see it in middle school, or even high school for that matter.

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Re: Really basic stuff that was never proven in class

Postby Spambot5546 » Wed Feb 23, 2011 2:14 pm UTC

New kinda interesting proof. I was considering getting a tattoo of the golden spiral when it occured to me that I couldn't think of any reason why [imath]{1\over{\varphi}} + {1\over{\varphi^2}} = 1[/imath]. It didn't seem likely that the diagram had been drawn wrong for thousands of years, but I decided to prove it anyway.

Given [imath]{\varphi = }{{1+\sqrt{5}}\over{2}}[/imath]
prove any number [imath]a = {a\over{\varphi}} + {a\over{\varphi^2}}[/imath]

[math]{a\over{{1+\sqrt{5}}\over{2}}} + {a\over{({{1 + \sqrt{5}}\over{2}})^2}} = {a\over{{1+\sqrt{5}}\over{2}}} + {a\over{{6 + 2\sqrt{5}}\over{4}}} = {2a\over{1 + \sqrt{5}}} + {2a\over{3 + \sqrt{5}}}[/math]

We cross multiply and get

[math]{{6a + 2a\sqrt{5} + 2a + 2a\sqrt{5}}\over{3 + \sqrt{5} + 3\sqrt{5} + 5}} = {{a(8 + 4\sqrt{5})}\over{8 + 4\sqrt{5}}} = a[/math]
Proven! Hooray, I proved a thing no one but me cares about!
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"But I like it
Because it is bitter,
And because it is my heart."

Mindworm
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Re: Really basic stuff that was never proven in class

Postby Mindworm » Wed Feb 23, 2011 2:40 pm UTC

Also, it kind of follows directly from the definition of phi, if you divide by phi² (I see phi+1=phi² as the definition of phi. That it can be expressed in terms of square roots is a mere coincidence).
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jestingrabbit
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Re: Really basic stuff that was never proven in class

Postby jestingrabbit » Wed Feb 23, 2011 4:13 pm UTC

Mindworm wrote:phi+1=phi²

and phi>0.
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