Math Proofs
Many people think math is really boring. This is because, for the most part, teachers are lame and don’t make the material interesting. There is a lot of cool and weird mathematics out there that kids unfortunately are not exposed to. Here are a couple of interesting mathematical oddities that will hopefully spark some interest in math:
You want to find the sum of the infinite series 1-1+1-1+1-1+1-… This pattern repeats forever. At first glance, you would likely say (1-1)+(1-1)+(1-1)+… = 0+0+0+0+… and conclude the sum is 0. At second glance you may say the 1+(-1+1) +(-1+1) +(-1+1) +(-1+1) = 1+0+0+0+0+… = 1. Turns out both of these are wrong and the sum turns out to be ½. Here is why:
Let’s call the sum of the series S, whatever it may be. So, S = 1-1+1-1+1-1+1-…
Now, look at 1-S. We get 1-S = 1- [1-1+1-1+1-1+1-…] = 1-1+1-1+1-1+1-… = S. This is the same as our original series. We just showed that 1-S = S which means that 1=2S or that S=1/2. Pretty crazy that you can add 1 and -1 infinitely many times to get ½.
Here is another cool little proof why 1=2:
Let a =b. Then a2 = ab.
So, a2+ a2 = a2+ab or 2a2 = a2+ab.
Now, Subtract 2ab from both sides of the equation. Doing so, we get:
2a2 -2ab= a2+ab-2ab
So, 2a2 -2ab= a2-ab
Now, we factor out a 2 from the left side of the equation which leave us with:
2(a2+ab) = a2+ab
Divide both sides by a2+ab leaves us with:
2=1.
Take a close look though. While everything seems to be right, we all know 2 does not equal 1. Can you find the erroneous step? If not, come to the Study Hut and we can show you what’s up.
Date posted: Thursday, May 20th, 2010 6:19 PM | Under category: 9-12, Algebra, Algebra 2, General, high school, Learning, math, Pre-Calculus, Teaching, Trigonometry, Tutor Tips
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There is a reason in Mathematically reason why dividing by 0 is theoretically, meaningless. The divide by zero operation comes from a2+ab =0 and you say to divide both sides by 0, which is literally means, both sides of your equation would approach infinity as you divide by an infinitesimal number close to zero.
As for the proof that the infinite summation of 1-1+1-1, is incorrect. The reason being that your logic is flawed and laws of infinite summations is manipulated.
Let me explain, the summation can either be 0 or a 1 given a finite number of periods n that denote the number of +1s and -1s, where if n is the total number of periods, then there are i +1s and n-i -1s. This means that at any point in time of the infinite series your solution can be 0 or 1.
Your reasoning of being 1/2 is simplified and is actually the mean of the result, which makes sense since the arithmetic mean of 0 and 1 is 1/2.
Now consider this. There are two equations you mention (1-1)+(1-1)+(1-1)+… = 0+0+0+0+, which would represent infiinite summation of (1-1)n, which will always equal 0.
and infinite sum of 1+(1-1) = 1+infinite sum (1-1) = always is 1.
The problem with your logic comes from the fact that you first mention that we have an equal number of +1 and -1, but then in your proof you present the possibility that the number of -1 and +1 are not equal in number which means i != n-i. However, no sequence or series can determine if you get a -1 or a +1, which means probability must be involved, and if there is an equal chance of getting a -1 or a +1 on the very next cycle, then your expected sum will = 1/2 by stochastic properties.
If you need help understanding this, come see The Study Hut Master.