Ok NAAites, its been awhile since we explored the new and amazing branch of mathematics, so the next couple of posts will be on that topic. Previously, we demonstrated that 1=0 (by dubious means – see Proof that Aliens Don’t Exist,) but today we are going to prove that the sum of all positive numbers is -1/12. While this seems like a crazy idea involving mushrooms of dubious origins, there are strong reasons to believe that there is another way of looking at this problem (besides the obvious answer of infinity.) In fact, we find this concept to be well documented in string theory and quantum physics.
Leonhard Euler was the first to demonstrate that the sum of positive numbers equaled -1/12. He was a bit of an outlaw at the time, but now is considered a pioneer of modern mathematical analysis. This was later modified to include similarly strange results for complex numbers with Riemann’s Zeta hypotheses and I can recommend this video for those of you wanting to learn about one of the six remaining unsolved mysteries of math for which there is a very large reward for its proof or disproof.
THE PROOF THAT THE SUM OF ALL POSITIVE INTEGARS IS NEGATIVE
OK on with the proof that the sum of all positive integers is a negative number. Yes even the dark elves are waiting breathlessly hoping I will fail completely and with maximum humiliation … but alas the proof is fairly simple.
Step 1 – Let us compute the value of C1 where C1 = 1-1+1-1+1 …
You can quickly see that the value depends on whether you stop on an even or odd position in the equation. If it is odd, the value is 1. If it is even the value is zero. Averaging the two you get ½. This can be shown in greater detail in any number of ways but here is a proof for the skeptical among you.
Step 2 – Let us compute the value of 2 times C2 where C2 = 1-2+3-4+5-6 …
One way to look at this is to add the two together. Thus you get:
Having shifted the second number over one you get 1-1+1-1+1 etc. Look familiar? It is (see step 1.) It is C1 which has a value of ½.
Thus 2C2 = 1/2 and C2 = 1/4
Step 3 – Finally let us create an equation C = 1+2+3+4+5 …
Step 4 – Subtract C2 from C
You get: C-C2= 1+2+3+4+5 …
C-C2 = 4+8+12 ….
Factor Out 4 yields
C-C2 = 4(1+2+3+4 …)
Step 5 – Substitute C Into The Right Side and 1/4 Into the Left Side for C2
You get C-1/4=4C or -3C = 1/4 or C=-1/12
For an actual demonstration of the math by actual mathematicians, see the video below:
CAN THIS REALLY BE TRUE?
The answer to this is it depends. For purposes of string theory and quantum physics it is apparently creditable and useful. As mentioned, such famous mathematicians as Euler and Riemann both got this result. On the other hand, this violates some principles of math involving what you can do with divergent series. For a criticism of the above video, watch below:
I found this little nugget while researching Riemann’s Hypotheses, which is one of the seven millennium math challenges that were posted in 2000 with an award of one million dollars each. One has already been solved and some think this may be solved or disproved one day as well. The fact that the sum of all positive numbers might actually be equal to -1/12 is mind blowing. In fact, it is truly new and amazing.
NEXT UP – The Greatest Challenge in Computer Science: P=NP