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#math proofs

The harmonic series are as follows:


And it has been known since as early as 1350 that this series diverges. Oresme’s proof to it is just so beautiful.


Now replace ever term in the bracket with the lowest term that is present in it. This will give a lower bound on S1.


Clearly the lower bound of S1 diverges and therefore S1 also diverges.
But it interesting to note that of divergence is incredibly small: 10 billion terms in the series only adds up to around 23.6 !

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Honestly this is such a good moment. In a few short hours, finals will be over and I’ll be watching Rogue One. Studying math proofs with a spacey playlist (ty ty @kaijuno) is so ideal and I have a large quantity of coffee in my fave mug and there’s snow outside. Just so ideal.

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-20 = -20

16 - 36 = 25 - 45

4^2 - (4)(9) = 5^2 - (5)(9)

4^2 - (4)(9) + (81/4) = 5^2 - (5)(9) + (81/4)

4^2  - 2(4)(9/2) + (9/2)^2 = 5^2 - 2(5)(9/2) + (9/2)^2

(4 - (9/2))^2 = (5 - (9/2))^2

4 - (9/2) =  5 - (9/2)

4 = 5

4 - 4 = 5 - 4

0 = 1                                                                             Q.E.D.

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I was just told that mathematical proofs are meant to be understood by anyone who understands basic math. 

I don’t get proofs (that involve an entire notebook page of notes to answer them or above college algebra level for that matter), the mathematical language, or the philosophy of mathematics. 

I feel in between upset and angry that I was basically called either lazy or stupid for not getting proofs. 

I’m a historian. A tired historian.

Math is not my focus and math is not always simple for those outside of the discipline. 

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so I’m procrastinating my Real Analysis homework by looking up the difference between a Rational and an Irrational number. 

Wikipedia says that an irrational number (squareroot 2, pi, and e) is an infinite non-repeating decimal which differentiates it from a rational number which will begin to repeat after the decimal.

Now my thought was “if Pi is truly infinite, then it would include every single number combination possible." 

That leads to two conclusions:

  1. Pi is not actually infinite, having a limit once every combination of numbers is satisfied.
  2. Pi is rational and repeating. Once it reaches that combination limit, it begins again.

So therefore, it could be theorized that irrationals are a subset of rationals, or that they are in fact the same thing as rationals only more complicated.

(Don’t mind me. I’m just a math major going insane.)

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Using 0*0=0 (a specific case of something proven in my last post, to be found here), and the distributive law (a*(b+c)=(a*b)+(a*c)), and the idea of an additive inverse, one can prove that -1*-1=1, which can in turn be generalized to prove that a negative times a negative is a positive, although I don’t think I’ll do that in this post.

  • 0*0=0
  • Substitute [1+(-1)]=0: [1+(-1)]*[1+(-1)]=0
  • Use the distributive law: [1+(-1)]*1+[1+(-1)]*(-1)=0
  • Use the distributive law again: [1*1]+[(-1)*1]+[(-1)*1]+[(-1)*(-1)]=0
  • Simplify (note that we can’t simplify [(-1)*(-1)] because that’s what we’re proving): [1+(-1)]+(-1)+[(-1)*(-1)]=0
  • Simplify: 0+(-1)+[(-1)*(-1)]=0
  • Add one to both sides: 1+(-1)+[(-1)*(-1)]=0+1
  • (-1)*(-1)=1
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