Math Changes your View on the World

Trying to make math cool is like trying to make "not holding your breath for five minutes" cool: It already is, and anyone thinking otherwise struggles through life with significantly reduced mental abilities. Anyone born into a world based entirely on numbers with little dollar signs in front of them, then deciding they don't need to know numbers, is volunteering to be a very small one adding up to someone else's very big one.

"Just 10 more short years, and you'll have that degree in Klingon paid off!"

But accounting is to math what diapers are to biochemistry: dealing with the stinking mess left by some unfortunate and immature side effects of being alive while doing it. Mathematics isn't cool in the same way spacetime isn't real estate. It's much bigger and more important than the ridiculous little structures we've erected on it. And anyone who doesn't understand how truly cool it is (2.725 K) simply doesn't appreciate the sheer scale of it.

Ymir's autopsy

We found reality's programming language. Physics is the operating system, but it's written in equations. E=mc^2 defines large parts of reality in fewer bytes than it takes to say "my penis" and has similar effects on local spacetime, ladies (~ 90 percent, + gentlemen ~ 10 percent). But that's physics. There are pure mathematical equations which are just as capable of blowing your mind.

#5. Infinite Pi

Pi is the ratio of a circle's diameter to its circumference, in the same way nuclear fission is a way of powering TVs to watch America's Got Talent: an appallingly simple effect of a reality-defining truth. Pi isn't a number, it's a startup constant of spacetime. Take a line in one dimension, rotate it around another, and the resulting ratio of lengths is a precise number. The existence of space has a numerical signature. It's called a transcendental number, because even attempting to think about how much it means is more mind-expanding than all the drugs.

Seen here in the three dimensions it glues together

Calculating pi has become the computer scientist equivalent of tuning muscle cars: we don't actually need more digits for anything useful, because the basics do everything we actually need it for, but we've spent years stacking up air-cooled hardware just because. In 1985 it was calculated to 17 million digits. Srinivasa Ramanujan found the formula used. Around 1910.

It wasn't the only such formula, but was incredibly useful because it converged exponentially compared to other algorithms, making it ideal for computers. Interesting note: at the time there was no such thing as computers. Srinivasa Ramanujan had pre-empted processors by decades. In 1985 his formula was used in the world record calculation of pi to 17 million digits, and a slight variant modified by David and Gregory Chudnovsky now holds the title at 10 trillion. We're not saying he's a robot, but if a perfect calculating machine ever goes missing in a time travel experiment we already know where it ended up.

Konrad Jacobs

It turns out that rogue Terminators like sharp suits and counting forever.

#4. Dividing By Zero

Dividing by zero gives you undefined, infinity, or who cares, depending on whether you're talking to a mathematician, physicist, or engineer. You can tell a lot about how disconnected someone is from reality by how they contemplate infinity. For example, cult leaders often explain how infinity after nothingness means you should obey them.

"Pay your entire life now for insurance against eternity with the King! No refunds!"

At its most basic, division means, "How many times can you take this out of that?" For example, you can take a human head from the members of One Direction five times before you get blessed silence. On the one hand you can take nothing out of something an infinite number of times, but on the other, no, you can't, because that would take forever. It's a mathematical problem so hard it once crippled a U.S. Navy cruiser. The USS Yorktown's experimental military computer control system tried it and was crippled by a buffer overrun. Making this the first ship to overflow without any water. On the upside, it's way easier than asking a computer to define love or telling it that truth is a lie, or asking if those are both the same question.

"These feelings are tearing me apart! Also the incoming torpedoes."

The solution is as brilliant as it sounds stupid. You deal with the impossible by sneaking up on it. It's nice to know calculus solves problems the same way SEAL teams do it: sneakily and permanently.

Taking the limit as X goes to zero means that you never actually get there, but you get as close as you need to be, no matter how close that is. Then the "limit" as you tend toward zero -- but never actually reach it -- gives you the answer. As X gets small, sin X is approximately equal to X, so you're always dividing something by itself and getting one. Then, with a particularly cunning flourish, you end up dividing nothingness by itself and becoming one. Which I think means mathematics is the king of zen.

His decades of calm are going to be wasted when he finds out mathematicians worked it out with a pencil.

There's a fairly easy proof using the squeeze theorem. Which doubles as a good line when you want to chat up a mathematician.

"... and you'll be with me because I give quite excellent dick, QED."

These limits as you hit zero, which is a much less depressing sentence in mathematics, are essential for calculus. And calculus is essential for everything. If math is the programming language of reality, calculus is the graphics processor working out things like explosions, lasers, and gravity, all the cool special effects. Or as we call them in physics, effects.

#3. Pick A Digit, Any Digit (of Pi)

If the Ramanujan formula was transcendent understanding of reality by the human brain, the Bailey-Borwein-Plouffe formula is outright sorcery. Developed by Simon Plouffe in 1995, this formula lets you skip straight to any digit of pi without working out the rest of the number.

You want the the 10-trillion-and-33rd digit of pi? No problem. There's a fundamental constant of reality and now, without any input apart from being in the same reality, this equation can read out any part of it. That's like pulling rabbits out of a hat where the rabbits are Star Trek energy beings and your hat is a beret.

The process of using the formula sounds like someone found a glitch in god's computer. If you want the Nth term, you split the infinite sum at the Nth term, and a bit of modulo math skims out the required digit in hexadecimal. That sounds like something you'd read on Mr Mxyzptlk's GameFAQ. Reality, mathematics, and the design of 8-bit computers lining up to accidentally output one of the universe's BIOS settings. The craziest part? It's still slower than Ramanujan for finding the whole thing. It really is just a cheat code for reality.

Binomial Expansion

Today we will learn how to expand a binomial into its fully expanded form. There are 2 ways. 

One way is the binomial therom

(A+b)^n

All you do is foil the entire thing to the nth time. Yah its a ton of work no fun. 

The other way is to use pasacals triangle. Much more fun cause it is a ton easier. 

A lot easier. How? Well you see that row 0,1,2... And so on well that corresponds to the n power of your equation. 

How do you do this. You take the coeffiecents that are needed then you multiply by the correct term. How do you eo this? Well it is quite simple. First take your right hand term and take the highest power and multiply the first coefficent by it. Next you move to the term right next to it and subtract one from the n power. Thats you right hand side. Next is your left hand side. All you do is the exact same just start at the right side and move to the left. After you have expanded simplify and you have your answers simple right. 

Here is an example. 

Well ordering principle

Well-ordering principle: Every nonempty subset T of N has a least element. Thatis,thereisanm∈T suchthatm≤nforalln∈T.

Intutively clear as it may seem at the first glance, this principle turns out to be logically equivalent to the mathematical induction, the fifth axiom of Peano, which is quite surprising.

Theorem 1. The mathematical induction is logically equivalent to the well-ordering principle.

Proof. Part I. We show the well-ordering principle implies the math- ematical induction.

LetS⊂Nbesuchthat1∈Sandk∈Simpliesk′∈S. Wewant to establish that S = N by the well-ordering principle.

Suppose N\S is not empty. Then by the well-ordering principle there is a least element m ∈ N\S. Since 1 ∈ S we know 1 ∈/ N\S. Therefore m ̸= 1 and so by one of the homework 2 problems there is some q ∈ N such that m = q′ = q+1, which implies q < m by the definition of <. Weconcludethatq∈S;orelsewewouldhaveq∈N\Sandso m would not be the least element of N \ S, which is absurd. However, since S has the property that k ∈ S implies k′ ∈ S, we conclude that m = q′ ∈ S because q ∈ S. This contradicts m ∈ N \ S.

The contradiction establishes that N \ S is empty. Hence S = N. Part II. We show the mathematical induction implies the well-ordering principle.

Let S(n) be the statement: Any set of natural numbers containing a natural number ≤ n has a least element. Consider the set

E ={m∈N:S(m) is true}.

1 ∈ E because 1 is the least element of N (why?).

Wenextshowm∈Eimpliesm′∈E. Nowm∈EmeansifXis a subset of N containing a natural number ≤ m, then X has a least element. From this we want to establish m′ ∈ E.

So let C be any subset of N containing a natural number ≤ m′. If C has no element < m′, then m′ is the least element of C and we are done. Otherwise, we can now suppose there is a natural number y ∈ C such that y < m′. In particular, y ≤ m because by one of the homework 3 problems we know there is no natural number strictly between m and m′. Therefore, C now has an element y ≤ m, so that the induction hypothesis given in the preceding paragraph implies that C has a least element. In any event, we have proven C has a least element, so that

m′ ∈ E. Hence, the mathematical induction implies E = N.

 In summary, the mathematical induction implies that the statement S(n) is true for all n ∈ N.

To establish the well-ordering principle, let T ⊂ N be a nonempty set. There is some m ∈ T because T is nonempty, so that T contains a natural number ≤ m, which is just m itself. Then T has a least element because the statement S(m) is true by the preceding paragraph 

Amaze the Teach

 
Who doesn't love magic tricks? Here, we're going to look at how you can amaze your friends with some simple tricks involving numbers. You don't need a long-sleeve shirt, a rabbit, or even know the slightest bit of magic – just a little math.

Let's start off with a simple math trick.

Lightning Addition
Proclaim to your friends: “I'm a human calculator! I can add five 3-digit numbers quicker than you can punch in the numbers in a calculator.” Then ask three friends to write down three separate and random 3-digit numbers. For instance:

240
520
842

Ask them to return the paper, upon which you add in the final two 3-digit numbers.

240
526
842
759
473

Race against your friends, who are armed with a calculator, to add all these numbers up, while you blurt out the answer within a matter of seconds: 2,840.

The secret: The last two numbers you throw in isn't random at all – it may look random to your friends, but you've carefully chosen them so that the fourth and first numbers add up to 999 (in the example above, it would be 759, since 240 + 759 = 999). Choose the fifth number so that it adds with the second number to give 999 (473 + 526 = 999).

Once you have that down, the rest is simple: the sum is 2,000, add the number in the middle, and subtract 2. And voila! 2,840.

Lightning Addition 2
If your friends weren't too impressed by that trick, try this one. Ask a friend to pick two different single-digit numbers (for example, 4 and 7) and add them up (4 + 7 = 11). Tell them to do it in secret, so that you can't see what numbers they chose, or what they're doing.

Now, ask them to take the sum of the first two numbers (11), and add it to the number prior to that (which is 7). So the chain of calculations now appears to be as such:

  4
+7  
+11
18

Continue on the chain of addition until he reaches 10 numbers.

  4
+7    
+11  
+18  
+29  
+47  
+76  
+123
+199
+322
????

Once done, ask your friend to show you the chain of numbers. Here comes the amazing part: challenge your friend, who's armed with a calculator, to figure the sum of all these 10 numbers, while you figure out the answer in less than five seconds – which is 836.

The secret: It turns out that this pattern of addition follows the Fibonacci sequence of numbers, which is the basis for the Fibonacci ratio. You might know it as the Golden Ratio, which often occurs in nature.

The trick here is that the seventh number in the sequence, multiplied by 11, will always equal the sum of the 10-number sequence. In this case, it's 76 x 11 = 836.

Wait a minute, you say. How does one multiply the number by 11 in under five seconds? Here's where you use another arithmetic trick: Take the two numbers, 7 and 6, and split it, leaving a blank space in the middle:

7 __ 6

In the space between, place the sum of the two numbers:

7 13 6

Because the space in between 7 and 6 can only have one digit, carry forward 1 to 7, and what you end up with is:

836

Simple? Try it with 48 x 11

4 __ 8

Place the sum of the two numbers

4 12 8

Then, carry forward the 1 to the 4

528

Note: If your friend starts off with 9 and 7, or 9 and 8, the seventh number would be a 3-digit number (101 and 109, respectively). In these special cases, just remember that the result after multiplying them by 11 is 1,111, and 1,199 respectively.

Arithmetic Sequences

What is an arithmetic sequence? It is when an identifiable difference can be found between two different terms. 

2,4,6,8

in this sequence the difference between wech term is +2 or the variable d. D is used to find the formula of the sequence.

next is how to find the sum of an entire sequence without finding each and every variable. this is accomplished by n/2(A1+An)

Sums with sigman notation

What is a sum? It is the combination of all terms added together. The easiest way to complete this is by sigmun notation. Although it is the easiest way there is some calrification that is needed such as what are the upper and lower limits. 

Now that we know what the signs are, how do we do this? You start with the first term or A1. the 1 is the variable n so you would plug it in to the formula. The n at the top is the upper limit. DO NOT PASS IT. 

lastly there are a few rules that must be applied.