5 Examples of Counter-Intuitive Results in Mathematics

Numbers, arithmetic, calculators and completely pointless word problems are what usually come to mind for most people when they think about math (I'm sorry, but I don't care enough about Sally to calculate how much her 72 watermelons are going to cost her!). Though this is true in high school, things get weird once you enter the realm of higher mathematics, due in large part to the fact that math eventually leaves the "real world" and enters a land of abstraction beyond the comprehension of our puny brains. But we can still sit back and scratch our heads at some of the more ridiculous mathematical results, such as…

5. 1 + 2 + 3 + 4 +….. = -1/12 

The above is what is called an infinite sum. It does exactly what it sounds like: it takes an infinite number of elements and adds them up. This sounds like a really weird idea, but many people (much smarter than you or I) have developed methods of actually performing infinite sums (even though it's actually impossible to punch an infinite number of values into a calculator).

Now, some infinite sums behave "nicely". These are typically called convergent. A convergent series is basically a sum where the values are getting closer and closer to some value (they are converging on that value). One of the more popular examples of a well behaved convergent series that makes sense is the sum of the reciprocals of powers of 2:

1 + 1/2 + 1/4 + 1/8 + 1/16 + …. = 2

As you keep on adding numbers in this series, it gets closer and closer to 2…and if you were to continue this process infinitely…you would get 2 (not really, really close to 2, but actually 2)!

NOW, on to divergent sums (the name itself is a hint that these guys are a bit trickier). These do not behave nicely and an example is the sum of the natural numbers (natural numbers being all positive whole numbers). Although the series 1 + 2 + 3 + 4 … doesn't converge on any one value, with some clever tricks (and some mathematics that is absolutely way beyond me, including Zeta function regularization and Ramanujan Sums), mathematicians are able to "assign" a value of -1/12 to it.

Here's a video that goes through the steps of how to get to such a ridiculous and mind numbing result. The following is not recommend for the faint of heart (or brain):

Now, as abstract and completely separate from reality as this sounds, this sum actually does have applications is areas such as String Theory and Quantum Field Theory. There is clearly some sort of physical and tangible representation of this counter-intuitive result in our world. Weird.

4. There Is (Mathematically) Correct Way to Cut a Cake 

One of the greatest problems of humanity that has puzzled even the best of our intellectuals is how we should go about cutting a cake so as to keep it dress for as long as possible. Most people will avoid the battle and simply slice the cake in triangular pieces from the centre to the edges. Unfortunately, you've been doing it "wrong" your entire life! The best way to keep the cake as fresh as possible for as long as possible, is to cut rectangular pieces across the diameter of the cake! 

Here's a great video that goes through the logic behind this: 

  If you'd rather not sit through that riveting video, the logic behind it is: 

- The idea starts with the premise that you'll have one slice per day, and that after cutting out your slice, the cake will be placed back in the fridge 

- Another premise is that the inside parts of the cake that are exposed and in direct contact with the air will become stale overnight. 

- The idea also assumes that we are speaking about a round cake 

- One final premise is that you (the consumer of the cake) do not enjoy stale cake (although I won't judge you if stale, old bread is your kind of thing)

The whole idea behind it is that when you cut out a triangular slice, you have some of the inside bread exposed (and it will thus, become stale), making your next slice on Day 2 very uninspiring. If you were to cut a rectangular slice across the diameter however, you would then be have two halves (almost like 2 semi circles separated). To keep the inside bread from being in contact with the air, you would (and here's the genius step!) push the two semi circles together and put a rubber band around the cake to keep it together! This way, the bread won't go stale and you can cut another rectangular piece the next day and repeat the same method! Thank the heavens - no one will ever face the shame of being forced to eat a stale slice of cake again! 

 3. 0.9999999…. = 1

The proof for this one is actually quite simple:

Let n = 0.999… (where 0.999… is "point 9 repeating" and implies an infinite number of nines after the decimal - most would assume this makes it infinitely close to 1, but…well, let's find out)

Let n = 0.999…
    10n = 9.999… (when multiplying by 10, you just move the decimal one space to the left)
    *Subtract equation 2 from equation 1 (left side minus left side and right side minus right side)
    10n - n = 9.999… - 0.999…
    *The left side goes to 9n and on the right side, the 0.999… cancels out with the decimals in 9.99…
    9n = 9 (Divide both sides by 9)
      n = 1
     But wait! At the beginning, we said that n = 0.999…. If our math is correct, then that means:
     0.999… = 1

A lot of people have a very hard time wrapping their heads around this one. Keep in mind that the equal signs means that those two numbers are unequivocally and absolutely the exact same! 0.999… is NOT just really, really close to 1…it is 1! This is what happens when you work with infinity!

Another quick and easy proof if you didn't understand the above method:

                      Let n = 1/3
It is assumed that you would agree that 1/3 = 0.333… (infinite number of 3's). Therefore:
                      0.333…. = 1/3
*Multiply both sides by 3
                     0.999…. = 1

Note that this also applies for other numbers (ex. 1.999…= 2 and 47.999… = 48 , etc.)

So really, instead of calling this entry on this article #3, I probably should have just gone with 2.99…

2. Turning a Sphere Inside Out 

Like some of the above examples, this one is absolutely leaps and bounds beyond me. This is another abstract field of mathematics called Topology, which deals with the mathematics and properties of surfaces. 

The eversion of a sphere (or more commonly referred to as "some sort of witchcraft that makes a sphere turn inside out") is a very old problem in mathematics because of its surprising results. The whole thing has been donned the name of "Smale's Paradox". 

I'm going to post another good video that explains how it all works (trust me, you're going to need visual aids to work through this one), so make sure to check it out! It walks us through the process of turning a 3-D sphere inside out without poking any holes in it, tearing it, or creasing it. 

You will never again look at a basketball the same way.

1. Complex Numbers 

Complex numbers are numbers of the form: 

a + bi 

- Where a and b are any real numbers (a real number being any number you could possibly think of that is on the number line - ex. 1, 4.4., root 2, pi, 3.33…., etc.) 

- And where i is the "imaginary number", defined as the square root of -1. 

"Now hold up!" you must be saying as you throw your hands up in discontent. "What do you mean 'square root of -1'? I learned that was impossible way back in 6th grade!" 

Well…you'd be kind of almost-right. When we're doing arithmetic with the real numbers (basically all of the math that most people did in grade school), a positive times a positive equals a positive and the same is true for a negative times a negative. Therefore, you could never find the square root of a negative because no number, multiplied by itself would result in a negative! 

But that's where things get interesting. You see, mathematics is a lot like a language in many ways. It is essentially, the language of the universe. The numbers, symbols and operations in math are really just a representation of the abstract language that nature speaks. Much like English, where new words can be created, new numbers or "types" of numbers can be created in math. 

So what did mathematicians do when they couldn't get the square root of a negative? They said "screw it" and decided to make a new number called i, standing for the imaginary number! 

And just because they're called "imaginary" (which was originally done by opponents of the imaginary number, in hopes of discrediting the idea), it doesn't make them any less "real" than the real numbers that we're so used to working with on a regular basis. Contrary to popular opinion, imaginary numbers absolutely have real applications in fields like quantum mechanics, electronics, electrical engineering, differential equations, and more. 

Now I know what you're thinking. "How can see i? Where is the imaginary number in the real world? It sounds like you just made it up!" 

In response to that, I ask you: Aren't all numbers "made up"?