Top 10 Most Uncomfortable Math Facts
The only other thing that might surpass the weirdness of physics is the mind-bending, headache-inducing nature of math. So, here’s my top 10 unsettling math facts, along with brief explanations of what they mean.
Imaginary numbers feel like they shouldn’t exist to begin with. After all, what would it mean to have 4+5i apples? Although they don’t make much sense in the real world, complex numbers are incredibly useful, and pave the way for even stranger math.
The function y=e^x is a strange one, because if you plotted the slope as a function of x, you would get y=e^x. Also, if you plotted the area underneath the line as a function of x, you would get y=e^x again! Aside from y=0, this is the only function that is a plot of its own area and slope.
8: 1 + ½ + 1/3 + ¼ + 1/5 + … = ∞
Even though each term in the harmonic series gets smaller and smaller, the sum still goes off to infinity. Although this seems counter-intuitive, there’s a simple proof for it. It’s easy to see that the harmonic series is larger than 1 + ½ + ¼ + ¼ + 1/8 + 1/8 + 1/8 + 1/8 + …, which can be written as 1 + ½ + 2/4 + 4/8 + …, or 1 + ½ + ½ + ½ + …, which clearly goes to infinity. Since the harmonic series is larger, it too must tend to infinity.
7: Some infinities are bigger than others
You’d think that there’s nothing bigger than infinity, but this isn’t exactly the case. If you tried to pair off every natural number (1,2,3, …) with an irrational number (sqrt(2), e, π, …), you would find that there would always be some irrational numbers left over. This means that the infinite number of natural numbers is smaller than the infinite number of irrational numbers.
Imagine there’s a program H that can check to see if another program will run forever or not. Now let’s say there’s another program P. When P runs, it will use H to check if P will run forever or not. P then takes the result, and does the opposite of what it says. But this means that H didn’t correctly predict what P would do, which is a contradiction! So, H must be an impossible program to start with.
Let’s say there’s a barber who only shaves everyone who doesn’t already shave themselves. Does the barber shave himself? If not, then he’s missing one person who doesn’t shave them self, but if so, he breaks his rule. This is the idea behind Russell’s paradox, which opened up a major contradiction in set theory that could only be fixed by changing axioms.
Fractals are probably one of the most beautiful parts of math, but they can be very difficult to describe. A main issue is that fractals don’t have a clear number of dimensions. For instance, a Koch curve has zero volume if you tried measuring it using 2 dimensions, but it has an infinite length if you tried using 1 dimension. This isn’t very useful, so we instead use something in between 1 and 2 dimensions. It turns out, a Koch curve has about 1.26 dimensions.
Euler’s identity is probably one of the most famous and beautiful math formulas that exist, combining e, i, and π all into a short equality. More generally, the formula is e^(ix) = cos(x) + i sin(x), derived from the series expansion e^x = 1 + x + x^2/2 + x^3/3! +… . Although this might be the most elegant, I wouldn’t say this is the most uncomfortable math fact that exists.
Going straight from Euler’s identity, we can quickly prove something even more strange. We know that e^(iπ/2) = cos(π/2) + i sin(π/2) = i, so we can raise each side to the power i to get i^i = e^(i*iπ/2), or e^(-π/2). From this, we can say that i^i ≈ 0.208. Somehow, raising an imaginary number to an imaginary number makes a real number!
1: Gödel’s incompleteness theorems
These two theorems are tied for first, both stating the limitations of math as a whole. The first incompleteness theorem states that there will always be true mathematical statements that can never be proven. This gives mathematicians nightmares, since some facts of math can never be known for certain, and we have no way of figuring out which ones they are.
The second theorem is even scarier, though, which states that arithmetic can never be proven to be consistent. This basically means that all of math could be wrong, but the only way to be certain would be to find some contradiction. If math does work, we’ll never know it.
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