Since “10 Mind Blowing Mathematical Equations” is one of my most popular articles, I decided to write another math list.
Plenty of things in math are downright uninteresting. Who cares that the area of a circle is πr², or that a negative times a negative is a positive? Why should this interest us at all? Perhaps the answer can be found in the most unexpected results, the counterintuitive facts that have sometimes eluded even the best mathematicians.
1. Birthday Paradox
The birthday paradox says that if there are 23 people in a room, there is a more than 50% chance that two people have the same birthday. It seems counterintuitive because the probability of having a birthday on any particular day is only 1/365.
But the difference relies on the fact that we only need two people to have the same birthday as each other. If, instead, the game was to get someone with a birthday on a particular day, such as March 14, then with 23 people, there is only a 6.12% chance that someone will have that birthday.
In other words, if there are 23 people in a room, and you choose one person X, and ask, “Does anyone else have the same birthday as X,” the answer will probably be no. But then repeating this on the other 22 people increases the probability every time, resulting in a net probability of more than 50% (50.7% to be more precise).
2. Mandelbrot Set (Looks Like This)
The Mandelbrot set is a set of complex numbers that, when iterated according to a certain formula, do not escape to infinity. Based on the simplicity of the formula itself, which is z -> z² + c, you would not expect such a complex figure to arise.
It really captures the idea of worlds within worlds, universes within universes. Here is a video of a zoom (among many on YouTube). I think it’s absolutely mind blowing.
If you still don’t think theoretical math is awesome after seeing that video, I don’t know what to say.
3. Banach-Tarski Paradox
The Banach-Tarski paradox says that you can split one shape into two perfect copies of itself. More specifically, it says that given a solid ball in 3-dimensions, it is possible to break it into a finite number of pieces and then arrange them back into two identical copies of the original ball.
Of course, it’s highly counterintuitive, and it’s considered by many to be the single most paradoxical result of mathematics. After all, in real life, we never see one object suddenly turning into two copies. In fact, it seems to defy the conservation of mass in physics, which says that mass should be preserved; shouldn’t the result, with two objects, have twice the mass of the original?
Well, not if the original mass was infinity. Then doubling infinity is still infinity, so there is technically no breaking of laws. For a layman explanation of the Banach-Tarski paradox, see this article I wrote in 2010.
4. Monty Hall Problem
This infamous problem is stated as follows:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
No one I know has gotten the correct answer on the first try. Surprisingly, the answer is that it’s better to switch!
Rather than trying to explain the details of the problem here, I will refer you to the Wikipedia article, which does a very good job at exposition. The story is pretty funny too:
Many readers of vos Savant’s column refused to believe switching is beneficial despite her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them claiming vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy (vos Savant 1991a). Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation confirming the predicted result (Vazsonyi 1999).
The lesson is, don’t trust your intuition.
5. Gabriel’s Horn and the Painter’s Paradox
Familiar perhaps to calculus students, Gabriel’s horn is a shape that has a finite volume but an infinite surface area (both are straightforward to check with integral calculus).
A popular way to make this into a real-world problem is to imagine painting the shape. The painter’s paradox states that it is possible to completely fill the horn with paint (finite volume), but it is impossible to completely paint the horn’s inside (infinite surface area).
The Koch snowflake is a shape, along similar lines, that has finite area but an infinite perimeter. In fact, the Mandelbrot set, from #2, also has finite area and infinite perimeter!
6. Basel Problem
The only item to appear both in the 10 equations list and in this list, the Basel Problem says that if you take the reciprocal of all the square numbers, and then add them all together, you get pi squared over six.
If you’re a normal, sane human being, it was probably completely unexpected that the stuff on the left side has anything to do with pi, the ratio of a circle’s circumference to its diameter.
7. Abel’s Impossibility Theorem
Most of you in high school have seen the quadratic equation, which tells you how to solve the degree 2 polynomial equation ax² + bx + c = 0.
But the story doesn’t end there. In the 1500s, mathematicians solved the cubic equation (degree 3), which is just one step up: ax³ + bx² + cx + d = 0. The corresponding solution is far more complicated:
Thank heavens you didn’t have to learn that in high school. But let’s go one step further. How do you solve a quartic equation (degree 4): ax⁴ + bx³ + cx² + dx + e = 0? At this point, the formula is absolutely ridiculous:
I dare you to click on that and scroll through the whole thing.
Now breathe a sigh of relief, because I’m not going to show you the formula for the next step up, the quintic equation (degree 5), ax⁵ + bx⁴ + cx³ + dx² + ex + f = 0, because it doesn’t exist! It’s not that we haven’t found it yet; we actually proved it’s impossible! In fact, for any polynomial with degree 5 or higher, there is no solution in roots.
8. There Are Different Levels of Infinity
Yes, some infinities are bigger than others. Technically, infinities have a property called cardinality, and an infinity with a higher cardinality than that of another infinity is the larger one. (Regular numbers have cardinalities too, but the cardinality of an infinity is always higher than that of a mere number.)
There are still many counterintuitive facts about cardinalities of infinity. For example, are there more integers than even integers? You would think that there are, since you’re missing all the odd integers. But the answer is no, they have the same cardinality. Are there more fractions than integers? Nope, there are just as many integers are there are fractions.
However, Georg Cantor showed that there are actually more real numbers than there are fractions. The real numbers are often referred to as the continuum, and for a long time, it was conjectured, but not known, that there is no level of infinity between integers and the continuum; this conjecture became known as the continuum hypothesis.
It turns out that the continuum hypothesis is neither true nor false in the normal sense. It was proved that it can be neither proved nor disproved. (Read that sentence again.) More precisely, Paul Cohen proved that the continuum hypothesis is independent of ZFC, the standard set of axioms for mathematics.
9. Gödel’s Incompleteness Theorem(s)
Basically, it was proved that some true things cannot be proved. There are various layman formulations of this result, and I’ll list a couple here:
- Any sufficiently powerful system has statements which can neither be proved nor disproved. (E.g, continuum hypothesis.)
- Any sufficiently powerful system cannot prove itself to be consistent, even if it is consistent.
10. Fermat’s Last Theorem
The Pythagorean theorem says that in a right triangle, a² + b² = c². Now suppose we force the variables to be integers. So the solution a=3, b=4, c=5 is allowed, but a=1.5, b=2, c=2.5 is not allowed, even though it fits the equation. It can be shown that there are an infinite number of solutions with a, b, c all integers.
But what happens if we take this one step up? How many integer solutions are there to a³ + b³ = c³? The answer is none. The same happens with a⁴ + b⁴ = c⁴: no solutions.
In fact, Fermat’s Last Theorem states that for any exponent higher than 2, this equation has no integer solutions. This famous problem, conjectured in 1637, took nearly four centuries to solve, being proved finally by Andrew Wiles in 1995.