Mathematics is way more than just simple arithmetic. Here are some mathematical oddities and paradoxes that we love!
1) Random Data Isn’t Actually Random
You read that correctly: random isn’t really random. If you have a list of numbers that represents for example city populations, death rates, street addresses, house prices, stock prices, or even electricity bills, there’s a pattern in how the numbers are distributed.
The law which states this is called Benford’s law, or the first-digit-law, and states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. About 30 percent of the numbers will begin with the digit , and the percentage will go down for every following number. Only one in twenty numbers will begin with the digit .
2) You Can’t Comb a Tennis Ball
Okay, you actually can comb a tennis ball. But however hard you try, it is impossible to comb all of the hairs in the same direction. In mathematical terms, there is no nonvanishing continuous tangent vector field on the sphere. This mean that if you try to comb a tennis ball, there will alway be a cowlick somewhere.
More interestingly, because of this principle, there is also at least one point on a planet at all times with no wind at all, which would be the tuft on the tennis ball.
3) Gabriel’s Horn and the Painter’s Paradox
Gabriel’s Horn, named after the Archangel Gabriel who is said to blow this horn to announce Judgment Day, is a neat little geometric figure which has an infinate surface area, but a finite volume. This means that you could fill it with a certain quantity of paint. However, that same amount of paint you used to fill it up with will never be enough paint to coat its inner surface, because for that you would need an infinite amount of paint.
You can find the volume and the surface area using the following definition:
The volume of the part of the horn between and will never exceed ; however, it will get increasingly closer to as gets larger. We can therefore say that the volume approaches as approaches infinity. As shown by the limit notation of calculus:
The surface area formula above gives a lower bound for the area as 2π times the natural logarithm of a. There is no upper bound for the natural logarithm of a as a approaches infinity. We can therefore say that the horn has an infinite surface area. Or as shown mathematically:
4) 0.999… is equal to 1
One of my favorites math facts is this one: 0.999…, which has infinite decimals is actually equal to the number 1.
The best way to explain this is to imagine it like this:
You could also think about it like this: 0.9… is smaller than 1, but what do you need to add to get 1? Well, there isn’t a number small enough. It simply doesn’t exist. And because it doesn’t exist, we can say that 0.999… equal 1.
To those interested in learning more: this fact is explained in our digital Precalculus course!
5) Birthday Paradox
This is a nice one to include because you try it out yourself at a party. If you have at least 23 people in the same room, the chance of (more than) two people sharing the same birthday is more than 50%.
You might think that the chance should be closer to 1/365, however you really only need two people to share the same birthday. It therefore operates on the assumption that every day of the year is equally probable for a birthday. Consequently, when you have 23 people the probability of two people sharing a birthday reaches 50%.
Can you guess how many people you need to reach 99.9%? The answer is only 70!
First, we define to be a set of people and let be the set of dates in a year.
Define the birthday function to be the map that sends a person to their birthdate. This way, everyone in has a unique birthday if and only if the birthday function is injective.
Next, we need to know how many functions, and how many injective functions, exist between and .
Since , and , it follows that there are possible functions.
Consequently, there are possible injective functions.
Now, let be the statement “Everybody in the set has a unique birthday.” By definition, is the fraction of injective functions out of all possible functions (i.e., the probability of the birthday function being one that assigns only one person to each birthdate), which gives
What are your favorite oddities?
Let’s talk again soon,