Most of the time, a mathematical equation is just something you memorize for a math test. But sometimes, an equation can be a lot more than that—it can be a work of art in its own right, with no real purpose but to be enjoyed. For today’s post, I have compiled together ten of the most startling, dazzling, and insane equations for that purpose. These ten equations should convince anyone that there is more to mathematics than the memorization of formulas.
1. Euler’s Identity
A very famous equation, Euler’s identity relates the seemingly random values of pi, e, and the square root of -1. It is considered by many to be the most beautiful equation in mathematics.
A more general formula is
When , the value of is -1, while is 0, resulting in Euler’s identity, as -1 + 1 = 0.
2. The Euler Product Formula
The symbol on the left is an infinite sum, while the one on the right is an infinite product. Theorized by Leonhard Euler once again, this equation relates the natural numbers (n = 1, 2, 3, 4, 5, etc.) on the left side to the prime numbers (p = 2, 3, 5, 7, 11, etc.) on the right side. Moreover, we can choose s to be any number greater than 1, and the equation is true.
The left side is the common representation of the Riemann zeta function.
3. The Gaussian Integral
The function in itself is a very ugly function to integrate, but when done across the entire real line, i.e. from minus infinity to infinity, it gives a bizarrely clean answer. It is certainly not obvious at first glance that the area under the curve is the square root of pi.
This formula is of extreme importance in statistics, as it represents the normal distribution.
4. The Cardinality of the Continuum
This states that the cardinality of the real numbers is equal to the cardinality of all subsets of natural numbers. This was shown by Georg Cantor, the founder of set theory. It is remarkable in that it states a continuum is not countable, as .
5. The Analytic Continuation of the Factorial
The factorial function is commonly defined as n! = n(n-1)(n-2)…1, but this definition only “works” for positive integers. The integral equation makes factorial work for fractions and decimals as well. And negative numbers, and complex numbers…
The same integral for n-1 is defined as the gamma function.
6. The Pythagorean Theorem
Probably the most familiar equation on this list, the Pythagorean theorem relates the sides of a right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse. It also relates triangles to squares.
7. The Explicit Formula for the Fibonacci Sequence
where (note that this number is the Golden Ratio). While many people are familiar with the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc., where each number is the sum of the previous two numbers), few know there is a formula to figure out any given Fibonacci number: the formula that we have above, where F(n) is the nth Fibonacci number. That is, to find the 100th Fibonacci number, you don’t have to calculate the first 99 numbers. You can just throw 100 into the formula.
Remarkably, even with all the square roots and divisions, the answer will always be an exact positive integer.
8. The Basel Problem
This equation says that if you take the reciprocal of all the square numbers, and then add them all together, you get pi squared over six. This was proved by Euler. Notice that this sum is just the function on the left hand side of Equation 2 (the Euler product formula) earlier in this post, with s = 2. That formula is the Riemann zeta function, we can say that zeta of 2 is pi squared over six.
9. The Harmonic Series
This is somewhat unintuitive, because it says that if you add a bunch of numbers that keep getting smaller (and eventually become zero), they still reach infinity. Yet if you square all the numbers, it doesn’t add up to infinity (it adds up to pi squared over six). The harmonic series, if you look carefully, is actually just zeta of 1.
10. The Explicit Formula for the Prime Counting Function
where is defined as
Here is the significance of this equation, in English:
Prime numbers are numbers that have no divisors other than 1 and themselves. The primes below 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. From this, it is already clear that there is no apparent pattern to the primes: in some runs of numbers you will get a lot of primes, in other runs you will find no primes, and whether a run has a lot of primes or no primes seems to be totally at random.
For a very long time, mathematicians have been trying to find a pattern to the prime numbers. The equation above is an explicit function for the number of primes less than or equal to a given number.
Here are what all the letters mean:
- — the prime counting function, which gives the number of primes less than or equal to a given number. For example, , as there are 3 prime numbers (2, 3, 5) less than or equal to 6.
- — the Möbius function, which gives 0, -1, or 1 depending on the prime factorization of n.
- — the logarithmic integral function, which is defined as the integral of 1/(log t) up to x.
- — any of the nontrivial zeros of the Riemann zeta function.
Amazingly enough, this formula will always give an exact integer! This means that, given any number, we can plug the number into this equation and obtain the number of primes less than or equal to that number. The fact that this equation exists means there is some pattern to the primes, though it may still be too early for us to understand.
More mathematics can be found at my other blog, Epic Math. Also, for one of my classes I wrote a slightly more detailed explanation (pdf) for #10, but beware though—it is for the mathematically inclined.
If you enjoyed this post, make sure to check out the follow-up, “10 Surprising Mathematical Facts.”