The equation above is an explicit function for the number of primes less than or equal to a given number. 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.įor a very long time, mathematicians have been trying to find a pattern to the prime numbers. Prime numbers are numbers that have no divisors other than 1 and themselves. Here is the significance of this equation, in English: The Explicit Formula for the Prime Counting Function The harmonic series, if you look carefully, is actually just zeta of 1.ġ0. Yet if you square all the numbers, it doesn’t add up to infinity (it adds up to pi squared over six). 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. That formula is the Riemann zeta function, we can say that zeta of 2 is pi squared over six. 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. 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. Remarkably, even with all the square roots and divisions, the answer will always be an exact positive integer. That is, to find the 100th Fibonacci number, you don’t have to calculate the first 99 numbers. 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. Where (note that this number is the Golden Ratio). The Explicit Formula for the Fibonacci Sequence 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. The same integral for n-1 is defined as the gamma function. And negative numbers, and complex numbers… The integral equation makes factorial work for fractions and decimals as well. The factorial function is commonly defined as n! = n(n-1)(n-2)…1, but this definition only “works” for positive integers. The Analytic Continuation of the Factorial Interestingly, this statement has a very strange property: it can be neither proved nor disproved.ĥ. Ī related statement is the Continuum Hypothesis, which states there is no cardinality between and. It is remarkable in that it states a continuum is not countable, as. This was shown by Georg Cantor, the founder of set theory. This states that the cardinality of the real numbers is equal to the cardinality of all subsets of natural numbers. This formula is of extreme importance in statistics, as it represents the normal distribution. It is certainly not obvious at first glance that the area under the curve is the square root of pi. from minus infinity to infinity, it gives a bizarrely clean answer. The function in itself is a very ugly function to integrate, but when done across the entire real line, i.e. The left side is the common representation of the Riemann zeta function. Moreover, we can choose s to be any number greater than 1, and the equation is true. 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. The symbol on the left is an infinite sum, while the one on the right is an infinite product. When, the value of is -1, while is 0, resulting in Euler’s identity, as -1 + 1 = 0. It is considered by many to be the most beautiful equation in mathematics. These ten equations should convince anyone that there is more to mathematics than the memorization of formulas.Ī very famous equation, Euler’s identity relates the seemingly random values of pi, e, and the square root of -1. For today’s post, I have compiled together ten of the most startling, dazzling, and insane equations for that purpose. 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. Most of the time, a mathematical equation is just something you memorize for a math test.
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