What is the longest known equation

10 of the most impressive formulas in mathematics

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For many students, a math formula is just something they memorize for a math test. But a formula is actually much more than that. Behind it is a story and a mostly extraordinary person. Each formula is a work of art in its own way, sometimes even without direct application, but always elegant and impressive. For this post we have put together ten famous formulas. These ten formulas should convince anyone that math is more than just stubborn memorization.

1. Euler's identity

Euler's identity is one of the most famous formulas, it contains the apparently random mathematical constants and as well as the imaginary unit with. Many say that it is the most beautiful of all mathematical formulas.

A more general formula is:

If so, one gets for and for, which gives Euler's identity:.

2. The Euler product

The left symbol (left term) describes an infinite sum and the right symbol (right term) stands for an infinite product. This formula was also developed or discovered by Leonhard Euler. It refers to the natural numbers () on the left and the prime numbers () on the right. Furthermore, we can substitute for any number greater than and the equation is always true.

Incidentally, the left side shows the usual representation of the Riemann zeta function (function).

3. The Gaussian error integral

The function alone is a difficult function to integrate, but when it is integrated over all of the real numbers, that is, from minus infinity to plus infinity, then we see a beautifully clear structure. At first glance, it is not evident that the enclosed area under the curve has the value of the square root of Pi.

By the way, this formula is extremely important in statistics because it represents the normal distribution.

4. The thickness of the continuum

The power set of.

This equation states that the cardinality of the real numbers is equal to the cardinality of all subsets of the natural numbers. The mathematician and founder of set theory Georg Cantor showed this in the 19th century. It is noteworthy that the formula says that a continuum is not countable. It applies.

A related proposition is the continuum hypothesis, which states that there is no set whose width falls between the widths of and. Interestingly, this statement leads to a very peculiar property: the continuum hypothesis can neither be proven nor disproved.

5. The analytical continuation of the faculty

The factorial function is generally defined as, but this definition only applies to positive integers (). The integral equation, on the other hand, can also be used for faculties that consist of fractions, decimal points, negative numbers and complex numbers.

Incidentally, the same integral with instead of is defined as a “gamma function”.

6. The Pythagorean theorem

This is probably the most famous formula on our list. The Pythagorean Theorem refers to the sides of a right triangle, where a and b are the legs and c is the hypotenuse, i.e. the longest side of the triangle. The formula also establishes a relationship between triangles and squares. If you want to go deeper, the article Theorem of Pythagoras is recommended.

7. The explicit formula for the Fibonacci sequence

Phi is hiding. The value of the number corresponds to the golden ratio.

Many are familiar with the Fibonacci sequence, where each number is the sum of the two preceding numbers. However, few people know that there is also a formula that can be used to calculate any Fibonacci number.

The above formula finds these Fibonacci numbers, where the -th is the Fibonacci number. For example, to find the 100th Fibonacci number, you don't have to write down all the numbers and calculate them, you can immediately use the formula and insert it. The result is:, see calculation here.

It is unusual that the formula always leads to a positive whole number despite the roots and divisions it contains.

8. The Basel problem

This formula says that if you take the reciprocal of all the squares and add them together, you get the value of what none other than Euler has proven. It should be noted that this sum is actually only the function (left side) of the already mentioned formula No. 2 (Euler product), namely with. This formula is the Riemann zeta function. We can say that zeta is equal to the value:

9. The Harmonic Series

This formula does not seem to be intuitive, because it says that if you add a lot of numbers together, which get smaller with each additional element and eventually decrease, their sum still goes to infinity. However, if you square all these numbers, the sum does not go towards infinity, but towards (see above). If you look closely, the harmonic series is actually nothing more than Zeta of:

But be careful, the zeta of is not defined as there is no finite limit here.

10. The explicit formula for the prime counting function

It is defined as:

The meaning of this function is explained: Prime numbers are natural numbers that have no divisors except and themselves. The prime numbers are smaller than:

When you look at these numbers, there doesn't seem to be a clear pattern of how these numbers can be calculated. Even if you set up a formula for the calculation, it will probably not work with larger numbers, because there are areas in which there are many prime numbers and in turn areas in which only a few prime numbers appear. Their distribution seems random.

Mathematicians have long been trying to identify the pattern behind the prime numbers. A first step is the formula above, because it is an explicit formula that calculates the number of prime numbers less than or equal to a given number.

The meaning of the terms in detail:

- the prime number counting function, which indicates the number of prime numbers less than or equal to a given number. For example, because there are prime numbers up to, and that.
- the Möbius function, which outputs, or, depending on the prime factorization of.
- the integral logarithm (logarithmic integral function).
- a non-trivial zero of the Riemann zeta function.

It's amazing that this formula always returns an integer. This means that we can insert any number in the function and it tells us the number of prime numbers up to (including) this number. The fact that this equation exists suggests that some pattern, some regularity in the prime number distribution does exist, even if it seems too early for us to understand.