10 Mind Blowing Mathematical Equations

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

e^{i\pi} + 1 = 0

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

e^{i x} = \cos x + i \sin x

When x = \pi , the value of \cos x is -1, while i\sin x is 0, resulting in Euler’s identity, as -1 + 1 = 0.

2. The Euler Product Formula

\displaystyle \sum_{n} \frac{1}{n^s} = \prod_{p} {\frac{1}{1 - \frac{1}{p^s}}}

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

\displaystyle\int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi}

The function e^{-x^2} 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

{\mathbb{R}} \sim {2^{\mathbb{N}}}

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 2^{\mathbb{N}} > {\mathbb{N}} .

A related statement is the Continuum Hypothesis, which states there is no cardinality between {\mathbb{N}} and {\mathbb{R}} . Interestingly, this statement has a very strange property: it can be neither proved nor disproved.

5. The Analytic Continuation of the Factorial

\displaystyle n! = \int_{0}^{\infty} {x^n e^{-x} \,dx}

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

a^2 + b^2 = c^2

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

F(n) = \frac{(\varphi)^n - (-\frac{1}{\varphi})^n}{\sqrt{5}}

where \varphi = \frac{1 + \sqrt{5}}{2} (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

\displaystyle1 + \frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdots =\frac{\pi^2}{6}

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

\displaystyle1 + \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots =\infty

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

\displaystyle\pi(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} J(\sqrt[n]{x})

where {J(x)} is defined as

\displaystyle J(x) = Li(x) + \sum_{\rho} Li(x^\rho) - \log 2 + \int_{x}^\infty \frac{dt}{t(t^2 - 1)\log t}

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:

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—it is for the mathematically inclined.

If you enjoyed this post, make sure to check out the follow-up, “10 Surprising Mathematical Facts.”

51 thoughts on “10 Mind Blowing Mathematical Equations

  1. Brilliant dude! I had a great time looking a lot of these up that I didn’t know about before. One thing that you didn’t mention is the sheer simplicity of many of these, a factor which really contributes to their beauty. I had no idea that e and pi could be related as simply as the Gaussian integral illustrates for example. (It also brings into question the fact that they are considered two separate fundamental constants of the universe.) And the Euler Product Formula is also great for its simplicity


      1. Are U mad? Lots of people like math… I bet U were one of those people that got their heads flushed down the lavoratory…good day to U!


      2. Y u mean just because someone else likes math doesn’t mean u can disown them. I bet that ur not even good at math. BTW if u didn’t like math y u click on this website


    1. Yep, that is a good one! It is normally written as i^(-i) = sqrt(e^pi). It is remarkable that raising i to the -i power results a real number, let alone being related to e and pi.


    1. Well, the quadratic formula is quite beautiful, but isn’t as mind blowing as the ones listed. The quadratic formula is quite… expected and reasonable, if you know what I mean


      1. No, no, no, no, no! The quadratic or trinomial equation is NOT beautiful. The equation which proves that infinity=-1/12 is gorgeous!


      1. The proof of it doesn’t really bother me, that actually seems pretty clear, albeit gorgeous and innovative. The final result really just blows my mind though.


  2. managing ur lyf x De best way 2 encourage othrs whlst u r in ur middle ages kk ..De Eula formula x De most formula ever I hv ever known…


  3. Re: Basel, it’s also cool (arguably slightly cooler?) to look at

    (4/1) – (4/3) + (4/5) – (4/7) + (4/9) – (4/11) + ….

    which converges very slowly, but steadily…. to exactly pi.

    Great post!


    1. To me Eulers identity does it (e(^pi*i)=0,which I believe it cannot be proved without using De moivre’s formula,that is ofcouse another mindblowiong formula.


  4. Новейшие базы данных фирм России 2017 года от производителей, а не посредников!

    Базы данных фирм городов России. как найти клиентов юристу

    Собираем сразу после заказа из открытых источников Интернета, БЕЗ ПРЕДОПЛАТЫ!

    Эффективные базы для поиска клиентов в сети Интернет.

    Стоимость со скидками от 500 рублей.

    На нашу почту ждем от Вас вопросы: bazy-gorodow(собака)yandex.ru

    Базы данных фирм городов России. найду клиентов бухгалтеру


  5. I come here searching for 10 Mind Blowing Mathematical Equations .
    Now, Mathematics comes from many different sorts of problems.
    Initially these were within commerce, land way of measuring, structures
    and later astronomy; today, all sciences suggest problems examined by mathematicians, and many
    problems occur within mathematics itself. For instance, the physicist Richard Feynman created the path important formulation of quantum technicians utilizing a combo of mathematical reasoning and physical information, and today’s string theory, a
    still-developing technological theory which tries to unify the four important
    forces of aspect, continues to encourage new mathematics.

    Many mathematical items, such as packages of volumes and functions, display internal structure because of procedures or relationships that are described
    on the set in place. Mathematics then studies properties of these sets
    that may be expressed in conditions of that composition; for instance quantity theory
    studies properties of the group of integers that may be expressed in conditions of arithmetic functions.
    Additionally, it frequently happens that different such organised sets (or constructions) show similar
    properties, rendering it possible, by an additional step of abstraction, to convey
    axioms for a school of buildings, and then research at once the complete class of set ups gratifying these axioms.

    Thus you can study groupings, rings, domains and other abstract systems; mutually such studies
    (for set ups identified by algebraic procedures) constitute the domains of abstract algebra.

    Here: http://math-problem-solver.com To be able to clarify the foundations of mathematics,
    the areas of mathematical logic and place theory were developed.
    Mathematical logic includes the mathematical analysis of logic and the applications of formal logic to the areas of mathematics; placed theory is the branch of mathematics that studies collections or series of items.

    Category theory, which offers within an abstract way with mathematical buildings
    and human relationships between them, continues to be
    in development.


  6. 9; 锟?5m fee with Monaco for midfield star Tiemoue Bakayoko on lucrative five-year dealinter and outInter Milan may be worst team ever for selling players before they reached their prime Paul Pogba posts funny instagram message for Antoine Griezmann's birthdayWayne Rooney 鈥?Manchester United to Everton 鈥?3.


  7. Correction on #1:
    When x = \pi , the value of \cos x is -1, while i\sin x is 0, resulting in Euler’s identity, as -1 + 1 = 0.
    When x = \pi , the value of \cos x is -1, while i\sin x is 0, resulting in Euler’s identity, as -1 + 0 = -1.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s