# 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:

• $\pi(x)$ — the prime counting function, which gives the number of primes less than or equal to a given number. For example, $\pi(6) = 3$, as there are 3 prime numbers (2, 3, 5) less than or equal to 6.
• ${\mu(n)}$ — the Möbius function, which gives 0, -1, or 1 depending on the prime factorization of n.
• ${Li(x)}$ — the logarithmic integral function, which is defined as the integral of 1/(log t) up to x.
• ${\rho}$ — 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—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

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1. are u mad bro? No one likes maths…I bet you were one of those people that got their heads flushed down the lavoratory…good day to you!

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1. If you don’t like math, why are you here? That is exceedingly redundant you incompetent moron.

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1. peepee says:

so much pretense. someone needs a brake from the thesaurus.

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2. peepee says:

*break

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3. JJ says:

I agree. If you are only here to whine, shut up.

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2. Patty ZH says:

I loved these equations!

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1. sama says:

I love math and i am only 12

Liked by 1 person

3. ellie says:

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!

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4. Suki says:

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

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2. Drew says:

Here’s a neat little ditty a friend of mine came up with.

“2Ln(1/(i^i)) = 3.1415..”

Liked by 1 person

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.

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2. bob says:

that’s wrong! the correct one is -3^5(55-d.o*~89)-0

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3. Al says:

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1. JG says:

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

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

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4. SOUMYAJIT NAG says:

ITS FREAKING MIND BOGGLING.THANKS MAN!

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5. SOUMYAJIT NAG says:

WELL CAN I GET RAMANUJAN’S INFNITE SERIES

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6. preet says:

I don’t think i’ll ever come to terms with the Basel Problem

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1. Yeah, I’ve gone through two proofs of it (well, one was only a semi-proof), but the equality is not any more apparent.

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1. preet says:

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.

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2. Nightmare says:

me who has no idea what im looking at:

mhm yes Ipq39rzxl uhh &#(% =360 noscopeG/0

im smart 🙂

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8. Steve says:

If you think think you guys are so smart then what is 1+2=?

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9. Steve says:

Sorry for me English

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1. sive says:

I would say duh!!! but I knowq such questions do not ned the real answer.

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10. Nana Amakye Ababio says:

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…

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11. nirvana says:

Hey math chaps this all seems funny i mean some high school maths ..yeah

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12. Kevin says:

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!

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This all equations are mindblowing. However, I think root of unity (Z^n=1) should have been accomodated in this list.

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1. sive says:

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.

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1. sive says:

i ment e^(i*pi)+1=0…………

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14. Ricardo Guzman says:

This one is neat: 1-1/2+1/4-1/8+1/16-1/32+1/64-1/128…….=1/3

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15. Gabriel says:

Great summary of the most beautiful identities !! Great job !

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16. chin says:

chin+chin=double chin or 🙂 + 🙂 = :))

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20. Jehosophat Wilkinson says:

Correction on #1:
Replace
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.
with
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.

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