Top 10 Most Beautiful Mathematical Equations
I base my subjective definition of a beautiful mathematical equation on two primary factors: the simplicity of the equation and the satisfaction I feel in using it. A couple of the choices I have made for this list will not be simple to novice mathematicians but will be beautifully simple to seasoned ones.
The choices on my list are limited to knowledge up to the BSc/MSc Mathematics modules which I have covered. If you feel you have suggestions that should have been on this list then feel free to comment.
The Google definition of an ‘equation’ lists one meaning as ‘the process of equating one thing to another.’
Therefore, if an equation, definition or otherwise features an ‘=’ sign then I have deemed it eligible to be included on my list, even if some may say that it is not mathematically an equation in the truest sense.
10. Derivative of x to the power of x
Perhaps I am alone in thinking the equation above is beautiful, but once you understand how to calculate the left-hand side it is very satisfying.
If you have studied math up to an advanced level then perhaps you see this derivative as rather hideous.
For me, solving this derivative feels like the mathematical equivalent of a five-star, three-course meal.
First comes the recognition of the beautiful relationship between x, the exponential and the natural logairthm.
Next, the application of this rule.
Now finally we can show that the left-hand side equals the right-hand side.
To me, the way this problem unfolds is both beautiful and rewarding. It combines powers, exponentials, logarithms and differentiation rules all in a way which is compact.
9. Definition of a group
The Group definition describes a set ‘E” and an operator ‘star’ to combine two elements of ‘E’ to produce an element of ‘E’ in adherence to the group axioms (associativity, existence of identities and inverses).
The ‘star’ operator can represent any operator such as multiplication, addition, subtraction or division — it’s basically the equivalent of ‘x’ but for operators.
I consider it to be beautiful as it visually represents what a group is in a way which accords to simplicity.
8. Aleph null, the number of natural numbers
Aleph null is defined as the number of natural numbers (that is the number of positive integers). Another words, to those who are unfamiliar with the set of natural numbers, it is the set of of whole, positive numbers counting upwards from 1 (i.e 1, 2, 3, 4, 5 and so on), which is why they are called the ‘counting numbers’:
So what are the number of numbers in this set? Infinity? An infinite quantity which we describe as aleph noel. It may be tempting to say that:
But this is not the full truth as there are infinite quantities larger than aleph-null. For instance, the number of real numbers (the number of whole and fractional numbers) is also infinite, however, it includes more than just the number of positive integers so it must be a larger infinite quantity than the number of positive integers. So when you hear mathematicians or physicists say that some infinities are larger than others, it truly is numerical reality.
7. P = NP (disputed)
The P VS NP Problem is the greatest open problem in Computer Science and Decision Mathematics. It poses the question of whether every language accepted by a nondeterministic algorithm in polynomial time is also accepted by a deterministic algorithm in polynomial time.
If all NP problems (easy to verify in polynomial nondeterministic time but difficult to solve) are in P (easy to solve and verify in deterministic time) then P = NP.
If ‘P = NP’ were accepted by the mathematical community, it would prove to be in my opinion, one of the most beautiful equations.
For those interested, my proof of P = NP can be found here: https://archive.org/details/p-np-mathematically-explained
6. Mass-energy equivalence
Einstein’s famous equation establishes the beautiful relationship between mass, energy and the speed of light, c.
5. Newton’s second law
As well as Newton’s second law being beautiful in simplicity it is also compact in the sense there are many versions of the same equation for different scenarios. It’s an equation which can be simple when it needs to be and complex when it needs to be.
My favourite edition of F = ma are the Navier-Stokes equations which are essentially a rewrite of F = ma used to describe the motion of fluids by modelling them as continuums. My favourite version of the equations are the following:
4. Zero factorial
I find this equation beautiful in the way it unites 0 and 1 in harmony without making them equal.
In addition to being the answer to zero factorial, this can equally be read by a programmer as ‘0 does not equal 1' which seems to be a recurring theme whenever I have in the past attempted to divide 1 by zero or calculate limits which tend to zero without first addressing the denominator.
3. Euler ‘s identity
Euler’s identity can be found using De Moivre’s theorem. We know that cos(π) = -1 and sin(π) = 0. We also know that the radius, r, is 1 using Pythagoras’ theorem. Therefore substituting these values into De Moivre’s theorem, the complex term isin(π) disappears and we get the beautiful Eulerian identity exp(iπ) = -1. This equation can be rearranged to the above which I believe is even more beautiful as it then includes zero.
It is considered a very beautiful identity by many in the mathematical community, as it displays a remarkable connection between π, i and Euler’s number, e.
2. One plus one is two
This may seem rather juvenile, but even the very foundations of mathematics cannot be taken for granted when we consider how long Mathematics has progressed in human history.
It is also one of the first equations in my life to introduce me to the field of mathematics.
1. The square root of -1
When I first saw this equation in a further maths textbook I was blown away by the sheer beauty and simplicity of this equation.
The idea of a complex number and learning of their applications in quantum mechanics truly made me feel as if I was entering an entirely new world of mathematics.
I remember one of the main reasons I wanted to study Further Maths when I was at school was just to learn what ‘imaginary numbers’ were about. It seemed completely intriguing at the time, and now complex numbers just feel like any other important aspect of Mathematics.
Thank you for reading the list of my top ten favourite equations in Mathematics.
