Which Is the Best Greek Letter?

Katie Steckles

The world of mathematics and science has a wealth of notation and symbols which we use to communicate ideas, lay out our thinking and share knowledge. Many symbols have specific meanings and uses, and one alphabet which gets more than its fair share of use for this is the Greek alphabet.

Thanks in part to the amount of mathematical thinking the Greeks contributed to our body of modern knowledge, their symbols have become emblematic of science and learning. (In early series of high-brow UK TV quiz Only Connect, the contestants had to choose a Greek letter to select a question; this was later deemed not to be silly enough, and changed to Egyptian hieroglyphics instead).

And throughout maths and science, there are specific meanings given to letters from the Greek alphabet – sometimes a single symbol can have multiple uses in different contexts. But which letter is the most useful? Here are some of the contenders: You be the judge.

A Slice of Pi 

Possibly the most famous Greek letter, by virtue of being drilled into every schoolchild when they learn about circles, π has become known as the ‘circle constant’ – the ratio between a circle’s circumference and its diameter.

And that value pops up in so many interesting places throughout mathematics, not just in finding circumferences – it is used for describing angles, as well as appearing in formulae for everything from the motion of a pendulum to the sum of the reciprocals of the square numbers (and in a slightly embarrassing spot on a fractal!). This means it is beloved by mathematicians, and secures its place as one of the most well-known numbers.

But the letter π is also used in other places too. You may be familiar with the notation that uses a capital sigma to denote a sum of terms:

\[ \sum_{n=1}^{\infty} \frac{1}{n^2} \]

The equivalent notation for a product uses a capital letter π:

\[ \zeta(s) = \prod_{p \textrm{ prime}} \frac{1}{1-p^{-s}} \]

It is also used in topology to denote the homotopy groups of a space – the first homotopy group of x, denoted \(\pi_1(x)\), roughly describes the number of possible loops you can make in that space.

For a sphere, all loops are equivalent – they can all be shrunk to a single point without leaving the surface of the shape, so this group is trivial. For more complex shapes like a torus, where there are two different ways to draw a loop on the surface which cannot be contracted back to a point, the group consists of all the combinations of those two loops a and b.

This can be generalised to higher homotopy groups, where \(\pi_n(x)\) is the equivalent but instead of loops we consider n-dimensional spheres (where a loop, or circle, is a 1-dimensional sphere).

Learn your Alpha, Beta, Gammas

Other potential contenders for well-used Greek letters in mathematics are the first couple of letters of the Greek alphabet (they are even used in creating the word ‘alphabet’!). Alpha and beta, along with the third Greek letter gamma, correspond to the letters A, B and C in the Roman alphabet, and are accordingly used to denote e.g. angles in a triangle, with their corresponding corners being labelled A, B and C (and the opposite sides often denoted a, b and c). 

These symbols also have some other uses in describing the physical world – for example, alpha is often used to denote angular acceleration (the rate of change of angular velocity), all three are used to denote types of radiation. Gamma is also used to denote the Euler-Mascheroni constant, and alpha-beta pruning is a technique in the analysis of tree structures, often used for algorithmic analysis of game trees. 

Gamma, in its capital form, is also used to denote the gamma function: an analogue of the factorial function that can be applied to non-integer inputs. Usually, n! (pronounced ‘n factorial’) denotes the product of all the numbers from 1 up to n; so \( 3! = 3 \times 2 \times 1\). The gamma function extends this to all numbers, taking the value \( (n-1)! \) for whole number values of n but also being defined for all the values in between (technically, via the analytic continuation of the integral of a complex function).

Don’t Forget About Delta 

Hot on the heels of alpha, beta and gamma, the fourth letter delta also has plenty of uses in mathematical notation.

Delta represents a change in a variable: \(\Delta x\) denotes the change in the value of \(x\). In calculus, the Roman equivalent symbol \(d\) is often used for the same purpose, or \(\partial\) (‘del’, which was defined as an inverted delta) for a partial derivative.

There are several functions known as ‘the delta function’, which have various uses in maths and physics:

The Kronecker Delta function takes two inputs i and j, and returns the value 0 if they are different, and 1 if they are the same:

\[ \delta_{ij} = \begin{cases}
0  & \textrm{if } i \neq j \\
1 & \textrm{if } i=j
\end{cases} \]

In a square identity matrix \(M\), the values in each entry \(m_{ij}\) are just \(\delta_{ij}\). It is also useful in calculating products of vectors, and in digital signal processing.

The Dirac delta function takes the value 0 everywhere – except at \(x=0\), where it is infinite. It is used in physics and engineering to model things like point masses (which only have mass at a single value) and impulses of energy taking place at a single instant:

\[ \delta(x) = \begin{cases}
0  & x \neq 0 \\
\infty & x=0
\end{cases} \]

There are many other strong contenders for Greek letters that are useful in maths: \(\zeta\) (zeta), giving us the Riemann Zeta function, relating to one of the most important outstanding problems in mathematics; there is also χ (chi), without which you cannot spell the maths typesetting language LaTeχ (if you did not notice that χ at the end was actually a chi, you have probably been pronouncing it wrong), perform a statistical chi-squared test, or define the Euler characteristic.

And we could not possibly forget φ (phi), which as well as being a useful notation for an angle when you have already used θ (theta), represents the Golden ratio, the totient function and the probability density function in statistics.

But if I have to pick a winner, there is a Greek letter with many useful and interesting uses I am calling my favourite, and that is the letter μ.

They Got Nothing on Mu, Baby

The Greek letter mu (μ): not just used for puns about cats! It is variously used to denote:

• The coefficient of friction in mechanics, describing how slippery or otherwise the interface between two surfaces is, by describing the relationship between the forces acting on it.

• The mean average in statistics, found by dividing the sum of a set of values by the number of values, and giving a rough measure of where the ‘centre’ of the set is. μ is often used for the mean over a whole population, rather than on a sample, which often cannot be measured directly.

• The prefix micro-, representing a factor of one millionth – a micrometre is a millionth of a metre, or a thousandth of a millimetre.

Mu also denotes the Möbius function \(\mu(n)\), which is defined as:

\[ \mu(n) = \begin{cases}1  & \textrm{if } n = 1 \\
(-1)^k & \textrm{if } n \textrm{ is the product of } k \textrm{ distinct primes}\\
0 & \textrm{if } n \textrm{ is divisible by a square } >1\\
\end{cases} \]

It is used in number theory and in studying functions, and also has applications in physics.

The symbol μ also has a huge range of meanings in science – among many other things, it is used to denote both magnetic permeability and linear density in physics, as well as being the symbol for a muon (one of the fundamental particles), and denotes viscosity in fluid mechanics.

Despite being associated with friction, without μ nothing would run smoothly. It certainly gets my vote for most useful Greek letter – but you may have your own opinions!

The post Which Is the Best Greek Letter? originally appeared on the HLFF SciLogs blog.