The Semicolon in Math: Three Possible Meanings

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The semicolon is a famously confusing piece of grammar.

You know what else is confusing? Algebra.

So what do you get when you put semicolons in algebra? Read on to find out!<  

What does the semicolon mean in math?

There are several different uses for the semicolon in mathematics, most of them in obscure equations normal people will never run across. The most common situation where a semicolon will be found is algebra, where the semicolon separates variables in a function from parameters. If a semicolon is being used in algebra, any variables after the semicolon are parameters. Less commonly, the semicolon can indicate a covariant derivative of a function in differential geometry and show the composition of relations in the calculus of relations.

 

The semicolon in algebra

To understand how to use semicolons in algebra, we need to briefly review algebraic functions.

Functions are kind of like mathematical sentences made up of numbers, mathematical operators like the plus or minus sign and algebraic variables like x or y. The purpose of a function is to show a relationship between two sets of numbers and variables.

For example, the function x + y = 1 describes the relationship between the number 1 and the total value of x+y as being identical.

 

Function variables

In algebra, a variable is a letter (usually x, y, and z) used to represent a number with an unknown value. Because this unknown value may be any number of things, it can be said to vary and thus is a variable.

Usually if you are doing algebra, you are trying to “solve for” the variable, which means to find out its value in the context of the equation.

However, sometimes the variables in an equation can also be used to help draw a line on a chart showing all possible values the equation may have (a technique called “plotting” the equation).

 

Function parameters

At first glance, parameters appear identical to variables. They are also represented by letters, and may even be represented by the same letters such as x.

In fact, a parameter is a type of variable. Unlike other variables, though, parameters are variables used to narrow down a specific group of solutions.

For example, if your teacher gives you an equation x + b = z and tells you that b is a parameter equaling seven, this allows you to turn the equation into a plot in a way you could not otherwise.

That’s because x + b = z doesn’t contain enough information to plot out, but x + 7 = z does.

 

Using a semicolon to separate variables and parameters

The main use of a semicolon in algebra is to separate variables from parameters in an equation.

If you see a semicolon, what comes before it is a variable and what comes after it is a parameter.

To reiterate, a variable is a letter that represents an unknown number, while a parameter is a variable for which you have been given a specific value or values.

The semicolon, then, allows someone to give you more information to either solve an equation or plot it on a chart.

Typically, the semicolon is only used when writing equations as a “function of x” like this: f(x).

We could then rewrite the example function y = my + b as f(x; m, b). Here it is clear that x is the variable, while m and b are parameters to keep in mind while solving for x or plotting the function on a chart.

 

Example: Feeding the Cat

Say you want a function to show how much food you should give a cat.

The instructions on the cat food say to give one cup a day per five pounds of cat. We’ll represent cups with the variable “c” and the cat’s weight with “p” for pounds.

Our function, then, looks like this: c = p / 5

If we have a ten pound cat, then we’ll feed 2 cups, because c = 10 / 5, so c = 2.

With us so far?

Now imagine we got a different type of cat food that says to feed one cup per four pounds of cat. We’d have to rewrite our equation as c = p / 4 instead.

Another option would be to turn the “per pounds” number into a parameter, something that changes depending on the type of food we’re giving.

We could represent this “per pounds” number as a variable, f (for “food type”) and write the equation like this: c = p / f.

Now, however, it’s hard to tell what’s a variable and what’s a parameter with a set value. We can use the semicolon to make that clear by saying c (p; f) = p / f.

Since you know that the “f” is a parameter, you should put your specific value into that variable based on the type of food you’re using and then solve the rest of the equation.
 

Other mathematical uses for the semicolon

There are two other cases where the semicolon appears in mathematics.

Since both are obscure and require an advanced understanding of complicated mathematical topics, this article will not go into detail about either. For basic descriptions of each, see below.

 

Differential geometry

Differential geometry is the study of geometry using differential calculus, integral calculus and linear algebra. Several free online textbooks exist for those who want to learn more about this topic.

In differential geometry, semicolons are used to describe covariant derivatives. Explaining what this means would take more than 1000 words by itself and would require a thorough understanding of differential geometry.

In any case, if you are looking at a differential geometry equation and you see a semicolon, it means that the index following the semicolon represents a covariant derivative.
 

The semicolon in the calculus of relations

Set theory is the study of collections of numbers or other objects. In set theory, it is possible to have a “binary relation,” or a group of paired objects that are related to one another in some way.

When using set theory, these relations can be defined using what is called the calculus of relations.

One such way to define relations is by talking about the “composition of relations,” a term that means forming new relations.

Enter the semicolon. In the calculus of relations, a semicolon is used to refer to the composition of relations or combining two groups of binary relations together.

Here, the semicolon is placed between both relations, each usually represented by a capital letter. For example, A ; B would mean the set of binary relations in sets A and B.