The Language of Algebra

All domains of modern mathematics use algebraic expressions and formulas. Because of this, algebraic notation has become quite universal, even if the meanings of expressions and formulas can be very different in different domains of mathematics. These differences can be very confusing, especially since mathematicians are rather parsimonious in the choice of their symbols. Even the most common symbols of arithmetic, such as +, -, >, and =, are used freely in other domains of mathematics, and they have meanings that are very different from their original meanings in arithmetic.

So we often say that algebra provides a language for all of mathematics. But use of the language is strongly regulated by its grammatical rules, which describe how individual words and phrases relate to each other, and how they form coherent sentences and texts. Does algebra as a language have a grammar? What are its words, and how do they relate to each other?

We will look only at algebraic expressions that are common to the part of algebra that is taught in schools before the general concept of function is taught, and before functional notation such as f(x), g(x), and so on, is introduced.

• numbers, which are usually written in decimal notation;
• variables and constants, such as x, y, area, height, z₁, z₂, c, π, and so on; and
• special symbols for operations and functions, such as +, *, -, sin, cos, √, and so on.

The grammatical rules of algebra govern which values are inputs to which operations. When algebraic expressions are used in different domains of mathematics, the meanings of variables and other symbols may change, but the grammatical rules remain the same. This is more than just a convenience. Many problems in mathematics can be solved without looking at the meaning, but just by looking at the rules. This part of mathematics has been called 'symbolic' or 'formal', but now it is most often called 'algorithmic'. This feature of mathematics has enabled the modern automatization of information processing. Computers never process meaning in any human sense. They process data, using only "formal" features of the data.

A word of warning. Just as different human languages have different grammars, so the grammar of school algebra is not the only possible one. Not only are other languages of algebra possible, but they are often used in the processing done by computers.

A good command of the grammar of algebra is a necessary condition for one to understand it. It is much more important that a command of the grammar of human languages. When we talk, most of the information is carried by words. In addition, our world knowledge allows us to understand texts even if they are very flawed. In algebra, and in mathematics in general, most of the information lies in not in the numbers, variables, and operations, but in the relationships among them. So grammatical errors always lead to conceptual errors. But the gain one gets from learning the grammar is very substantial, because it provides a sufficient basis for a large part of algebraic processing. Fortunately the grammar of algebra is easy. It is much easier than grammars of human languages.

Remarks.

There is a trend in math education, to postpone "formal" mathematics until students learn the topics on an intuitive level. Delaying the introduction of formal mathematics is at least partially possible in arithmetic and geometry, because we all have a good intuitive grasp of the concepts of quantity and space. But algebra is predominantly a language, and the only way we can master a language is by learning how to use it.

The wisdom of postponing the use of algebraic notation until middle school can be also questioned. There is no lower bound on the age at which children may start learning their second language. So it is hard to see why algebra should be an exception. One possible argument for postponement, that algebra is only a written language, is due to an unfortunate tradition in teaching. Skills in correctly reading algebraic formulas, and in smoothly embedding them in informal English (or Spanish or ...) are not taught at any level. Starting algebra as early as arithmetic would require a departure from this unfortunate tradition.

The structure of algebraic expressions.

Each algebraic (and arithmetic) expression has a strictly hierarchical structure, which is called in mathematics a tree (drawn upside down). At the bottom are numbers and variables, which form the tree's leaves. Their values are inputs to functions (operations) which provide values to other functions, until the final value is reached. (See diagram.)

Sine and the last multiplication provide input to the second addition, which together with 3, provides input to the middle multiplication. The first and middle multiplications provide inputs to the top addition, which returns the value of the whole formula.

Notice that we do not need to know what the values of x, y, and z are, or even how operations are performed, or what the sine function is, in order to understand how all these "pieces" relate to each other.

The usual algebraic expression for this tree is:

2*x + 3*(sin y + 4*z)

Knowing the grammar of algebra is knowing how to construct a tree for a given algebraic expression. To write this tree, one has to know that precedence of operations puts sine at the top, before multiplication, which is higher than addition, and that parentheses overrule precedence of operations.

This can be read, "2 times x plus 3 times the sine of y plus 4 times z." This reading is ambiguous because it does not show parentheses. You can indicate parentheses by intonation, but when you are dictating, it is better to read the parentheses too: "2 times x plus 3 times, left paren, sine of y plus 4 times z, right paren."

There are alternate readings that are sometimes less ambiguous, such as "2 times x plus 3 times the sum of the sine of y and 4 times z." But there is no sure way to avoid ambiguity in reading algebraic expressions without mentioning parentheses.