Expressions Versus Sentences

Click here for a diagram that summarizes the ideas in this section.

People sometimes have trouble understanding mathematical ideas: not necessarily because the ideas are difficult, but because they are being presented in a foreign language—the language of mathematics.

The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is:

precise (able to make very fine distinctions)
concise (able to say things briefly)
powerful (able to express complex thoughts with relative ease)

Every language has its vocabulary (the words) and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception.

As a first step in studying the mathematical language, we will make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts).

DEFINITION expression

An expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest.

Some MathJax

Inline math: For $\,a\ne 0\,,$ $\,ax^2 + bx + c = 0\,$ is equivalent to $\,x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\,.$

Display math: For $\,a\ne 0\,,$ $$ \begin{gather} ax^2 + bx + c = 0\cr \text{is equivalent to}\cr x = \frac{-b\pm\sqrt{b^2-4ac}}{2a} \end{gather} $$ Try to keep display lines short to accomodate small screens.

Some JSXGraph

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Expressions Versus Sentences

Some MathJax

Some JSXGraph

Examples

Practice