What is a variable? Part I: Symbols.

A better initial question is: What is a symbol, and what are the types of symbols used in mathematics?  A symbol is an object, a mark, a sound, etc. that represents an idea, a process, or a physical entity.  Simple examples include stop signs, clock faces, company logos, the music used for the olympics games, badges, red roses for love, etc.

Symbols in mathematics are usually marks on paper, or special characters or pictures on computers.   One of the most important set of symbols in mathematics are numerals, which are symbols for numbers.  For example, the numeral “3” stands for and names the idea of “threeness” (for mathematicians: “threeness” is the equivalence class of all sets that have the same order as the set {*,**,***}).  You could also use three tick marks, | | |, to denote that number, but denoting the number 3,431 using tick marks would be very unproductive!

There are three important types of symbols used in mathematics: constant symbols, placeholder symbols, and operator symbols.

• Constant symbols.  These symbols denote numbers, as in

0,   1,   2,   3,  4,  5,  -12,  3.14,    π,  $\sqrt{2}$.

The key for understanding constant symbols is that they specify a unique number or   object.  For example, if we say, “Let $x$ stand for the number 3,” then $x$ is being used as a constant symbol—not as a variable or placeholder symbol.

• Placeholder or variable symbols.  Letters such as $x$, $y$, and $z$ are often used as placeholders for where we might expect to see or place a number.  (Note that ONLY 1 number is expected—a variable, contrary to popular belief, does not “vary.”)

A variable symbol is “waiting” to be supplied with a unique number.  Context is everything in distinguishing between variables and constants.  By analogy, the word he in, “He is the President of the United States,” is like a variable.  With no context or clarifying sentences  surrounding this sentence, we cannot determine who “he” is, and so we cannot determine if the sentence is true or false (determining when a sentence is true or false becomes very important in later posts on variables).  However, a little context and the word he becomes a constant symbol, “Look son, there goes Barack Obama.  He is the President of the United States.”  The word his no longer a variable.

• Operator symbols.   These symbols include the addition symbol “+” and the multiplication symbol “×”.  There are usually one or two blank spaces associated to operator symbols in mathematics.  For example, the addition symbol in a mathematical sentence is really  “___ + ___” with the two blank spaces to be filled in with constants or variables.   Here are some other familiar operator symbols:

__ – __      |__|      -(__).

Maybe surprisingly, the division symbol is not considered an operator symbol in formal  mathematical logic because not ALL numbers can be placed in the blank spaces associated to the division symbol.  For example, placing a 0 in the blank of 3÷___ is not allowed.  However, the division symbol is defined in terms of the multiplication symbol, i.e., when $b\ne 0$,  a÷b=c if and only if a=b×c.  In school mathematics, it is customary to include the division symbol in the list of operators with the following caveat: the set of numbers that can be substituted must now be specified (like, 3 ÷ for all x ≠ 0).

Operator symbols are also called function symbols—they are, in fact, shorthand notation for functions (recall that a function is a rule that assigns a unique output of the range set for each input or inputs from a domain set).

Next Up:  Part II: Expressions.

CHANNEL: Engineering School Mathematics