Now, let's just say that there are situations where one knows that a value exists, but one does not know exactly what that value is. For example, suppose you know that there are some salt shakers in this box:

** How many** salt shakers that are in that box
is a

However, none of this stops us from doing mathematics with the value of the number of salt shakers in this box. We just need to understand how to represent that value.

Symbols called variables are used to represent the concept of 'how many' when that concept is considered to exist, although the exact value of that concept may be unknown at the present, (and may continue to be unknown).

Well, let's just say that variables ** act**
like numbers, but you may not know which number that a variable is
acting like.

Variables usually look like ** letters** of the
alphabet. So, if we want to do mathematics with the number of
salt shakers in that box above, we need to decide upon a
letter of the alphabet to represent the 'how many' for that
value. Let's let the letter 'n' be that letter. Then we would
say:

'Let n be the number of salt shakers in the box.'

If we wish to consider that there are three salt shakers in the box, then we would make statements like:

Let n equal 3.

Or:

Let n = 3.

Or simply:

n = 3

It is important to know, though, that we
can do mathematics with the variable ** n** without ever knowing
how many salt shakers are in the box. For example, if the box
doesn't weigh too much, it's possible to show that the total
weight of it and its contents about doubles if the number of
salt shakers inside of it doubles. One could find out that
fact mathematically without ever knowing how many salt
shakers were actually in the box.

So, values can be represented by numbers and variables.

Numbers are made up of digits and maybe a decimal point. Sometimes one considers a positive or negative sign part of a number. The value of a number is clearly represented by itself. Its location on the number line is not up to question.

Variables are symbols which are usually
single letters of the alphabet. Variables are often called * 'unknowns'*
because the exact value for a variable may be unknown in the
problem you are facing. In fact, knowing the exact value of a
variable may not actually enter into the work you are doing
involving that variable. For example, this is true:

5 = 3 + 2

And this is true, too:

5 + n = 3 + 2 + n

The above statement is true no matter what
the value for ** n** may actually be. Here we have done some
algebra using a symbol,

So, we now have two methods for showing a value in a mathematics expression:

- Use a number, say, like 3.
- Use a variable, for example, n.

There is another manner in which a value can be introduced into a mathematics expression: the concept of a function. Currently, we will not go into depth regarding the definition of a function. As far as notation and algebra go, however, let's say that functions work in mathematics expressions much like a number or a variable.

The symbol for a function can have slightly different meanings in different contexts. Here, let's say that it is a symbol which represents a value.

A symbol for a function often looks like this:

f(x)

The above symbol is pronounced like this:

'f of x', ('ef of ex')

This entire symbol, f(x), acts like a value. That is, since this is true:

5 = 3 + 2

Then so is this true (adding f(x) to each side):

5 + f(x) = 3 + 2 + f(x)

What value does this symbol, f(x), represent? That is a good question! It's value can change, so it acts a bit like a variable. Actually, though, its value, called its output, is determined by what is known as its input. And the rule that ties the value of its output to the value of its input is called the definition of the function.

In the symbol ** f(x)** the

If we know the function definition, then we can determine the output value once we know the input value. For example, suppose the function is defined as:

f(x) = 5x

This states that the function will have a value, or output, equal to five times the input value. For example if 2 is input to this function, it will output 10. Starting again with the definition and placing a 2 in for the x, the math looks like this:

f(x) = 5x

f(2) = 5(2)

f(2) = 10

For the above we would say:

'f of two equals ten'

Of course, a different input would yield a different output. For example:

f(6) = 30

For the above we would say:

'f of six equals thirty'

Now, there is * a lot*
more to understanding functions than this. They are mentioned
here because it is important to understand the common ways in
which values enter mathematics expressions right from the
start when working with expression evaluation.

So, now we know that there are three common ways that values are represented in mathematics expressions:

- Numbers
- Variables
- Functions

Where are you?

*Here: Other Representations Besides Numbers for Values*

Introduction to Operators and Operands

More about Operators and Operands

And Now Including Variables and Functions