Any expression that require some work to evaluate can be represented as a machine that takes input and produces output. Here is an illustration:

For example, consider a machine called “adder” that accepts 2 inputs, adds them, and outputs the sum. Let’s say that the inputs are 1 and 2. This machine would then output 3:

This machine is equivalent to the following expression:

1 + 2

From our machine analogy, we can now say that the expression 1 + 2 has 2 input values, 1 and 2, and a ‘+’ operator that will add the two inputs and produce output, or 3. A more general term for operator (which is a math construct) is function. Also note that “output” is synonymous with “evaluated value.” In summary, most expressions contain one or more inputs and a function that consumes input and produces output.

We can change the inputs to produce different expressions:

7 + 11

1024 + 256

Now we have 3 different expressions, but they still share the same add function. Note that the output depends on the inputs; in other words, when we vary the input, the output also changes, even if the function stays the same. When the inputs were 1 and 2, the output (evaluated value) is 3. But when the inputs are 7 and 11, the output is 18.

Here is another example:

5 is greater than 8

The inputs are 5 and 8, the function is “greater” and the output is “false” because 5 is not greater than 8.

Let’s consider a non-mathematical example:

It will be cloudy in New York City tomorrow.

There are many possible inputs for this expression, but for simplicity, let’s say there is just 1 input, "New York City." What is the function? Let’s try varying the input:

It will be cloudy in Dallas tomorrow.

It will be cloudy in Seattle tomorrow.

It appears that the function takes a location as input and determines whether it is cloudy at that location tomorrow. So the output may be true or false depending on the input.

Unlike the previous examples, we don't know the output of these expressions. In the real world, we frequently encounter expressions with unknown inputs and/or outputs. Such expressions can still be very useful. In fact, unknown, or variable, inputs are an essential part of properties.