An expression is a formula that evaluates to and can be expressed as a literal value.

The word expression, from the Latin, literally means the "pressing out" (out being "ex").

A facial expression, such as a smile or frown, is pressed out from the face.

A computer code expression is a formula whose value is "pressed out" (or reduced) to a literal value.

Be aware that mathematicians often call an expression a formula while computer scientists often call a formula an expression.

• Does a literal express a value? Yes. A literal is an expression.
• Does a variable express a value? Yes. A variable is an expression.
• Is 3 + 4 an expression? Yes. A formula is an expression. This one evaluates to 7.

Integer arithmetic operators
Operator	Operation	Expression	Value
+	addition	4 + 3	7
-	subtraction	4 - 3	1
*	multiplication	4 * 2	12
/	division	4 / 3	1
%	modulus	4 % 3	1

Note: Floating point operations have no modulus and division yields a number with a fractional part.

Expression evaluation
Expression	Value
156 / 16	9
156 % 16	12
3 + 4 * 5	23
3 + (4 * 5)	23
(3 + 4) * 5	60
60 / 6 * 5	50
(60 / 6) * 5	50

Here are the expressions that we evaluated in converting the number 19 from decimal to binary.

A floating point number is an approximation of a real number.

What is 1.0/3.0?

The answer in math class is 0.3333333....

Mathematicians are rarely interested in efficiency. In computer terms, we do not have an infinite number of bits, and we need the division operation to compute fast, so we need to approximate the value. In such cases, a floating point approximation is used.

A double is a C/C++ floating point number. There is also a float type with less precision. Unless told to do so, always use double and not float.

Something is recursive if it directly or indirectly refers to itself.

• Is no your answer to this question?
• The next sentence is true. The previous sentence is false.

Problems sometimes result if the recursive relationship is circular.

A procedure or function is recursive if the procedure and/or function calls itself either directly or indirectly.

Remember the message passing model.

An expression is a formula that evaluates to a value. That is, an expression expresses a value.

• All literals are expressions.
• All constants are expressions.
• All variables are expressions.
• If E1 and E2 are expressions, then so is E1 op2 E2 where op2 is a binary operator.
• If E1 is an expression, then so is op1 E1 where op1 is a unary operator.
• If E is an expression, then so is (E).
• A function call is an expression since the value returned by the function can be used in the expression.

The recursive definition of expressions provides a simple and, yet, powerful way to recognize and evaluate expressions.

Consider the expression (3 + 4) * 5 that evaluates to 35. Here is the tree diagram for this expression.
Mathematicians think in terms of parentheses. Computer scientists write parentheses in programs, but think in terms of tree structures. One works up from the leaves to the root to determine the value of the expression.

Let us change the parentheses and use the expression 3 + (4 * 5) which evaluates to 23. Here is the tree diagram for this expression.
Notice that tree structures have no parentheses. To a computer scientist, a program is just a big tree structure to be evaluated (and executed).

The precedence rule for 3 + 4 * 5 is to do multiplication first, as in 3 + (4 * 5) instead of addition first as in (3 + 4) * 5.

Whenever one is in doubt, just use parentheses to get the desired expression evaluation.