Arithmetic Operations on a Computer
Order of Operations
A computer can add, subtract, multiply, and divide numbers. The symbols for these operations are +, , *, and / respectively. Some computer languages, such as spreadsheets, also allow exponentiation. Its symbol is ^ .
Most computer languages use the same precedence of operations as mathematics. Exponentiation is done first in a string of operations, with multiplication and division next, and addition and subtraction last. When operations have the same precedence, such as addition and subtraction, they are done from left to right. This is also called the hierarchy of operations.
Highest ^
Next highest *, /
Lowest +, -
For example, consider the algebraic expression:
x + yz + w2
Most people would recognize that this
means to square w,
multiply y and z
,
and add x, yz, and
w2 together.
If x = 1, y
= 2, z = 3, and
w =
4, for instance, the expression evaluates to
1 + 2*3 + 42 = 1 + 6 + 16 = 23 .
Parentheses are used to change the
order of operations.
Example: Evaluate
10 – 6 / 2 , (10
– 6) / 2 ,
and 10 – (6 / 2) .
Solution: In the first expression,
do the division
first, because it has higher precedence than subtraction. The result is
10 - 3 = 7 .
In the second expression, the
parentheses force the
subtraction to be done first: 4
/ 2 = 2
.
The third expression evaluates to 10 – 3 = 7 , the same as
the first
expression. The
parentheses are not
needed, although it is not incorrect to include them.
In-line Expressions
On a computer, there are limitations to the manner in which expressions can be entered. We must create in-line expressions that are just a sequence of characters (keystrokes at the keyboard). Some rules:- No juxtaposition. In algebra, x y means to multiply x and y together. For the computer, we must place an asterisk to indicate multiplication. EG: x * y
- Exponentiation: x2 is not a sequence of characters. Thus, we insert the carat: x ^ 2
- Fractions are written for the computer as in-line
expressions, not in built-up form:
1 / 2
, not ½, for
example.
| Built-up algebraic expression: | Creating the in-line expression equivalent. |
| x + yz + w2 | The inline expression equivalent would be: x + y * z + w^2 |
 |
Parentheses are required around the numerator. Otherwise, only y would be divided by 3 ! The inline form is (x + y) /3 . |
 |
Exponentiation has the highest precedence, and division and multiplication have the same precedence. For emphasis, we could place parentheses around y2 and 4/3 giving (4/3) * x * (y^2): but they are not needed. The expression 4/3 * x * y^2 is simpler and is equivalent. |
 |
Parentheses are needed around the numerator and around the denominator. Do not forget the multiplication symbol between 2 and y. The result is (x+3)/ (2*y-5) |
| in-line expression | built-up algebraic expression |
| x - y / z + w | Since division has precedence over subtraction
and addition, only y is divided by z :  |
| (x - y) / z + w | x - y is divided by z
. w is not part of the fraction  |
| (x - y) / (z + w) | w is part of the denominator in this
expression:  |
Exercises
Exercises for this are available in a pdf document.
Credits and licensing
This article is by Robert P. Webber and Scott McElfresh, licensed under a Creative Commons BY-SA 3.0 license.
Version 2015-Aug-24 12:30