Calculators and the Order of Operations

The Order of Operations is known to most modern calculators:

1. Groupings
2. Exponents
3. Multiplication and Division from left-to-right
4. Addition and Subtraction from left-to-right

This order is generally clear and unambiguous but problems do occur. One problem is a bug in Casio calculators made before 2007. For example, the expression should evaluate to 16 according to the order of operations. The division should be done first. It is evaluated incorrectly to 1 by earlier Casio calculators. If you instead enter the Casio will give the correct answer. TI calculators do not have this problem. Casio has fixed this problem in newer calculators.

A common problem students often have is failing to recognize an implied grouping. In the expression the numerator forms an implied grouping. We know that we must do the addition first, but the calculator is not aware that there is a horizontal fraction bar (a vinculum). A horizontal fraction bar causes an implied grouping. The calculator must be told there is a grouping by entering . If we naively enter into the calculator, we will get the wrong answer because the calculator will perform the division first. Another example of an implied grouping that often causes problems is . In this expression we must perform the multiplication before the exponent. Again, the calculator must be told there is a grouping .

I also often see students incorrectly evaluate compound fractions, such as occurs in the test statistic for the t distribution hypothesis test. To evaluate this on a calculator, the fraction in the denominator must be put in parentheses: These same grouping rules apply if you are programming in a computer language, or if you are posting a problem in an email or on a math forum web site. For example, the rational expression must be written as (x^2 – 3x – 10) / (x^2 + 5x + 6) with parentheses to demark the groupings that are implied by the horizontal fraction bar.

Rick Castrapel Imperial Valley College 2007