Common Types of Mistakes: Arithmetic
Arithmetic mistakes can be avoided.
First, know the basic addition and multiplication facts. Some facts are not universally known. If you're thinking "Why bother? - I have a calculator," here are two answers. Picking up a calculator is a big waste of time for computing 6 × 9; you should know the answer instantly. Wasting time using a calculator on a timed test means you have less time to think about the real questions on the test. And what are you going to do when you have a test on which calculator use is not permitted?
Other arithmetic mistakes occur because parentheses or the rules about orders of operation are misunderstood or ignored. You can remember the correct order of operations rules by the mnemonic PEMDAS, which says to compute anything inside Parentheses first, then compute Exponential expressions (powers) next, then compute Multiplications and Divisions from left to right, and finally compute Additions and Subtractions from left to right. The highest priority for parentheses means that you should follow the remaining rules for anything inside the parentheses to arrive at a result for that part of the calculation.
Example 1: Compute 8 - 2 × 3.
Mistake:
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The mistake is to compute a subtraction first. The correct order of operations is to compute the multiplication first. In this case that means not computing in left to right order. (Warning: some calculators compute left to right when you enter the calculation. Others understand correct order of operations. To be safe, use parentheses to force the calculator to compute correctly - in this case, enter 8-(2*3) into the calculator.)
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Example 2: Compute -22.
Mistake:
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The rules for order of operation say that the exponentiation must be computed first, that is, compute 22 first. Then take the negative of the result (which we can think of as multiplication by -1). The mistake is the result of computing (-2)2 instead (note the parentheses).
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There is one common situation in which parentheses are assumed to be present but don't usually appear explicitly.
Example 3: Compute
In this situation the fraction line (also called a vinculum) indicates that the entire numerator and entire denominator must each be computed separately before the division indicated by the fraction line is carried out.
If you compute this on a calculator you must explicitly include parentheses around each of the numerator and denominator. You would enter: (2*4-3)/(√(9)+5). Note carefully the parentheses.
There's no reason to make arithmetic mistakes. Learn the facts and the rules and use them. Be especially careful how you enter calculations in your calculator - you may need to use lots of parentheses to make sure the order of operation rules are followed.
