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Operations on the Real Numbers
The five most common operations on the set of real numbers are: addition, subtraction, multiplication, division, and exponentiation. "Exponentiation" means the raising of a real number to a power; for instance, 2^{3} = 2^{.}2^{.}2 = 8.
When we write an expression involving two or more of these operations, such as
2(3 - 5) + 4 ^{.} 5, | or | 4 - (-1) |
, |
we agree to use the following rules to decide on the order in which we do the operations:
Standard Order of Operations
1. Parentheses and Fraction Bars Calculate the values of all expressions inside parentheses or brackets first, using the standard order of operations, and working from the innermost parentheses out. When dealing with a fraction bar, calculate the numerator and denominator separately and then do the division. 2. Exponents Next, raise all numbers to the indicated powers. 3. Multiplication & Division Next, do all the multiplications and divisions from left to right. 4. Addition and Subtraction Last, do the remaining additions and subtractions from left to right. |
(a) A valid first step in the calculation of (2^{3} - 4) ^{.}5 is
(6 - 4) ^{.}5 | (8 - 4) ^{.}5 | 2^{3} - 20 |
(b) Thus, the complete calculation gives (2^{3} - 4) ^{.}5 =
20 | -12 | 36 |
(c) | The quantity 2/3^{2}-5 | is/is not the same as | 3^{2} - 5 |
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Entering Formulas
Any good calculator or computer program will respect the standard order of operations. However, we must be careful with division and exponentiation and often must use parentheses. The following table gives some examples of simple mathematical expressions and their calculator equivalents in the functional format used in most graphing calculators and computer programs. It also includes some for you to do.
Note on Accuracy Here is one more fact about calculators (and calculations in general): A calculation can never give you an answer more accurate than the numbers you start with. As a general rule of thumb, if you have numbers measuring something in the real world (time, length, or gross domestic product, for example) and these numbers are accurate only to a certain number of digits, then any calculations you do with them will be accurate only to that many digits (at best). For example, if someone tells you that a rectangle has sides of length 2.2 ft and 4.3 ft, you can say that the area is (approximately) 9.5 sq ft, rounding to two significant digits. If you report that the area is 9.46 sq ft, as your calculator will tell you, the third digit is probably meaningless.
Now go over the examples and try some of the exercises in Section 1 of the Algebra Review of Calculus Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.
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