It is very useful to picture the real numbers as points on the real line, as shown here.
Note that larger numbers appear to the right: if a < b then the point corresponding to b is to the right of the one corresponding to a.
Some subsets of the set of real numbers, called intervals, show up quite often and so we have a compact notation for them.
Here is a list of types of intervals along with examples.
The points a and b of the closed interval [a, b] are called its endpoints. Open intervals have no endpoints, and each half-open interval has a single endpoint; for example (-1, 3] has 3 as its endpoint..
The five most common operations on the set of real numbers are:
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
3. Multiplication & Division
4. Addition and Subtraction
Notes on Technology
Q1 A valid first step in the calculation of (23 - 4) .5 is
|A||(6 - 4) .5||B||(8 - 4) .5||C||23 - 20||HELP|
Q2 Thus, the complete calculation gives (23 - 4) .5 =
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 A.1 of the Algebra Review of Applied Calculus and Finite Mathematics and Applied Calculus
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