Page 16 - Calculus Demystified

Basics
CHAPTER 1
In summary: There are three types of real numbers: (i) terminating decimals, 3
(ii) non-terminating decimals that repeat, (iii) non-terminating decimals that do not
repeat. Types (i) and (ii) are rational numbers. Type (iii) are irrational numbers.
You Try It: What type of real number is 3.41287548754875 ... ? Can you express
this number in more compact form?

1.2 Coordinates in One Dimension

We envision the real numbers as laid out on a line, and we locate real numbers from
left to right on this line. If a< b are real numbers then a will lie to the left of b on
this line. See Fig. 1.1.
_ 3 _ 2 _ 1 0 1 2 3 4

a b
Fig. 1.1

EXAMPLE 1.1
On a real number line, plot the numbers −4, −1, 2, 6. Also plot the sets
S ={x ∈ R:− 8 ≤ x< −5} and T ={t ∈ R: 7 <t ≤ 9}. Label the plots.

SOLUTION
Figure 1.2 exhibits the indicated points and the two sets. These sets are called
half-open intervals because each set includes one endpoint and not the other.

_ _ _
9 6 3 0 3 6 9

_ _ _
9 6 3 0 3 6 9
S T
Fig. 1.2

Math Note: The notation S ={x ∈ R:− 8 ≤ x< −5} is called set builder
notation. It says that S is the set of all numbers x such that x is greater than or equal
to −8 and less than 5. We will use set builder notation throughout the book.
If an interval contains both its endpoints, then it is called a closed interval.Ifan
interval omits both its endpoints, then it is called an open interval. See Fig. 1.3.
closed interval open interval

Fig. 1.3