Download 0.1 Numbers and Sets Real Numbers

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0.1 Numbers and Sets
Real Numbers
We will spend the majority of our time in Calculus exploring the set of ____________________. The set
of all real numbers can be denoted as _____.
Positive and negative whole numbers are called __________________. Ex) -3, -2, -1, 0, 1, 2, 3…
A _______________ number is a real number that can be written as a quotient of the form
p
for some
q
integers p and q, with q ≠ 0.
Sets and Intervals of Real Numbers
The entire real number line is the interval _______________________.
*ALWAYS put parenthesis next to an infinity sign!
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The set containing no numbers is called the _________________ or the ___________________.
This can be denoted by ____________.
Intervals are a special type of ______________ of the set of all real numbers.
There are two ways to write out subsets:
1. Using curly brackets which either include all numbers or a common pattern.
Ex)
{0, 3, 6, 7}
2. Set notation: S = { x ∈ category | test determining whether x is in the set }.
*x is an element
Ex) { x ∈ ℝ | x 2 > 4 }
This reads “the set of real numbers x such that x 2 is greater than 4.”
*The empty set is a subset of the set of real numbers!
Unions and Intersections of Sets
Throughout our study of calculus we will often have to consider combinations of intervals of real
numbers.
The _______________________of two sets A and B is the set of elements that are in either A or B.
A ∪ B = { x | x ∈ A or x ∈ B}.
The ____________________________of two sets A and B is the set of elements that are in both A and B.
A ∩ B = { x | x ∈ A and x ∈ B}.
*A way to remember how the “or” statement works is to say the elements exist in A or B or both.
*It is helpful to draw number lines when working with unions and intersections.
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Absolute Value and Distance
The ________________________________ of a real number, a, is the magnitude, or size, of a. You can also
remember it by saying it is the distance from zero on the number line.
This is denoted as ____________.
The absolute value of a real number a is a =
Ex) |3| = 3
|-2| = 2
Theorem 0.7
Given any real numbers a and b, ab  a b and
a
a

b
b
The ____________________ between two real numbers a and b is dist(a, b) = |b – a|.
Ex) |x – 2| < 3
Ex) 0 < |x – 2| < 1
The Cartesian Plane
An _____________________________ of real numbers x and y is denoted ______________.
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The distance between two points P =  x1 , y1  and Q =  x2 , y2  in the plane is given by the distance
formula:
dist(P, Q) =
 x2  x1    y2  y1 
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2
The coordinates of the midpoint between two points P =  x1 , y1  and Q =  x2 , y2  in the plane are
 x1  x2 y1  y2 
,

2 
 2
given by the midpoint formula:midpoint (P, Q) = 
Exercises:
1. Express the rational number -4.16 in the form
p
for some integers p and q.
q
2. Describe the set A of all real numbers x between 1 and 9, inclusively, using (a) set notation, (b) a
number line, and (c) interval notation.
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3. Calculate the length and midpoint of the segment from P = (2, -7) to Q = (7, 5).
4. For the sets A = { x ∈ ℝ | -1 ≤ x ≤ 5 } and B = { x ∈ ℝ | -3 < x ≤ 3 }, write A ∩ B in interval
notation.
Answers:
1.
2.
104
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(a) A = { x ∈ ℝ | 1 ≤ x ≤ 9 }
(b) a number line with closed circles on 1 and 9, darkened line between
(c) A = [1, 9]
9
2


3. 13 ,  , 1 
4. [-1, 3]
Homework: p. 12-14: 1ab, 7, 9, 11, 23, 31, 39, 41, 43, 53, 57, 59, 61, 63, 65, 69, 73(follow ex. 6), 78
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