Download Real Numbers and Their Graphs

Name __________________________________
Period __________
Date:
Essential Question: What is the significance of a point on a
number line?
Topic: Real
Numbers and Their
Graphs
Standard: 9-12.A.1.3
Objective:
Determine the relative position on the number line and the relative
magnitude of integers, decimals, rationals, irrationals, and numbers in
scientific notation.
The student will learn to graph real numbers on a number line,
to compare numbers, and to find their absolute values.
Natural Numbers: {1, 2, 3, …}
Also called the “counting numbers”
The set of natural numbers is represented by the
symbol, .
Whole Numbers: {0, 1, 2, 3, …}
The set of whole numbers is the union of the set of
natural numbers and the set containing only zero:
* +.
Integers: {…,-3, -2, -1, 0, 1, 2, 3, …}
The set of integers is represented by the symbol .
Summary
Rational Numbers: {
}
e.g. 1,
, 0.5,
, 0
The set of rational numbers is represented by the
symbol, (for quotient).
}
Irrational Numbers: .{
e.g. √ ,
√ ,
Real Numbers: The set of real numbers is the union of the set of rational
numbers and the set of irrational numbers.
The set of real numbers is represented by the symbol,
.
Value of Real
Numbers:
The value, or magnitude, of a real number can be represented
by placing a dot on a number line.
Number Line: A number line is a horizontal line divided into equal segments
by short, vertical lines. Each vertical line represents a real
value. We usually represent integral values by the vertical
lines.
-3
-2
-1
0
1
2
3
We usually indicate the position of zero by a larger line, and
we call that location the origin.
2
Graphing Real
Numbers:
Each point on a number line is paired with exactly one real
number. That number is called the coordinate of the point.
The dots on the number line below have coordinates,
3.1,
, 1, and √ .
Each real number is paired with exactly one point on the
number line. That point is called the graph of the point.
The dots on the number line below are the graphs of
the real numbers, 3.1,
, 1, and √ .
origin
1 √
-3.1
-3
Exercise:
-2
-1
0
1
2
3
Graph the following points on the number line below:
,
-3
-2
-1
√ , ,
.
0
1
2
3
3
Exercise:
What are the coordinates of points A, B, and C on the number
line below?
B
C
-3
-2
-1
0
1
A
2
3
A:
B:
C:
4
Distance between
points on a number
line:
The distance between two points on a number line is the
algebraic difference between their coordinates.
What is the coordinate of a point that is ⁄ of the way from
point C, below, to point D?
C
-3
P
-2
-1
D
0
1
2
3
Coordinate of point D: 2
Coordinate of point C: 3
Distance between D and C:
(
)
Two fifths of distance
Coordinate of point is,
(
)
This point is indicated by point P above.
5
Exercise:
What is the coordinate of a point that is ⁄ of the way from point C,
below, to point D?
C
-3
D
-2
-1
0
1
2
3
Graph the point.
What is the coordinate of a point that is ⁄ of the way from point C,
below, to point D?
C
-3
-2
-1
D
0
1
2
3
Graph the point.
6
Order on a Number
Line:
On a number line, the larger a number is, the farther to the
right its graph is.
C
-3
D
-2
-1
0
1
2
3
The number line clearly shows that negative three is smaller
than two.
Moreover, two is larger than negative three.
The order of two numbers, as seen on a number line, can be
expressed as an inequality.
Inequalities:
In the above example, we can write,
.
This is read, “Negative three is less than two.”
Moreover, the relationship between these two numbers can
also be written as,
.
These two inequalities are equivalent.
7
Exercise:
Use the number line to fill in the blank in the expressions
below with the correct inequality symbol.
-3
-2
-1
0
1
2
3
2 ____ 1
2 ____ 1
3 ____ 0
1 ____ 2
1 ____ 3
0 ____ 1
0 ____ 3
2 ____ 3
8
Opposite of a Real
Number:
Every real number has an opposite.
Opposite real numbers lie equidistant from and on opposite
sides of the origin.
2
-3
Exercise:
-2
2
-1
0
1
2
The opposite of two is negative two.
( )
The opposite of negative two is two.
(
3
)
On the number line below, graph and label the following
points:
-3
The opposite of one
( )
The opposite of two
( )
The opposite of negative three
(
The opposite of zero
( )
-2
-1
Zero is its own opposite:
0
1
)
2
3
( )
9
Absolute Value of a
Real Number:
The absolute value of a real number is the distance between
the graph of that number and the origin.
2
-3
-2
3
-1
0
1
2
3
The absolute value of three is three, because the distance
between the origin and three is three.
The absolute value of negative two is two, because the
distance between the origin and negative two is two.
Symbolically, absolute value is written using two vertical
lines:
| |
|
Definition of
Absolute Value:
|
𝑥
𝑥
𝑥
𝑥
⇒ |𝑥|
⇒ |𝑥|
⇒ |𝑥|
𝑥
𝑥
10
Exercise:
On the number line below, graph and label the following
points:
| |
|
|
|
|
| |
-3
-2
-1
0
Class work:
Page 3
Oral Exercises 1-19
Homework:
Page 4
Written Exercises 1-39 odd
Page 5
Mixed Review 1-12
1
2
3
11