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1-8A Number Systems
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Algebra 1
Glencoe McGraw-Hill
Linda Stamper
What are real numbers?
Pretend you are in
the first grade.
Your teacher asks
you to count. What
would you say?
REAL NUMBERS
Rational Numbers: Any number that can be
a
written in the form of . As a decimal they repeat or
b
terminate.
1
ex:
= 0.3333... Repeats
3
ex: 1 = 0.25 Terminates
4
Integers: Whole numbers and their opposites
(this means positive and negative whole numbers).
ex: … ‫ ־‬4 ,‫ ־‬3 ,‫ ־‬2 ,‫ ־‬1 ,0 ,1 ,2 ,3 ,4 …
Whole Numbers: Natural Numbers and
zero. ex: 0,1,2,3…
Natural or Counting Numbers
ex: 1,2,3,4,…
Irrational
Numbers:
ex:
 and 2
These must
be
represented
by a symbol
(ex: ), or as a
rounded
number, or in
radical form
because the
decimal
doesn’t repeat
or terminate
(stop).
So what isn’t a real number? When you divide by zero
and get no solution (  ), and -1 = i (imaginary numbers).
A rational number is any number you can write as a
a
quotient of two integers, where b is not zero.
b
Two numbers that are the same distance from 0 on a
number line but on opposite sides of 0 are opposites.
•
-2
-1
0
1
•
2
The numbers –2 and 2 are opposites because each is
2 units from zero.
Integers are the whole numbers, including zero, and
their opposites.
Zero is neither positive nor negative, and zero has no
opposite.
Name the set of numbers to which each real number belongs.
Example 1
6
11
Example 2
rational
irrational
 2
Example 3
81
Example 4
5

1
natural
rational
whole
number
integer
integer
rational
Square Roots
You will be allowed to use a calculator for tomorrow’s lesson but
NOT on the CHAPTER 1 test! NO GRAPHING CALCULATORS!
You know how to find the square of a number. For
instance, the square of 3 (written as 32) is 9.
32  9
3
3
The square of –3 is also equal to 9 because (–3)2 = 9.
The inverse of a square number is the square root. Square
roots are written with a radical symbol
. The number
or expression inside a radical symbol is called the radicand.
radical
symbol
9
radicand
All positive real numbers have two square roots:
positive square root (principal square root)
What two
 9  32
identical factors =
9 is 3
 3 read as the positive square root of 9?
negative square root
What two identical
factors = 9?
 9   32
  3 read as the negative square root of 9 is –3
This may be written together:
 9   32
 3 read as plus or minus the square root of 9
is plus or minus 3.
All negative real numbers do NOT have square roots because
two negative numbers multiplied produce a positive number.
9 =
undefined
The square root of a negative radicand is undefined!
Zero has only one square root and that is zero!
What two identical factors = – 9?
When two negatives are multiplied
the result is positive.
0 0
The square of an integer is called a perfect square.
32
3 is an
integer
2
3
3
3.52
3.5 is not an integer
(integers are whole
numbers)
9 3
3
Therefore 9 is a
perfect square.
3 is an
integer
12.25  3.52
 3.5
The figure is a
square but it is not
composed of square
sections.
The square of an integer is called a perfect square.
3 is an
integer
32
9  32 therefore 9 is a
perfect square.
3
3
3
12
(3.4641016…)2
not an integer
(irrational number)
What two
identical factors =
12?
(3.4641016…)
(3.4641016…)
12 is not a perfect
square.
Determine whether the number is a perfect square.
Example 5
49
yes
Example 6
 36
no
Example 7
7
no
What two identical
factors = the given
number?
Is your answer an
integer?
Example 8
144
yes
Evaluate the expression.
Example 9
Example 10
81
 81
9
9
Example 11
 81
9
Example 12
 81
undefined
To graph a set of numbers means to draw, or plot, the points
named by those numbers on a number line. The number that
corresponds to a point on a number line is called the
coordinate of that point.
Graph –1, 2 and – 3 on a number line. Order the numbers
from least to greatest.
-4
•
-3
-2
•
-1
0
1
•
2
3
Draw a number line. Label the number line.
Plot the points on the number line.
List the integers from least to greatest. –3, –1, 2
Example 13 Graph – 4, 4, – 6 and 0 on a number line.
Order the numbers from least to greatest.
•
-6
-5
•
-4
Draw a number line.
Label the number line.
-3
-2
-1
•
0
1
– 6, – 4, 0, 4
Plot the points on the number line.
List the integers from least to greatest.
•
2
Graphing Inequalities
For this part of the lesson, you will
need a ruler and a colored pencil.
The graph of an inequality in one variable is the set of
points on a number line that represent all solutions of the
inequality.
x  4 endpoint
•
O
ray
If the endpoint on the graph is not a solution, draw an open
dot.
If the endpoint on the graph is a solution, draw a solid dot.
Then draw an arrowhead to show that the graph continues to
infinity.
What is the name
for the geometric
figure that
represents the
solution?
Reading and Graphing an Inequality in One Variable
All real numbers greater
than or equal to 2.
All real numbers less
than 0.
All real numbers less
than or equal to – 5.
x > 2
•
a <0
0
O
–5>y
Rewrite as y < – 55
•
When the variable is before the inequality symbol,
what do you notice about the direction of the ray
and the direction of the inequality symbol?
Graphing an Inequality in One Variable
1. Write inequality.
Rewrite with variable first.
7 > x
2. Draw a line (use arrowheads).
•
3. Draw open or solid dot and
label the endpoint.
4. Draw the ray in the direction
of the inequality symbol.
You do NOT
need to
draw in the
tick marks.
x < 77
Example 14 Graph the solutions of each inequality on
a number line.
–4
a) x > – 4
b)
•
y < 15
O
c) –3 > x
Rewrite as x < – –3
3
•
1-A12 Pages