1. NUMBER SYSTEMS Ch 1-1 The Integers Ch 1-2 The Rational Numbers
C. N. Colón
Algebra
St. Barnabas H.S.
Bronx, New York

2. OBJECTIVE To recognize integers and the subsets of
integers found in the Set of Real Numbers
To define and evaluate absolute value
expressions
To define and identify a domain
To define and evaluate inequalities

3. What is a Set? A set is a collection of objects called elements.
The elements have something in common.

4. Let’s look at a Set? Living Things Plants Animals Bacteria Mammals
Reptiles
Fish
Whales
Humans
Horses
Barnabites
Tree
Diagram
Ariana Grande
Taylor Swift

5. Venn Diagram for Quadrilaterals
Concave
Trapezoids
Convex
Parallelograms
Rhombuses
Rectangles
Squares

6. Decimals will terminate or repeat
Set of Real Numbers
Rational
Irrational
Decimals: non-terminating and non-repeating
any number which
can be written as a
fraction.
, 7, -0.4
Integers
Ex:Fractions/Decimals
…-3, -2, -1, 0, 1, 2, 3...
, -0.33, - 2.1
Whole
Ex: Negative Integers
Decimals will terminate or repeat
0, 1, 2, 3...
…-3, -2, -1
Natural
Zero
1, 2, 3...
(counting numbers)

7. Real This Venn Diagram displays the Sets of Real Number
(Rational, Irrational, Integers, Wholes, and Naturals)
Real
Rational
-2.65 5
Integers -3
-19
Wholes 5
Irrational
Naturals
1, 2, 3...

8. You will learn about imaginary numbers in Algebra II
Real
Rational
-2.65
Integers -3
-19
Wholes
Irrational
Naturals
1, 2, 3...

9. The square root of any negative number will yield “No Real Solution”
RULE
The square root of any negative
number will yield “No Real Solution”

10. Identify all of the sets to which each number belongs.
(Real, Rational, Irrational, Integers, Wholes, Naturals)
1) -6
Real, Rational, Integer 2)
Real, Rational
3) 14
Real, Rational, Integer, Whole, Natural
4) 6
Real, Irrational

11. RATIONAL NUMBERS
The rational numbers are all numbers that can be expressed in the form of a fraction and the denominator does not equal 0.
Remember that a ratio is the comparison of two numbers by division.

12. RATIONAL NUMBERS The set of rational numbers is everywhere DENSE.
This means that if you had any two rational numbers, you would always be able to find an infinite number of rational numbers in between them.
You can find the exact number in between any two rational numbers by finding their mean, or average.

13. What is the exact number that lies in between 24.2 and 20.6?
To find the answer just take the average of both numbers which means add them together and then divide the sum by two!

14. RATIONAL NUMBERS
Rational numbers can be expressed as decimals that will either REPEAT or TERMINATE.
 = or  = 2.75

15. RATIONAL NUMBERS
An integer can become a decimal if you simply place a decimal after it.
8 = 8.0
Any fraction can be become a decimal if you drop and dot, then divide. 7 5 .
2 8 20 20

16. TYPES OF SETS
An infinite set is a set whose elements cannot be counted since there is no end to the set. An example is the set of whole numbers or the set of counting numbers. They keep going on and on… infinitely…
Let W = the whole numbers
W = {0, 1, 2, 3…}
Let C = the counting numbers
C = {1, 2, 3…}

17. TYPES OF SETS
A finite set is a set whose elements can be counted. An example is the set of digits. Since we only have the digits 0 to 9, then there are only a specific or finite number of elements in that set.
Let D = the set of digits
D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

18. TYPES OF SETS The empty set or null set is a set that has no
elements and is written using the symbols
{ } or ø
Let S = engineers in room 204
S = { } since there are none in this room
S = ø another way of writing it

19. Domain: the set of values which may be
meaningfully substituted for a given variable.
Which set of numbers would best describe the variable?
1) Let x = the number of Barnabites in this room.
D: x Naturals
“ is a member of ”
2) Let w = your weight
D: w Positive Real Numbers
3) Let t = the low temperature in New York today
D: t Integer
4) Let n = the lowest recorded temperature in Alaska.
D: n Integers

20. Symbols of Inequality < > < >
If two numbers are not equal the relationship between
them can be expressed as an inequality.
An inequality can be written in several different ways.
INEQUALITIES
< > < >
less than
greater than
at most
at least
fewer than
more than
no more than
no less than
less than or equal to
greater than or equal to
Note that the symbols point to the smaller number and open up to
the larger number

21. The hungry alligator always eats the largest number.

22. Example: How to graph an inequality
Graph t < 3 over the given domain.
1) D: Reals
2) D: Integers
3) D: Wholes

23. ASSIGNMENT
p. 9 #4-56 (multiples of 4) + #53
p. 15 #4-64 (mo4)