Transcript Real Number Properties and Basic Word Problems

Real Numbers and Properties
Natural Numbers……

Known as “Counting” Numbers

Example: 1, 2, 3, 4, 5,…….
Whole Numbers……


You add the number 0 to the natural
numbers.
Example: 0, 1, 2, 3, 4, 5…….
Integers……


Integers are made up of whole numbers
and their opposites.
Example: …-4,-3,-2,-1,0,1,2,3,4….
Rational Numbers……

The set of rational numbers is made up
of all of the following
a. Natural Numbers
b. Whole Numbers
c. Integers
d. Plus every repeating and
terminating decimal.
Examples of Rational
Numbers……

A. ½ = 0.5 (Terminating Decimal)

B. 1.23232323 (Repeating Decimal)

C. 0.256256256 (Repeating Decimal)
D. 2.735 (Terminating Decimal)
Irrational Numbers….


Consists of Non-Terminating and NonRepeating Decimals.
Example: 0.9482137507264
Real Numbers (ℝ)
Rational Numbers (ℚ)
Integers (ℤ)
Whole Numbers
Natural Numbers (ℕ)
1, 2, 3, …
0, 1, 2, 3,
…
…-3, -2, -1, 0, 1, 2, 3,
…
Decimal form either
terminates or repeats
All rational and irrational numbers
Irrational Numbers
Decimal form is
non-terminating
and nonrepeating
The Number Line……


A number line consists of positive numbers
(right of 0) and negative numbers (left of 0).
A real life example of a number line is a
temperature thermometer.
Negative
Positive
0
For example…..


-5 would represent 5 degrees below
zero.
+4 would represent 4 degrees above
zero.
Make the Comparison……


7 degrees below 0 is
(warmer/colder)
than 4 degrees
above 0.
7 degrees below 0 is
a (lower/higher)
temperature than 4
degrees above 0.

Colder

Lower
Coordinates on a Graph….

Find the best estimate of the point.
a. -2
b. 2
c. -1.75 d. -1.5

2
1
0
Answer: -1.75
1
2
Sets and Subsets……



A set is a group of numbers.
Example: Set A = {1,2,3,4,5}
A subset is a group of numbers in which every
member is in another set.
Example: Set B = {1,2,3}
So, B is a subset of A.
Which of the following would represent a
subset of integers?




States Sales Tax Rate
Amount of Gas in a Car
Number of Students in Class
A Dinner Receipt




Strategy: Eliminate
those that are NOT
integers.

7.5% - NO
6.5 Gallons – NO
12 – YES
$10.31 - NO
You Try…Which of the following would
represent a subset of integers?




Costs of a TV
# of miles on the
odometer of a car
A person’s weight
Number of residents in
South Carolina

No

No

No

Yes
Inequalities…..


We use inequalities to compare
numbers.
The following are inequalities:




Examples…….


“4 is less than 7” - 4  7
“9 is greater than or equal to 5” -
95
You Try……Insert the appropriate
inequality sign.
1.
-5
-2
1.
<
2.
-7
2
2.
<
3.
4
-12
3.
>
Least to Greatest……


This means to arrange numbers in the
order from the smallest to the largest.
HINT: If there are fractions it might be
easier to convert to decimals first.
Which Number is Smaller?
3
2
 or 
7
9
3
  0.42857
7
2
  0.22222
9
3
 is smaller
7
Which Number is Larger?.......
5

or  0.32
13
5
  0.3846
13
 0.32  0.32
So,  0.32 is l arg er.
You Try…Compare

2
3
4

5
 0.68

-0.67 > -0.68
6

7

-0.8 > -0.86
Which Set is Ordered from
Least to Greatest?
1.
{-3/2, -3, 0, 2/3}
1.
{-3/2, -3, 0, 2/3}
2.
{-3, -3/2, 0, 2/3}
2.
{-3, -3/2, 0, 2/3}
3.
{0, 2/3, -3/2, -3}
3. {0, 2/3, -3/2, -3}
4.
{0, -3/2, -3, 2/3}
4. {0, -3/2, -3, 2/3

What kinds of
numbers are used to
represent numbers
below zero?

Answer:

NEGATIVE Numbers

Make -8
-4 a
true statement.

Answer:

<
Quick Review
-400
-200
0
200
1) Coordinate of A:
a) -250
b) -300 c) -325 d) -500
2) Coordinate of B:
a) -210
b) -350 c) -100 d) -50
3) Coordinate of C:
a) 350
b) 425
c) 325
d) 275
400
Quick Review
4) Use  ,  : -8
5
5
5) Which is smaller?  7
or
6) Write from smallest to largest:
-3, -3.8, -5, 5.6, -5.6
3

8
Number Properties
Commutative PropertyChanges Order

For Addition

For Multiplication
A+B = B+A
Ex. 2+3 = 5
3+2 = 5
2+3=3+2
AB = BA
Ex.
4(8) = 32
8(4) = 32
4(8) = 8(4)
THIS IS NOT TRUE FOR
SUBTRACTION OR DIVISION!
Associative PropertyChanges Grouping

For Addition
A + (B + C) = (A + B) + C
Ex.
5 + (2 + 4)
=5+6
= 11
(5 + 2) + 4
=7+4
= 11
5 + (2 + 4) = (5 + 2) + 4

For Multiplication
A(BC) = (AB)C
Ex.
2 (3 5)
= 2(15)
= 30
(2 3) 5
= (6)5
= 30
2 x (3 x 5) = (2 x 3) x 5
This is not true for subtraction or division!
Which Property?
1)
2)
3)
4)
5)
6)
3x 4 = 4 3x
6y + (7 + 3z) = (6y +7) +3z
(5x + 7) + 8y = 5x + (7 + 8y)
(3x)(2x + 5) = (2x + 5)(3x)
10x + 4y = 4y + 10x
(2x 5)(10y) = (2x)(5 10y)
Distributive Property
A (B + C) = AB + AC A (B – C) = AB – AC
(B + C) A = BA + CA (B – C) A = BA – CA
Ex. -3 (4 – 2x)
Strategy: Think -3 (4 – 2x) means -3 (4 + -2x)
= -3(4) + (-3)(-2x)
= -12 + 6x
TRY THESE:
A) 4 (6 +2a)
B) -7 (-3m – 5)
Which Property?
1)
-3x(y + 2) + 4y = -3x(y) – 3x(2) + 4y
2)
-3y + 4x(y + 2) = -3y + 4xy + 4x(2)
3)
6x + (3y + 1) = (3y +1) + 6x
What is an example of the
commutative prop. of addition?
A)
B)
C)
D)
3
3
3
3
+
+
+
+
5m
5m
5m
5m
=
=
=
=
3 + (1 + 4)m
5m + 3
5 + 3m
3m + 5
A)
B)
C)
D)
7 + 4m = (7 +4)m
(5 + 2) + 4m = 7 + 4m
7 + 4m = 4 + 7m
7 + 4m = 4m + 7
Homework