Download Review: The real Number and absolute Value

MAT 111: Intermediate Algebra
THE REAL NUMBER AND ABSOLUTE VALUE
Set: A set is collection of objects whose contents can be clearly determined. The objects in a set are called the
elements of the set. There are several ways to represent a set and one of the most common one is using curly brackets.
for examples: {1, 2, 3, 4}, {a, b, r,t, u, o}.
Real Numbers
Rational numbers can be written either one ( or more) of the following three forms:
a
(a and b are integers and b 6= 0)
Terminating Decimal
b
9 3
,
11.23
2 4
Repeating Decimal
7.12
Irrational numbers are those which are not rational. The decimal form of
√an irrational number is non terminating
and non repeating decimal number. Examples: 2.71828182846..., π and 2.
√
13 √
EX 1) Consider the set of numbers {3.56, − 5, − 58 , −1, , 0, 0.8, π, , 25, 111.12}, tell which numbers are
2
a Integers
b Rational numbers
c Irrational numbers
We graph real numbers using a line called number line.
Ex 2) Graph each number on the same number line. a. -2.5
b.
9
4
c.
√
2
d. 5
d Real numbers
e. − 5
For any real number a, the additive inverse of a is −a. These numbers are at the same distance from 0, but on opposite
sides of 0, on number line.
Ex 3) Find the additive inverse of the following numbers.
1. −41
2. −8 21
3. 1.7
1
4. − 52
Written by B. Paudyal
MAT 111: Intermediate Algebra
The absolute value of a number is its distance from 0 on a number line. Absolute value of number is never negative.
The notation uses two vertical lines: |number|.
Some facts about absolute value:
• absolute value of a positive number is the number
• absolute value of a negative number is its additive inverse
• absolute value of 0 is 0
• If |x| = a then either x = −a or x = a
2
b. |0| =
Ex 4) Find the value. a. =
5
Some Symbols:
= is equal to
< is less then
≤ is less then or equal to
6=
>
≥
c. | − 2.9| =
d. − | − 9| =
is not equal to
is greater than
is greater than or equal to
Ex 5) Indicate whether the inequality is true or false.
a. 0 < −6
c.
b. −7 ≤ 0
10 5
≥
6
3
d. −1 > 0.3
1
Ex 6) Graph {−1.4, 3, − , −2.6} on the same number line. Then rank them in ascending and descending order.
2
Properties of Real Numbers:
For any real numbers a, b, and c
For Addition
a+b = b+a
Name of Property
Commutative Property
Associative Property
(a · b) · c = a · (b · c)
(2 · 3) · 4 = 2 · (3 · 4)
(2 + 3) + 4 = 2 + (3 + 4)
a+0 = a
a·b = b·a
2·3 = 3·2
2+3 = 3+2
(a + b) + c = a + (b + c)
For multiplication
Identity Property
a·1 = a
2+0 = 2
2·1 = 2
a + (−a) = 0
1
a·
= 1, a 6= 0
a
Inverse Property
2·
2 + (−2) = 0
2
1
2
=1
Written by B. Paudyal
MAT 111: Intermediate Algebra
The Distributive Property: a · (b + c) = a · b + a · c
Example:
2 · (3 + 4) = 3 · 3 + 2 · 4
Zero factor property: a · 0 = 0
Example:
2·0 = 0
Division and 0:
A number
is NOT defined
0
Ex 7) Find each of the following.
a. The additive inverse of 9.1
c. The multiplicative inverse of −5
b. The multiplicative inverse of 12
The additive inverse of −5
d.
Ex 8) Indicate the definition, property, or number fact that justifies each statement.
a. 5(−3) = −3(5)
b. (3 + 13) + 27 = 3 + (13 + 27)
c. 2 · 3 + 2 · 5 = 2(3 + 5)
0 = 6.3 + (−6.3)
d.
Addition, subtraction, multiplication and division
EX 9) Add and Subtract
1. -3 + 6
2. -27 (-31)
3. -9 + (-25) + 4
To Multiply Real Two Numbers
• positive · positive = positive
• negative · negative = positive
• positive · negative = negative
4. -7 2 + (-23)
To Divide Two Real Numbers
• positive ÷ positive = positive
• negative ÷ negative = positive
• positive ÷ negative = negative
a
Note • Different notations of division: a ÷ b, a/b and .
b
• Different notations of multiplication: a · b, a × b and (a)(b).
• Decimal notation (.) and dot product notation (·).
EX 10) Multiply and divide
1. 4 · (−5)
4.
−12
4
2. 3 · (−4) · (−1)
5.
−1
0
1
3. ( ) · (−1)(1)(−3)(−1)(−1)
3
6.
0
−800008
3
Written by B. Paudyal