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WARM UP
1
Write fractional notation for each number 5.15
2
Write the decimal notation for
3
Calculate a. 6.79 + 3.4 10.19 b. 3 – 1.53
c. 2.21 x 3.978
1.8
e.
2
1
1 3
3
4
g.
3
11
4
12
3
2

8
3
4. What is 25% of 25?
2
1
4
6.25
3
3
5
103
20
3.6
1.4
7
d. 2.7 ÷ 7.2
0.375
f. 8 4  7 5
4
6
h. 1 9  5
16
8
130% of 64
11
12
2
1
2
83.2
5. Write the correct symbol =, <, >. 0.074 _< 0.703
6. Write the correct ratio:
a. 28 astronauts for 4 missions 7 to 1 b. 52 wks in 12 mon 13 to 3
ALGEBRA 1 REVIEW
REAL NUMBERS & OPERATIONS
OBJECTIVES
Identify rational numbers & distinguish
between rational & irrational numbers
Add positive & negative numbers
Subtract positive & negative numbers
KEY TERMS & CONCEPTS
 Absolute value
 Rational numbers
 Additive inverse
 Real numbers
 Difference
 Subtraction
 Integers
 Subtrahend
 Irrational numbers
 Whole numbers
 Natural numbers
 |x|
 Number line
 -x
 Opposite
REAL NUMBERS
The most important set of numbers in algebra is the set of real
numbers.
There is exactly one real number for each point on a number line
The positive numbers are shown to the right of zero and the
negative to the left. Zero is neither positive nor negative.
-2.5
-½
2
π
The sets of natural numbers [1, 2, 3, 4,……..], whole numbers [0,
1, 2, 3, 4,……] and integers [-2, -1, 0, 1, 2, 3,….] are all subsets of the
set of real numbers.
RATIONAL NUMBERS
The real numbers consist of the rational and
irrational numbers
DEFINITION
Rational numbers are those that can be expresses
as a ratio a , where a and b are integers and b ≠ 0
b
These are rational numbers: 4, 9.6, 0
Since they can be written as: 4/1, 96/10, 0/1
IRRATIONAL NUMBERS
If a real number cannot be expressed as a ratio of integers
1/b, b ≠ 0, then it is called irrational.
For instance, we can prove that there is no irrational
number that is a square root of 2.
We can come close but there is not rational number whose
square is exactly 2.
Thus, √2 is not a rational number. It is irrational.
Unless a whole number is a perfect square, its square root is
irrational.
IRRATIONAL NUMBERS
The following numbers are irrational: √2, √8, -√45,
√11, π.
Decimal notation for a rational number either ends or
repeats.
Decimal notation for an irrational number never ends
and never repeats.
EXAMPLES
Determine which are rational and which are irrational
numbers:
1. 8.974974974…
(numerals repeat) Since they repeat, the number
is rational. We can express it as 8.974.
2. 3.12112111211112…
(numerals do not repeat) Since they do not
repeat and does not end the number is irrational.
3. 4.325
Since the number ends, it is rational.
4. √17
Since 17 is not a perfect square, it is irrational.
TRY THIS…
a.
7.42
b. √49
c.
0.47646464… (numerals repeat)
d. -√32
e.
59/37
f.
2.5734107656631…(numerals do not repeat)
ABSOLUTE VALUE
The absolute value of a number a is the distance
between a and 0 on a number line. The symbol
represents the absolute value of a.
If a is positive, then
If a is 0, then
=a
=0
If a is negative, then
= -a
Example:
=2
=0
= -(-2) = 2
ABSOLUTE VALUE
•
The absolute value of a number is its distance from 0
on a number line.
•
We denote absolute value of x as |x|.
- 6
-5
-4
-3
6 units
-2
-1 0 1
2
3
4
5
6
4 units
•
Since 4 is four units from 0, |4| = 4. Since -6 is six
units from 0, |-6| = 6.
•
The absolute value of 0, |0| = 0.
REAL NUMBERS ADDITION
•
One way to add or subtract two real numbers is to use
the number line.
To add a positive number move right
- 6
-5
-4
-3
-2
-1 0 1
To add a negative number, move left
2
3
4
5
6
REAL NUMBERS ADDITION
Recall the rules of signs for adding real numbers.
RULES FOR ADDITION OF REAL NUMBERS
1. To add when there are like signs, add the absolute
value. The sum has the same sign as the addends.
2. To add when there are unlike signs, subtract the
absolute values. The sum has the sign of the addend
with the greatest absolute value.
Examples: Add
-5 + (-9) = -14
23 + (-11) = 12
Adding absolute values; the sum is negative.
Subtracting absolute values; the positive addend has
greater absolute value
TRY THIS…
a. -8 + (-9)
b. -8.9 + (-9.7)
c. -6/5 + (-23/10)
d. 14 + (-28)
e. -4.5 + (7.8)
f.
3/8 + (-5/6)
ADDITIVE INVERSE
Every real number has exactly one additive inverse or
opposite.
The additive inverse of a number is the number added to it
to get 0.
The additive inverse of x is –x.
PROPERTY OF ADDITIVE INVERSES
For every real number a, there is exactly one number b for which a
+b=0
The additive inverse of a number is the number opposite it, with
respect to 0 on the number line.
ADDITIVE INVERSE
The additive inverse of a number is the number
opposite it, with respect to 0 on the number line.
 To find the additive inverse of 5, we reflect the opposite
side of 0. the additive inverse of 5 is -5.
 To find the additive inverse of a number quickly, just
change the sign.
REAL NUMBER SUBTRACTION
Subtraction is defined in terms of addition.
Subtraction and addition are inverse operations.
SUBTRACTION
The difference a – b is the number c such that c + b = a.
We can always subtract by adding an inverse.
The number subtracted is called the subtrahend.
To subtract, we can change the sign of the subtrahend and then
add it to the other number.
14 – 8 = 14 + (-8)
EXAMPLES
1) -12 – 19 =
You can then use the number line to start at negative 12 and go
left 19 units because you are adding a negative number.
2) 5 – (-4) = 5 + 4 = 9
TRY THIS…
a. 8 – (-9)
b. 23.7 – 5.9
c. -11/16 – (23/12)
CH. 1.1 HOMEWORK
Textbook pg. 7 & 8 #12, 16, 18, 24, 28, 30 & 32