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Section 1.1
Real Numbers
Rational Numbers
a/b with a and b are integers
with b ≠ 0
Irrational Numbers
√
Non-terminating and
repeating decimals
Integers
…,− ,− , , , ,…
Whole Numbers
, , , …
Natural numbers
, , …
non-repeating decimals
Example 1: Given
− , , √ , √ , .
a. Natural Numbers
b. Whole Numbers
c. Irrational
d. Real Numbers
Order
>
⟹ greater than
<
⟹ less than
=
⟹ equals
Example 2: Order
a.
15
√
b. 2.1
c.
4.5
d.
e.
0
,
Number Line:
The additive inverse of a number x is the number that when added to x equals zero.
Example 3: Find the additive inverse for;
a.
−
b.
23
c.
0
Section 1.2
Absolute Value: The absolute value of a real number is the distance from zero on the
number line. The numbers 2 and −2 are both 2 units away from zero.
That is
|2| = 2 and |−2| = 2
Absolute value is never negative!
|−1.2| = 1.2
|0| = 0
|−1| = 1
Adding integers
Same sign − just add the numbers and keep the sign
Different signs – subtract the smaller absolute value from the larger absolute value and
take the sign of the number with the larger absolute value.
Example 1: Add
a. 8 + 4
b. 8 + $−3&
c. −4 + $−6&
d. 6 + $−6&
e. −25 + $−31&
Subtracting integers
One method is to change the problem to an addition problem.
Example 2; Subtract
a. 3 − 9
b. 6 – (-10)
c. −8 − $−3&
d. 7 − 10 + 4
e. −7 − 4
f. −22 − $−18& + 4
g. 10 − 20
Multiplying and Dividing Integers
•
•
Multiply and divide normally
If two numbers have:
• Have the same sign then the answer is positive
• Have opposite signs the answer is negative
Note: An even number of negative signs the answer will be positive. An odd number
of negative signs then the answer will be negative.
Example 3: Multiply or divide
a.
−8$2&
b. −2$−10&
c.
+ ,
,
d. −4$−7&$1&$−3&$0&
e. −2$−3&$−4&
f.
−1$−6&$−2&$−1&
Section 1.3
Greatest Common Factor GCF
Least Common Multiple
LCM
GCF
1. Write the number as a product of prime numbers.
2. The GCF of two or more numbers is the numbers written as a product of prime
numbers. Then find the factors that are common to all the numbers.
Example 1: Find the GCF
a. 18, 27, 45
b. 252, 700
LCM
1. Write each number as a product of primes.
2. Take the greatest power of each prime and multiply.
Example 2: Find the LCM.
15, 18, 36
Example 3: Find the GCF and LCM
a. 60, 120
b. 98, 105
Adding and Subtracting Fractions
1. Find the least common denominator.
2. Multiply by 1= , so the LCD appears in each fraction
Example:
⋅ + ∙
3. Add or subtract the numerators, keep the same denominator.
Finish the example from above:
⋅ + ∙ =
Example 4: Add or Subtract
0
a.
−
b.
1 +
c.
12 +
12
1 −
+
=
Multiplying Fractions
1. Simplify the fraction
2. Multiply the numerators and denominators together. Paying attention the
number of negative signs.
Example 5: Multiply
a 3
b.
5
7.1
0
4 3 4
3−2 4
6
2
10
8 9
25 33
Dividing Fractions
1. Change the problem to a multiplication by taking the reciprocal of the
denominator of the fraction.
Example of a reciprocal:
5
the reciprocal is
:
the reciprocal is
5
:
2. Multiply fractions.
Example 6: Divide
a.
;
6
÷8
=
>
b. ?@
=
c.
d.
6
÷ ;;
;
33 4 ÷ 3−2 :4
Example 7: Simplify
a.
b.
=
B
C
A
C@
+ A
=
D
C
A
C>
+ A
+