1 2 3
Basic Math Review
Numbers
NATURAL NUMBERS
{1, 2, 3, 4, 5, …}
WHOLE NUMBERS
{0, 1, 2, 3, 4, …}
INTEGERS
{…, 3, 2, 1, 0, 1, 2, …}
RATIONAL NUMBERS
All numbers that can be written in the form , where a
and bare integers and .
IRRATIONAL NUMBERS
Real numbers that cannot be written as the quotient of two
integers but can be represented on the number line.
REAL NUMBERS
Include all numbers that can be represented on the number
line, that is, all rational and irrational numbers.
PRIME NUMBERS
A prime number is a number greater than 1 that has only
itself and 1 as factors.
Some examples:
2, 3, and 7 are prime numbers.
COMPOSITE NUMBERS
A composite number is a number that is not prime. For
example, 8 is a composite number since
.8 =2#2#2=23
Rational Numbers
Real Numbers
23, 22.4, 21 , 0, 0.6, 1, etc.
2
4
_
5
25VN
Irrational
Numbers
p 23, 22, 21, 0, 1, 2, 3, pIntegers
0, 1, 2, 3, pWhole Numbers
Natural Numbers 1, 2, 3, p
3,
2, p, etc.
VN
bZ0
a>b
–5–5 – 4– 4 –3–3
Negative integersNegative integers Positive integers
The Number Line
Zero
–2–2 –1–1 01 2 3 4 5
ISBN-13:
ISBN-10: 978-0-321-39476-7
0-321-39476-3
9 780321 394767
90000
Integers (continued)
MULTIPLYING AND DIVIDING WITH NEGATIVES
Some examples:
Fractions
LEAST COMMON MULTIPLE
The LCM of a set of numbers is the smallest number that is a
multiple of all the given numbers.
For example, the LCM of 5 and 6 is 30, since 5 and 6 have no
factors in common.
GREATEST COMMON FACTOR
The GCF of a set of numbers is the largest number that can
be evenly divided into each of the given numbers.
For example, the GCF of 24 and 27 is 3, since both 24 and
27 are divisible by 3, but they are not both divisible by any
numbers larger than 3.
FRACTIONS
Fractions are another way to express division. The top num-
ber of a fraction is called the numerator, and the bottom
number is called the denominator.
ADDING AND SUBTRACTING FRACTIONS
Fractions must have the same denominator before they can
be added or subtracted.
, with .
, with .
If the fractions have different denominators, rewrite them as
equivalent fractions with a common denominator. Then add
or subtract the numerators, keeping the denominators the
same. For example,
.
2
3+1
4=8
12 +3
12 =11
12
dZ0
a
d-b
d=a-b
d
dZ0
a
d+b
d=a+b
d
or 36
2 18
2 36
18
1-242>1 -82=3
1-721-62=42
-3#5=-15
-a,b=-
a
b
-a
-b=a
b
-a#-b=ab
-a#b=-ab
Important Properties
PROPERTIES OF ADDITION
Identity Property of Zero:
Inverse Property:
Commutative Property:
Associative Property:
PROPERTIES OF MULTIPLICATION
Property of Zero:
Identity Property of One: , when .
Inverse Property: ,when .
Commutative Property:
Associative Property:
PROPERTIES OF DIVISION
Property of Zero: , when .
Property of One: , when .
Identity Property of One:
Absolute Value
The absolute value of a number is always ≥0.
If , .
If , .
For example, and . In each case, the
answer is positive.
ƒ5ƒ=5
ƒ-5ƒ=5
ƒaƒ=aa 60
ƒaƒ=aa 70
a
1=a#1
aZ0
a
a=1
aZ0
0
a=0
a#1b#c2=1a#b2#c
a#b=b#a
aZ0a#1
a=1
aZ0a#1=a
a#0=0
a+1b+c2=1a+b2+c
a+b=b+a
a+1-a2=0
a+0=a
Key Words and Symbols
The following words and symbols are used for the
operations listed.
Addition
Sum, total, increase, plus
addend addend = sum
Subtraction
Difference, decrease, minus
minuend subtrahend = difference
Multiplication
Product, of, times
factor factor = product
Division
Quotient, per, divided by
dividend divisor = quotient
Order of Operations
1st:Parentheses
Simplify any expressions inside parentheses.
2nd:Exponents
Work out any exponents.
3rd:Multiplication and Division
Solve all multiplication and division, working from
left to right.
4th:Addition and Subtraction
These are done last, from left to right.
For example,
.
Integers
ADDING AND SUBTRACTING WITH NEGATIVES
Some examples:
-19 +4=4-19 =-15
-3-17 =1-32+1-172=-20
a-1-b2=a+b
-a+b=b-a
-a-b=1-a2+1-b2
=12
=15 -6+3
=15 -2#3+27 ,9
15 -2#3+130 -32,32
aba
ba>bba
a*b, a#b, 1a21b2, ab
more➤
Rates, Ratios, Proportions,
and Percents
RATES AND RATIOS
A rate is a comparison of two quantities with different units.
For example, a car that travels 110 miles in 2 hours is mov-
ing at a rate of 110 miles/2 hours or 55 mph.
A ratio is a comparison of two quantities with the same
units. For example, a class with 23 students has a
student–teacher ratio of 23:1 or .
PROPORTIONS
A proportion is a statement in which two ratios or rates are
equal.
An example of a proportion is the following statement:
30 dollars is to 5 hours as 60 dollars is to 10 hours.
This is written
.
A typical proportion problem will have one unknown
quantity, such as
.
We can solve this equation by cross multiplying as shown:
.
So, it takes 60 minutes to walk 3 miles.
PERCENTS
A percent is the number of parts out of 100. To write a per-
cent as a fraction, divide by 100 and drop the percent sign.
For example,
.
To write a fraction as a percent, first check to see if the
denominator is 100. If it is not, write the fraction as an
equivalent fraction with 100 in the denominator. Then the
numerator becomes the percent. For example,
.
To find a percent of a quantity, multiply the percent by the
quantity.
For example, 30% of 5 is
.
30
100 #5=150
100 =3
2
4
5=80
100 =80%
57% =57
100
x=60
20 =3
20x=60 #1
1 mile
20 min =x miles
60 min
$30
5 hr =$60
10 hr
23
1
Fractions (continued)
Equivalent fractions are found by multiplying the numerator
and denominator of the fraction by the same number. In the
previous example,
and .
MULTIPLYING AND DIVIDING FRACTIONS
When multiplying and dividing fractions, a common
denominator is not needed. To multiply, take the product
of the numerators and the product of the denominators:
To divide fractions, invert the second fraction and then
multiply the numerators and denominators:
Some examples:
REDUCING FRACTIONS
To reduce a fraction, divide both the numerator and denom-
inator by common factors. In the last example,
.
MIXED NUMBERS
A mixed number has two parts: a whole number part and a
fractional part. An example of a mixed number is . This
really represents
,
which can be written as
.
Similarly, an improper fraction can be written as a mixed
number. For example,
can be written as ,
since 20 divided by 3 equals 6 with a remainder of 2.
6 2
3
20
3
40
8+3
8=43
8
5+3
8
5 3
8
10
12 =10 ,2
12 ,2=5
6
5
12 ,1
2=5
12 #2
1=10
12 =5
6
3
5#2
7=6
35
a
b,c
d=a
b#d
c=ad
bc
a
b#c
d=a#c
b#d=ac
bd
1
4=1#3
4#3=3
12
2
3=2#4
3#4=8
12
more➤
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