Notesale: Turn your study into money

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Basic Math Review
Numbers

Important Properties

NATURAL NUMBERS
{1, 2, 3, 4, 5, …}

Identity Property of Zero: a + 0 = a

WHOLE NUMBERS

Inverse Property: a + 1 a2 = 0

{0, 1, 2, 3, 4, …}

Commutative Property: a + b = b + a

INTEGERS

Associative Property: a + 1b + c2 = 1a + b2 + c

{…, —3, —2, —1, 0, 1, 2, …}
The Number Line

–5 – 4 –3 –2 –1 0 1 2 3 4 5
Negative integers

Positive integers

a

Zero

RATIONAL NUMBERS
All numbers that can be written in the form a>b, where a
and b are integers and b G 0
...


0
a
a

a

= 0, when a G 0
...


REAL NUMBERS
Include all numbers that can be represented on the number
line, that is, all rational and irrational numbers
...


Real Numbers
Irrational
Numbers

Rational Numbers 23, 22
...
26, 1, etc
...


Whole Numbers 0, 1, 2, 3, p
Natural Numbers 1, 2, 3, p

PRIME NUMBERS
A prime number is a number greater than 1 that has only
itself and 1 as factors
...


COMPOSITE NUMBERS
A composite number is a number that is not prime
...
2
...


1

Key Words and Symbols

Integers

(continued)

The following words and symbols are used for the
operations listed
...
5 = - 15
1 - 721 - 62 = 42
1 - 242>1 - 82 = 3

factor × factor = product
Division
Quotient, per, divided by
a
a ÷ b = = a>b = bq a

—18
—36
or
= —18
2q—36

Fractions

dividend ÷ divisor = quotient

LEAST COMMON MULTIPLE
The LCM of a set of numbers is the smallest number that is a
multiple of all the given numbers
...


Order of Operations
Simplify any expressions inside parentheses
...


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
...


Solve all multiplication and division, working from
left to right
...
The top number of a fraction is called the numerator, and the bottom
number is called the denominator
...

a +b
a
b
+
=
, with d Z 0
...
Then add
or subtract the numerators, keeping the denominators the
same
...


= 12
...
In the
previous example,
1
1
...
4
8
=
...

and
3
3 4
12
4
4 3
12

RATES AND RATIOS

A rate is a comparison of two quantities with different units
...

A ratio is a comparison of two quantities with the same
units
...

1

MULTIPLYING AND DIVIDING FRACTIONS
When multiplying and dividing fractions, a common
denominator is not needed
...

a c

...
c = ac
b d
b d
bd

PROPORTIONS
A proportion is a statement in which two ratios or rates are
equal
...

This is written
$30
$60
=

...

20 min
60 min

To divide fractions, invert the second fraction and then
multiply the numerators and denominators:
a
c
a d
ad
÷ =
...
2
6
=
5 7
35
5
1
5
...
1
60
x =
= 3
...
In the last example,
10
10 ÷ 2
5
=
=
...


PERCENTS

MIXED NUMBERS

A percent is the number of parts out of 100
...

For example,
57
57% =

...
An example of a mixed number is 583
...

8
8
8

To write a fraction as a percent, first check to see if the
denominator is 100
...
Then the
numerator becomes the percent
...

5
100

Similarly, an improper fraction can be written as a mixed
number
...

For example, 30% of 5 is
30
...

100
100
2

since 20 divided by 3 equals 6 with a remainder of 2
...
The decimal point separates the whole number part from the fractional part
...
9 represents

To change a number from a percent to a decimal, divide by
100 and drop the percent sign:
58% = 58/100 = 0
...


9
...
Then add or subtract as
usual, keeping the decimal point in the same place
...
37 =
23
...
5% for 5 years:
p = $3800
r = 5
...
055
t = 5 years

— 0
...
63

MULTIPLYING AND DIVIDING DECIMAL NUMBERS
To multiply decimal numbers, multiply them as though they
were whole numbers
...
For example, 3
...
5 is

I = 13800210
...

The amount of interest is $1045
...
72
× 4
...
740

1 decimal place

Scientific Notation
Scientific notation is a convenient way to express very large
or very small numbers
...
For
4 are expressed
example, 3
...
2
in scientific notation
...
If that number is
positive, move the decimal point to the right
...
The number tells you how
many places to move the decimal point
...
If it is not, move the decimal place to the right
(multiply by 10, 100, and so on) to make it a whole number
...

For example,
0
...
2 = 4
...
35
= 12q 4
...


To change a number to scientific notation, move the decimal point so it is to the right of the first nonzero digit
...
If
the decimal point is not moved, n is 0
...
0000216 = 2
...


The decimal point in the answer is placed directly above the
new decimal point in the dividend
...
S
...
= inch
ft = foot
min = minute
sec = second
gal = gallon
yd = yard
pt = pint

To multiply or divide numbers in scientific notation, we can
change the order and grouping, so that we multiply or divide

Statistics

oz = ounce
c = cup
mi = mile
hr = hour
lb = pound
qt = quart
T = ton

Metric Units

There are several ways to study a list of data
...

Median is the number that separates the list of data into two
equal parts
...
If the number of data is odd, the median is
the middle number
...

Mode is the number in the list that occurs the most frequently
...

For example, consider the following list of test scores:
{87, 56, 69, 87, 93, 82}

mm = millimeter
cm = centimeter
km = kilometer
m = meter
mL = milliliter
cL = centiliter
L = liter
kL = kiloliter
mg = milligram
cg = centigram
g = gram
kg = kilogram

To find the mean, first add:
87 + 56 + 69 + 87 + 93 + 82 = 474
...
S
...
S
...


12 in
...

To find the median, first list the data in order:
56, 69, 82, 87, 87, 93
...
5
...
001 m = 1 mm
0
...
001 L = 1 mL

The median score is 84
...

The mode is 87, since this number appears twice and each
of the other numbers appears only once
...
01 m = 1 cm
0
...
01 L = 1 cL

Geometry

Geometry

The perimeter of a geometric figure is the distance around it
or the sum of the lengths of its sides
...


L
W

c

a

P = 2L + 2W
b

The perimeter of a square is 4 times the length of a side:
s

Area: A =
r2
Circumference: C =

s

d=2

r

P = 4s
and p is approximately 3
...


Area is always expressed in square units, since it is twodimensional
...
W
...
s

or

r

A = s2
...

A straight line has an angle of 180 degrees
...

For example, the number of days until your vacation
changes every day, so it could be represented by a
variable, x
...
For
example, the number of days in the week, 7, does not
change, so it is a constant
...
For example,
x + 7 is an expression
...
For example,
x + 7 = 18 is an equation
...
For example, in using the above
equation, x + 7 = 18, we know that the statement is true
if x = 11
...
h
2

The sum of all three angles in any triangle always equals
180 degrees
...
The
hypotenuse of a right triangle is the side opposite the right
angle

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