Download Math 208 -- Number Sense

Math 208 -- Number Sense
Quick Reference Card
This Quick Reference Card contains information about our number system with which you should be familiar before you start
Math 208. Other concepts with which you should be familiar are fractions and decimals. For more information about any of
these concepts, visit the Center for Math Excellence (CME -- under SERVICES on your University eCampus Web page). At the
CME, click on "Running Start," then "Pre-Algebra Review," and finally on "Whole Numbers." Even more information about
number sense (sometimes called numeracy) can be found on the Web at sites such as www.JoleneMorris.com and
www.purplemath.com. As soon as your Math 208 class begins, you will also have access to FREE tutoring at the CME. Ask for
tutoring in any of the Number Sense concepts that are not clear to you.
Expanded Form of a Number
Ordering Large Numbers
A number in standard form is
collected into groups of three
digits using commas. Each of
these groups is called a period.
Place Value
Expanded form is a way to write a large
number by showing the sum of values of each
digit of a number. Thus, a number in expanded
form shows each digit along with the digit's
place value. Here is an example:
Standard form:
Within each group, the place
values are always the 100's
place, the 10's place, and the
1's place (from left to right).
Expanded form:
7 x 100,000,000 + 2 x 10,000,000 +
5 x 1,000,000 + 2 x 100,000 + 9 x 10,000 +
8 x 1,000 + 5 x 100 + 7 x 10 + 1
Understanding place value is
key to understanding our
number system. Decimal
numbers simply extend the
place values to the right and use
"ths" to identify the places (e.g.
100 millionths place).
Expanded form can also be written using
exponents:
7 x 108 + 2 x 107 + 5 x 106 + 2 x 105 +
9 x 104 + 8 x 103 + 5 x 102 + 7 x 10 + 1
When you are asked to order large
numbers, write them above one another
with the place values lined up. Then,
starting from the left, look for the largest
value. For example, if you are asked to
order:
5,139 986,733 3,950 77,922
Write them above each other with the
place values lined up as
you would if you were
5,139
986,733
going to add the
3,950
numbers. Looking at the
77,922
place values from left to
right, the largest number
is 986,733. The next largest number is
77,922. Both the first and third numbers
start in the same place value but 5 is
larger than 3 so 5,139 is larger than
3,950.
725,298,571
Rounding
Averaging
Parentheses
Rounding a number requires
that you understand place value
(see above). Rounding a
number is a type of estimation.
Rounding is also called
"rounding off."
The common meaning of an average is to find
the arithmetic mean. To average a group of
numbers, add all the numbers together and
divide by how many numbers there are. For
example, the average of 5, 7, 12, and 8 is
(5 + 7 + 12 + 8) / 4 = 8.
To round a number, look at the
digit to the right of the place
being rounded. If that digit to
the right is 5 or higher, add 1 to
the place being rounded.
Change all places to the right of
the place being rounded to
zeroes.
Some related statistical measures are median,
mode, maximum, and minimum.
 Median is the middle number when the list is
ordered in size
 Mode is the number that appears most often
in the list
 Maximum is the largest number in the list
 Minimum is the smallest number in the list
Parentheses are a way to group numbers.
Other grouping symbols are braces { },
square brackets [ ], and the vinculum
or fraction bar.
To remove parentheses, we distribute
the number immediately outside the
parenthesis (with its sign). We distribute
by multiplying by the number and its
sign. If the outside sign is "hidden," it is
understood to be positive (see Example 1
below). If the outside number is "hidden,"
it is understood to be a 1 (see Example 2
below). For example:
3(3x +1) = 9x + 3
-(2x - 5) = -2x + 5
Order of Operations
When you are asked to simplify or evaluate an
expression, you must follow the Order of Operations:
1. Simplify inside the parentheses
2. Simplify any numbers or expressions with exponents
3. Perform all multiplication & division starting at the
left
4. Perform all addition & subtraction starting at the left
Several algebra textbooks teach one or both of the
following mnemonics to remember the Order of
Operations:
PEMDAS
Please Excuse My Dear Aunt Sally
Properties of our Number System
PROPERTY
ADDITION
MULTIPLICATION
Closure
a + b is a real number
ab is a real number
Commutative
a+b=b+a
ab = ba
Associative
a + (b + c) = (a + b) +c
a(bc) = (ab)c
Inverse
a + (-a) = 0
a • 1/a = 1
Identity
a+0=a
a•1=a
Distributive
a(b + c) = ab + ac
This Quick Reference Card prepared by Jolene M. Morris ([email protected])
Divisibility Rules
Knowing the divisibility
rules is not critical to your
knowledge of algebra;
however, the divisibility
rules will make it easier for
you to find factors, GCF,
LCM, and LCD.
2 = Even numbers (ending
in 0, 2, 4, 6, and 8)
3 = If repeated sums of the
digits result in 3, 6, or 9
4 = If the last two digits
are divisible by 4
5 = If the last digit is 0 or 5
6 = If the number is
divisible by both 2 and 3
7 = Repeatedly double the
last digit and subtract it
from the remaining digits -if the result is 0 or ±7
8 = If the last three digits
are divisible by 8
9 = If repeated sums of the
digits result in 9
10 = If the last digit is 0
11 = If the sum of the
every other digit in the
number minus the sum of
the alternate digits is
divisible by 11
Factors
Factors are numbers that
divide evenly into other
numbers -- without a
remainder. For example, 5
divides evenly into 40 so 5 is
a factor of 40.
We often create a factor tree
or a prime factorization of
numbers to help us recognize
the factors of a number:
GCF
LCM
The Greatest Common
Factor (GCF) of two or more
numbers is the largest
number that is a factor of
all the numbers.
The Lowest Common Multiple
(LCM) of two or more numbers
is the smallest number that is a
multiple of all the numbers.
One way to find the GCF is
to write the prime
factorization of each of the
numbers above each other.
Then "bring down" those
factors that are in common
and multiply them:
Find the GCF of 12, 18, 30:
12 = 2 × 2 × 3
Thus, the prime factorization
of 120 = 2 x 2 x 2 x 3 x 5
Check out the National Virtual
Manipulatives Web site to help
you create factor trees:
http://nlvm.usu.edu/en/nav/f
rames_asid_202_g_2_t_1.ht
ml
A knowledge of factors is
helpful when reducing
fractions and in Math 209
when you need to factor a
trinomial.
18 = 2
× 3 × 3
30 = 2
× 3
GCF = 2
× 3
The factors common to all
three numbers above are 2
and 3. 2 x 3 = 6 so 6 is the
GCF of 12, 18, and 30.
The GCF is used to simplify
(reduce) fractions to
lowest terms.
Composite & Prime Numbers
Our Number System

Complex numbers = a + bi
Imaginary numbers = i (the square root of negative one)
Real numbers = the numbers that are located on a number
line
Irrational numbers = all real numbers except the rational
numbers; those numbers that are square roots of nonsquare numbers or are non-terminating, non-repeating
decimal numbers ; examples are




5
and
 but not 64
because that is a square root of a squared number.
Rational numbers = the integers and fractional numbers;
those decimal numbers that terminate or repeat; examples
are 12.5 and 23.666666...
Integers = the counting numbers, their negatives, and zero
Whole numbers = the counting numbers and zero
Natural numbers = the counting numbers
9, 18, 27, 36, 45, …
12, 24, 36, 48, 60, …
18, 36, 54, 72, …
The LCM of 9, 12, and 18 is 36.
Another way to find the LCM is
to write the prime factorization
of each of the numbers above
each other with the factors all
lined up. Then "bring down"
one of each factor and multiply
them:
9 =
3 × 3
12 = 2 × 2 × 3
18 = 2 ×
3 × 3
LCM = 2 × 2 × 3 × 3 = 36
The LCM is used to find a
common denominator when
adding and subtracting
fractions.
Squared & Cubed Numbers
Prime numbers are those numbers that have no factors
except for 1 and the number itself. Composite numbers are
numbers that are not prime. Here is a list of prime numbers
from 2 to 1000:



× 5
One way to find the LCM is to
count by each of the numbers
and find the first number that is
a multiple of all. For example,
find the LCM of 9, 12, and 18: