Download Classifying Numbers

Chapter 1
Classifying Numbers
Tip
How is 0 classified?
There are an awful lot of numbers—ranging from
familiar numbers used to count everyday things,
like two apples, to huge abstract amounts that
one can barely comprehend, such as Bill Gates’
net worth. That vast range, however, only
encompasses a miniscule fraction of the numbers
that exist. Ironically, you cannot count the
number of numbers. That being the case,
mathematicians have classified numbers into
more manageable groups, or number systems.
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2
3
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POSITIVE NUMBERS
• A positive number is a
number that is greater
than 0 .
• Positive numbers
become larger the
farther they are from
zero. For example, 6 4
is larger than 22 .
16
5
6
7
8
Talk about a loaded question.
Zero is a number with a lot
of peculiar properties that
mathematicians have been
debating for centuries. For our
purposes, 0 can be classified as
an even number because it can
be evenly divided by 2. In terms
of being classified as positive or
negative, 0 sits in a numeric
limbo between the groups and
is considered neither positive
nor negative.
You will already be familiar with some of these
groups, like positive, negative and odd numbers,
while others may be new concepts, such as prime
numbers, integers and irrational numbers. It is
good to be on friendly terms with all of these
number systems as they are all utilized in
algebra. Many have quirky traits and
relationships to other numbers that will come
in very handy as you learn more about them.
Define each of the following statements
as true or false. You can check your
answers on page 250.
1) 5 is a positive, prime and composite
number.
2) 3 is a positive, odd and prime number.
3) –20 is a negative, odd and composite
number.
4) –7 is a negative, odd and prime
number.
5) 24 is a positive, even and composite
number.
6) 43 is a positive, odd and composite
number.
Positive Number Examples
0
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Algebra Basics
Even Number Examples
Prime Number Examples
2, 4, 6, 8, 10, –2, –4, –6, –8, –10
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Odd Number Examples
Composite Number Examples
1, 3, 5, 7, 9, –1, –3, –5, –7, –9
4, 6, 8, 10, 12, 14, 15, 16, 18
Negative Number Examples
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• A positive number can
either be written with a
positive sign (+) in front
of the number or no sign
at all. If a sign does not
appear in front of a
number, consider the
number to be a positive
number. For example, +4
can also be written as 4.
-10
-9
-8
-7
-6
NEGATIVE NUMBERS
• A negative number is
a number that is less
than 0.
• Negative numbers
become smaller the
farther they are from
zero. For example, –6 4
is smaller than –22.
-5
-4
-3
-2
-1
0
• A negative number is
always written with a
minus sign (–) in front
of the number.
EVEN NUMBERS
• An even number is a
number that you can
evenly divide by 2.
ODD NUMBERS
• An odd number is
a number that you
cannot evenly divide
by 2 . When you divide
an odd number by 2,
you get a left over
value, known as the
remainder.
PRIME NUMBERS
•
COMPOSITE NUMBERS
A prime number is a
positive number that
you can only evenly
divide by itself and
the number 1.
• A composite number is
Note: The number 1 is
not considered a prime
number since it can only
be evenly divided by one
number. The number 2
is the smallest prime
number.
• The numbers you can
a number that you can
evenly divide by itself,
the number 1 and one
or more other numbers.
evenly divide into another
number are called factors.
For example, the factors
for the number 1 2 are 1 ,
2 , 3 , 4 and 6.
CONTINUED
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Chapter 1
Classifying Numbers
continued
Tip
Why do some numbers
have a line drawn over
the decimal places?
Think of these number classification groups as if
they were clubs. Some clubs are relatively small
because very few people meet their criteria for
membership. Other clubs are huge due to the
fact that they will let just about anybody join in.
The prime number club, for example, is relatively
exclusive and far smaller than the real number
club, which will accept just about any number
that walks in off the street.
A line drawn over decimal
places indicates a repeating
pattern. For instance, 16
equals 0.166666666666... ,
with the 6 s repeating
on forever. To make the
number easier to handle,
it is expressed as follows:
It’s important to remember, however, that just
because a number belongs to an elite club,
that does not keep it from belonging to other
less-exclusive groups. In fact, as you are likely
beginning to notice, most numbers usually
meet the criteria to belong to a variety of
these number clubs. The number 5 for example
is a prime number, but it is also a positive
number, a whole number and a real number.
ctice
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Algebra Basics
Classify each of the following numbers
into the natural, whole, integer, rational,
irrational and real number groups. You
can check your answers on page 250.
1) –1
2)
√ 34
3) –18
4)
1
0.16
5) 3
6) 0
Rational Number Examples
Natural Numbers
5, –5, 7, 55, 12 , 13 , 34
1, 2, 3, 4, 5, 6, 7, 8, 9, 10....
9.25, 5.23, 0.2424242424...
Real Number Examples
Integer Examples
3, –3, 0, –55, 23 , 16 , 7.35, 4.1313...,
0, 1, 2, 3, 4, 5, –1, –2, –3, –4, –5
Whole Numbers
Irrational Number Examples
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10....
(pi),
√2
√3
NATURAL NUMBERS
WHOLE NUMBERS
• Natural numbers are
• Whole numbers are the
• Natural numbers are
• Whole numbers are the
the numbers 1, 2 , 3, 4, 5
and so on.
also called the counting
numbers.
numbers 0 , 1 , 2, 3 , 4 , 5
and so on.
same as natural numbers
with the addition of the
number 0 .
Note: Some mathematicians
consider 0 to be a natural
number instead of a whole
number.
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√ 2, INTEGERS
• An integer is a whole
number or a whole number
with a negative sign (–) in
front of the number.
(square root of 3 )
RATIONAL NUMBERS
• Rational numbers
(square root of 2 ),
include integers and
fractions. A rational
number is also a
number that you can
write with decimal
values that either end
or have a pattern that
repeats forever.
IRRATIONAL NUMBERS
• Irrational numbers do
not include integers or
fractions. An irrational
number has decimal
values that continue
forever without a
pattern.
REAL NUMBERS
• Real numbers include
natural numbers,
whole numbers,
integers, rational
numbers and
irrational numbers.
• Real numbers can
include fractions as
well as numbers with or
without decimal places.
• The most well-known
irrational number is
pi ( ), which is equal
to 3. 141 59 26 . . .
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