Numbers
Before you take the Math IIC, you should become familiar
with some common types of numbers. Understand their properties and
you will be well served.
- Whole Numbers: the set of counting
numbers, including zero {0, 1, 2, 3, . . .}.
- Natural Numbers: the set of all whole numbers
except zero {1, 2, 3, 4, 5, . . .}.
- Integers: the set of all positive and negative
whole numbers, including zero. Fractions and decimals are not included
{. . . , –3, –2, –1, 0, 1, 2, 3, . . .}.
- Rational Numbers: the set of all numbers
that can be expressed as a quotient of integers. That is, any numbers
that can be expressed in the form m
/n,
where m and n are integers. The
set of rational numbers includes all integers, and all fractions
that can be created using integers in the numerator and denominator.
- Irrational Numbers: the set of all numbers
that cannot be expressed as a quotient of integers. Examples include
, 
,
1.01001000100001000001. . . . The sets of irrational numbers and
rational numbers are mutually exclusive. Any given number must be
either rational or irrational; no number can be both.
- Real Numbers: every number on the number
line. The set of real numbers includes all rational and irrational
numbers.
- Imaginary Numbers: numbers that do not appear
on the real number line. We explain the ways in which imaginary
numbers appear on the Math IIC in the Miscellaneous Math chapter
later in this book.
On the Math IIC, integers and real numbers will appear
far more often than any of the other types.
Even and Odd Numbers
Even numbers are those numbers that are divisible by two
with no remainder.
Only integers can be even or odd, meaning decimals and
fractions cannot be even or odd. Zero, however, is an integer and
divisible by two, so it is even.
. . . , –6, –4, –2, 0, 2, 4, 6, . . .
Odd numbers are those numbers not evenly divisible by
two.
. . . , –5, –3, –1, 1, 3, 5, . . .
The set of even numbers and the set of odd numbers are
mutually exclusive.
A more rigorous definition of even and odd numbers appears
below:
- Even numbers are numbers that can be written
in the form 2n, where n is an integer.
Odd numbers are numbers that can be written in the form 2n +
1, where n is an integer.
This definition is nothing more than a technical repetition
of the fact that even numbers are divisible by two and odd numbers
are not. It may come in handy, though, when you need to represent
an even or odd number with a variable.
Operations of Odd and Even Numbers
There are a few basic rules regarding the operations of
odd and even numbers that you should know well. If you grasp the
principles behind the two types of signed numbers, these rules should
come easily.
Addition:
even + even = even
odd + odd = even
even + odd = odd
Subtraction:
even – even = even
odd – odd = even
even – odd = odd
Multiplication and Division:
even
even = even
odd
odd = odd
even
odd = even
Positive and Negative Numbers
Positive and negative numbers are governed by rules similar
to those associated with even and odd numbers. First, for their
quick definitions:
- Positive numbers are numbers that are greater
than zero. Negative numbers are numbers that are less than zero.
The number zero is neither positive nor negative.
Operations of Positive and Negative Numbers
The following rules define how positive and negative numbers
operate under various operations.
Addition and Subtraction
When adding and subtracting negative numbers, it helps
to remember the following:
Adding a negative number is the same as subtracting its
opposite. For example:
Subtracting a negative number is the same as adding its
opposite. Again, for example:
Multiplication:
positive
positive = positive
negative
negative = positive
positive
negative = negative
Division:
positive
positive = positive
negative
negative = positive
positive
negative = negative
The rules for multiplication and division are exactly
the same since any division operation can be written as a form of
multiplication:
a
b =
a/b = a 
1/b.
Absolute Value
The absolute value of a number is the distance on a number
line between that number and zero. Or, you could think of it as
the positive “version” of every number. The absolute value of a
positive number is that same number, and the absolute value of a
negative number is the opposite of that number.
The absolute value of x is symbolized
by |x|.
Solving an equation with an absolute value in it can be
particularly tricky. As you will see, the answer is often ambiguous.
Take a look at the following equation:
We can simplify the equation in order to isolate |x|:
Knowing that |x| = 2 means that x =
2 and x = –2 are both possible solutions to the
problem. Keep this in mind; we’ll deal more with absolute values
in equations later on in the Algebra chapter.
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