Before you take the Math IC, you should know the common types ofnumbers. Of these types, the most important ones to understand areprobably integers and real numbers. They can be spotted in nearly everyquestion on the test and will be explicitly mentioned at times.
- 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 wholenumbers, 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 number that can be expressed in the form m⁄n , where m and nare integers. The set of rational numbers includes all integers and allfractions that can be created using integers in the numerator anddenominator.
- Irrational Numbers. The set of all numbers that cannot be expressed as a quotient of integers. Examples include π,
,1.01001000100001000001 . . . . The sets of irrational numbers andrational numbers are mutually exclusive. Any given number must beeither 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. See the “Miscellaneous Math” chapter later in this book.
On the Math IC, integers and real numbers will appear far more often than any of the other number 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, meaningdecimals and fractions are not included. Zero, however, is an integerand thus a member of the set.
. . . , –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 2
n, where
n is an integer. Odd numbers are the numbers that can be written in the form 2
n + 1, where n is an integer.
This definition is nothing more than atechnical repetition of the fact that even numbers are divisible bytwo, and odd numbers are not. It may come in handy, though, when youneed to represent an even or odd number with a variable.
Operations of Odd and Even Numbers
There are a few basic rules regarding theoperations of odd and even numbers that you should know well. If yougrasp the principles behind the two types of signed numbers, theserules should all 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 byrules similar to those that have to do with even and odd numbers.First, for their quick definitions:
Positive numbers are numbers that are
greaterthan zero. Negative numbers are numbers that are less than zero. Thenumber 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. 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 divisionare exactly the same since any division operation can be written as aform of multiplication:
a
b = a/b =
a
1/
b.
Absolute Value
The absolute value of a number is the distanceon a number line between that number and zero. Or, you could think ofit as the positive “version” of every number. The absolute value of apositive number is that same number, and the absolute value of anegative number is the opposite of that number.
The absolute value of
x is symbolized by |
x|.
Solving an equation with an absolute valuein it can be particularly tricky. As you will see, the answer is oftenambiguous. 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 = –2are both possible solutions to the problem. Keep this in mind; we’lldeal more with absolute values in equations later on in the Algebrachapter.
