Numbers
4.1 Order of Operations
 
4.2 Numbers
 
4.3 Factors
 
4.4 Multiples
 
4.5 Fractions
 
4.6 Decimals
 
 
4.7 Percents
 
4.8 Exponents
 
4.9 Roots and Radicals
 
4.10 Logarithms
 
4.11 Review Questions
 
4.12 Explanations
 
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:
eveneven = even
oddodd = odd
evenodd = 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:
positivepositive = positive
negativenegative = positive
positivenegative = negative
Division:
positivepositive = positive
negativenegative = positive
positivenegative = negative
The rules for multiplication and division are exactly the same since any division operation can be written as a form of multiplication: ab = 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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