Number Properties
Number Definitions
Real numbers: All numbers that do not involve i or the square root of a negative number. In
other words, all numbers other than complex or imaginary numbers. Real numbers include all
positive numbers, negative numbers, and zero, and include decimals and fractions.
Integers: Positive and negative whole numbers and zero. …-2, -1, 0, 1, 2…
Rational numbers: Numbers that can be expressed as the ratio of two integers. Another way of
saying the same thing is that rational numbers include all integers, all fractions, and all numbers
that include a terminating or repeating decimal. 8, -3, 0, -.555…, 4/3, -1/2, 3.489489489…
Irrational numbers: Numbers that do not fit the above definition for rational numbers. When
written as a decimal, irrational numbers neither terminate nor repeat. In other words, the
decimal goes on infinitely with no repeating pattern. Irrational numbers fall into two main
categories: (1) special numbers such as and e, and (2) roots of numbers that do not resolve to
integers such as √ , √ , 3√ , and√ .
Prime numbers: Numbers greater than one that have no factors other than itself and one. Two
is the only even prime number. Prime numbers include 2, 3, 5, 7, 11, 13, and 17.
Imaginary numbers: Numbers involving i, such as i, 3i, and -2i. By definition, i = √
that i2 is a real number because i2 = -1.
. Note
Complex numbers: Numbers that have a real component and an imaginary component, such as
3 + 4i and 5-2i.
Non-negative numbers: All positive numbers and zero.
Non-positive numbers: All negative numbers and zero.
Factors: Numbers that another number can be evenly divided by to form an integer. Factors of
a number include the number itself and 1. For instance, the factors of 24 are 1, 24, 2, 12, 3, 8,
4, and 6.
Multiples: A multiple of a number can be formed by multiplying the number by any positive
integer. For instance, multiples of 8 include 8, 16, 24, 32, 40, and so on.
Number Properties
even
even
even = even
odd = odd
odd odd = even
even • even = even
even • odd = even
odd • odd = odd
even number raised to any integer power is even
odd number raised to any integer power is odd
positive • or positive = positive
positive • or negative = negative
negative • or negative = positive
positive number raised to any power is positive
negative number raised to an even power is positive
negative number raised to an odd power is negative
Find the Even
When asked which expression must be even for all integer values of x, your target is x or a
positive integer power of x multiplied by an even number, since an even number times any
integer must be even. It is optional to then add or subtract an even number, as doing so will
keep the expression positive. Do not make the common mistake of assuming that x2 must
always be even; remember that odd numbers squared are odd. Examples of expressions that
must be even for all integer values of x include 4x, 6x2, 2x3, 8x5 – 2, and 12x3 + 20.
Find the Odd
When asked which expression must be odd for all integer values of x, begin by finding an
expression that must be even by multiplying x or a positive integer power of x by an even
number. Then, add or subtract an odd number to make the expression odd. Examples of
expressions that must be odd for all integer values of x include 2x – 1, 4x2 + 3, 6x5 – 7, and 8x +
5.
Find the Positive
When asked which expression must be positive for all values of x, your target is x raised to an
even power, since any nonzero number raised to an even power must be positive. It is optional
to then multiply or divide by a positive number, as doing so will keep the expression positive.
Examples of expressions that must be positive for all nonzero values of x include x2, x4, 6x20,
10x2, and x2/4.
Find the Negative
When asked which expression must be negative for all values of x, begin by finding an
expression that must be positive by raising x to an even power. Then, multiply or divide by a
negative number to make the expression negative. Examples of expressions that must be
negative for all nonzero values of x include -3x2, -5x6, -x10, and -3x24/5.
Find the expression that must be true given another equation or inequality.
Often, you are asked to find which expression must be true given a certain equation or
inequality in the question. On this type of problem, think carefully about what you know given
the equation or inequality in the question before looking at the answer choices. Usually, you
should know exactly what your target is before ever looking at the answer choices. Often, an
algebraic manipulation of the initial expression is useful in figuring out your target. Observe the
following two examples.
Example: If 8a6b5 0, which of the following statements must be true?
Before looking at the answer choices (not included here as they are not truly relevant to how
you should approach the problem), figure out a target based on the inequality in the question.
The expression on the left hand side of the inequality must be negative. Because a 6 must be
positive for all a, 8a6 must be positive too. Therefore, b5 must be negative, so b itself must be
negative. Your target answer is going be b 0 or some variant thereof, such as 6b 0 or
8b3 0.
Example: If a4 + 5 is odd, which of the following statements must be true?
Just as in the previous example, before looking at the answer choices (not included here as they
are not truly relevant to how you should approach the problem), figure out a target based on
the information in the question. If a4 + 5 is odd, a4 must be even, which is only the case when a
itself is even. Therefore, your target answer is either “a is even” or some other statement that
must be true if a is even, such as “a – 4 is even” or “a3 + 1 is odd.”