Number Rules
Consecutive Numbers
Divisibility
Fractions
Decimals
Exponents
Roots & Radicals
Note: this section is designed as an introduction for students scoring in the middle to low ranges.
Number Definitions
Integer
a member of the set of positive whole numbers {1, 2, 3, . . . }, negative whole numbers {-1, -2, -3, . . . }, and zero. Fractions and decimals are not integers. Integers are also called "whole numbers".
Positive number
a number greater than zero, such as +5 (usually written simply as 5).
Negative number
a number less than zero, such as -5.
Zero
is neither positive nor negative and is an integer.
Rational number
a number that may be expressed as integers, decimals or fractions (a ratio of integers), as opposed to an...
Irrational number
a number that cannot be expressed by the ratio of two integers, such as π or √2. Very mysterious! Note: anything divided by zero is irrational.
If you forget a rule (this happens all the time under test day pressure) consider using Experiments or Backsolving/Plug-In to get around it (we discuss these advanced strategies in II. General Math Strategies).
ORDER OF OPERATIONS - PEMDA
Take a look at:
5 + 2 × 3, what is it?
(5 + 2) × 3 = (7) × 3 = 21
Add first, then multiply
or?
5 + (2 × 3 ) = 5 + (6) = 11
Multiply first, then add
To address this issue we have the "Order of Operations", which is the priority list for calculations. Parentheses go first, and then followed by exponents. Multiplication and division are next, but are interchangeable depending on which comes up first when going from left to right in a problem. The same is true for addition and subtraction. There is a handy mnemonic to memorize these rules.
Rule I: Please Excuse My Dear Aunt Sally
Please
Parentheses
(-5 + 2)(-3) = (-3)(-3) = 9
Excuse
Exponents
22 + 4 = 4 + 4 = 8
My
Multiplication
5 + (2 × 3) = 5 + (6) = 11
Dear
Division
8 ÷ 2 + 2 = 4 + 2 = 6
Aunt
Addition
4 + 6 - 10 = 10 - 10 = 0
Sally
Subtraction
7 - 5 - 2 = 2 - 2 = 0
Rule II: Combine all like terms in an expression.
2x + x - y + 4y
3x + 3y
The expression is simplified by combining the like terms (x's and y's).
Rule III: Distribute numbers to eliminate parenthesis. This means multiply everything within the parenthesis.
2(x - y) = 2x - 2y
-2(x - y) = -2x + 2y
Multiply the negative through to make a positive number.
Rule IV: Eliminate inner parentheses first and the outermost parentheses last.
In the expression:
combine like terms (the x's) inside the parenthesis.
7x2 + 5x
remove the last parentheses through distribution.
Example
Simplify 16 ÷ 2(8 - 3(4 - 2)) + 3
Solution
16 ÷ 2(8 - 3(4 - 2)) + 3
16 ÷ 2(8 - 3(2)) + 3
combine the inner parentheses: (4 - 2) = 2
16 ÷ 2(2) + 3
combine the inner parentheses: 8 - 3(2) = 2
16 ÷ 4 + 3
multiply out the 2(2) = 4
4 + 3 = 7
16 ÷ 4 = 4. Answer = 7
Example
Simplify 16 - 3(8 - 3)2 ÷ 15
Solution
16 - 3(8 - 3)2 ÷ 15
16 - 3(5)2 ÷ 15
combine the inner parentheses: (8 - 3) = 5
16 - 3(25) ÷ 15
multiply out the exponent: (5)2 = 25
16 - 75 ÷ 15
multiply out 3(25) = 75
16 - 5 = 11
75 ÷ 15 = 5. Answer = 11
Odd / Even Rules
Even number
an integer that is divisible by 2 (0, 2, 4, 6, 8, 10). No, don't email us saying that's a typo, zero is an even integer.
Odd number
an integer not divisible by 2 (3, 5, 7, 9).
Prime number
a positive integer with exactly two different positive divisors: 1 and itself. For example, 2, 3, 5, 7 are prime numbers. Note: 1 is not considered a prime number since it has only one positive divisor.
Adding, Subtracting and Multiplying Odd/Even: The following is true of even and odd whole numbers (use example numbers in your mind to illustrate).
Rule
Example
Even + Even = Even
4 + 4 = 8 (even)
Odd + Even = Odd
3 + 4 = 7(odd)
Odd + Odd = Even
3 + 3 = 6 (even)
Even × Even = Even
2 × 2 = 4 (even)
Even × Odd = Even
2 × 3 = 6 (even)
Odd × Odd = Odd
3 × 3 = 9 (odd)
Even - Even = Even
16 - 8 = 8 (even)
Even - Odd = Odd
16 - 5 = 11 (odd)
Odd - Odd = Even
9 - 5 = 4 (even)
Examples (with k as an odd number)
a) k + k + k
(k + k) is even. Thus (k + k) + k is an even plus an odd, which is odd.
b) k × k × k
k × k is odd. Thus (k × k) × k is an odd times an odd, which is odd.
c) k + 2k
k + 2k is an odd plus an even, which is odd.
d) 2k × k
2k is even. An even times an odd is even.
Positive/Negative Rules
Adding a negative number is the same as subtraction.
4 + (-5) = 4 - 5 = -1
Subtracting a negative number is the same as addition.
4 - (-5) = 4 + 5 = 9
ALWAYS REMEMBER: DOUBLE NEGATIVE = POSITIVE. If you see two negative signs you can cancel them out and make a plus sign.
3 + 4 - (4 - 6)
3 + 4 - (-2)
Combine numbers in parenthesis (4 - 6).
3 + 4 + 2 = 9
double negative = positive: - (-2) = + 2.
Multiplication/Division When multiplying or dividing: if they have the same sign, then it is POSITIVE. If they have different signs, then it is NEGATIVE.
Positive × Positive = Positive
2 × 2 = 4
Positive × Negative = Negative
2 × -2 = -4
Negative × Negative = Positive
-2 × -2 = 4
800score Strategies
Use Plug In or Backsolving on odd/even and positive/negative questions.
Make up numbers to substitute in for variables on questions that ask if variables result in a positive or negative number. For example, if x < 0, is x2 positive? Sub in -2 for x and you can see that (-2)2 is positive. This is also very useful for double checking.
Jot down all possibilities when dealing with discrete sets If a positive, negative or prime number question asks "How many two digit prime numbers" or "How many two digit numbers less than 50" then you have a finite (small) number of possibilities. Here it is useful to sometimes make a list of all the number possibilities rather than try to solve the question using algebra.
If you forget an odd/even addition/subtraction rule, use Experiments to derive it in the middle of the test.
Absolute Value
Theabsolute value of a number, written as | -5 |, is the distance of a number from zero on the number line:
| +5 | = | -5 | = 5.
This means that absolute value questions usually have two possible answers (one positive and one negative).
|x + 5| = 1
x + 5 = 1
x + 5 = -1
To get rid of the absolute value symbol, create negative (-1) and positive scenarios (+1). Subtract 5 from both sides of equations to solve.
x = -4
x = -6
Two solutions: x = -4/-6
Powers
Writing a number raised to a certain power
is a shorthanded way of expressing multiplication of a number by itself.
For example:
52 = 5 × 5 = 25
43 = 4 × 4 × 4 = 64
In these two cases, 5 and 4 are the respective
bases, and 2 and 3 are the powers,
or the number of times we multiply a number by itself.
