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Definitions

Integers

  • a number such as -1, 0, 1, and 2 that has no fractional part.
  • Integers can be positive {1, 2, 3, …}, negative {-1, -2, -3, …}, or zero {0}.
  • The GMAT will define whole numbers as positive integers and zero {0, 1, 2, 3, …}.
  • In other sources the term “whole numbers” may refer to all integers or just positive integers.

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. Zero is an integer and an even number.

Rational number

  • a number that can be expressed as a fraction of two integers.

Irrational number

  • a number, such as π or √2, that cannot be expressed as a fraction of two integers.

Division and Zero

  • Addition, subtraction, and multiplication of integers will always result in an integer.
  • However, division is different. Sometimes a quotient is not an integer, though it is a rational number:
    12 ÷ 4 = 3, but 4 ÷ 12 = 1/3
  • The result of a number divided by zero is undefined. For example, 8/0 = ? since there is no number 0 × ? = 8. Zero divided by a number is equal to zero. For example, 0/3 = 0 since 0 × 3 = 0.

Variable

  • a letter used to represent one or more numbers. A variable expression has numbers, variables, and operators.

Order of Operations (PEMDAS)

Order of Operations

Video Courtesy of Kaplan GMAT

  • Take a look at:
    5 + 2 × 3
  • (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 you need to know the “Order of Operations,” which is the priority list for calculations.

The correct order of operations is Parentheses – Exponents – Multiplication – Division – Addition – Subtraction. An easy way to remember the order is by the initials PEMDAS or the mnemonic phrase Please Excuse My Dear Aunt Sally.

Rule 1: 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

Parentheses can be used for notating fractions and negative numbers.

One-half can be written as \dfrac{1}{2} or (1 / 2).

  • \dfrac{8 + 4}{5 \,–\, 2} = (8 + 4) / (5 – 2) = 12/3 = 4
  • 25 × (1/5) = 5
  • 5 + (–2) = 3

Note:
Addition and subtraction have equal precedence and subtraction is the same as addition of a negative number.
5 – 3 = 5 + (-3) = 2.
When you change all of your subtraction to addition by negatives, these operations can be done interchangeably.
Example: 15 + 8 ? 6 ? 5 is easier when you do 15 + (-5) + 8 + (-6) = 10 + 2 = 12.
Just make sure you first change subtraction to addition of negatives if you change the order (8 – 6 is not the same as 6 – 8).

Similarly, multiplication and division have equal precedence and division is the same as multiplying by the reciprocal.
When you change all your division to multiplication by reciprocals, then these operations can be done interchangeably.
Example: 18 × 2 ÷ 3 ÷ 2 is easier when you do 18 × 2 × 1/2 × 1/3 = 18 × 1 × 1/3 = 18 ÷ 3 = 6.
Just make sure you first change division to multiplication by reciprocals if you change the order (18 ÷ 3 is not the same as 3 ÷ 18).

Rule 2: 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 3: Distribute numbers to eliminate parentheses.

  • 2(x – y) = 2x – 2y
  • -3(x – y) = -3x + 3y      Be sure to distribute the negative to each term.

Rule 4: Evaluate expressions' grouping symbols from the inside out.

  • x(x + 2(3x + 4) – 3)
  • x(x + 6x + 8 – 3) …Distribute 2 into the inner parentheses: 2(3x + 4) = 6x + 8
  • x(7x + 5) …Combine the like terms (the x’s) inside the parentheses
  • 7x2 + 5x … Distribute to remove the last parentheses.

Example

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 next parentheses: 8 – 3(2) = 2
  • 16 ÷ 4 + 3 …Multiplication before division: multiply out 2(2) = 4
  • 4 + 3 = 7 …Division, 16 ÷ 4 = 4 before addition

Example

10 – 3(8 – 3)2 ÷ 15 =

Solution

  • 10 – 3(8 – 3)2 ÷ 15
  • 10 – 3(5)2 ÷ 15 …Combine inside the parentheses
  • 10 – 3(25) ÷ 15 …Calculate the exponent
  • 10 – 75 ÷ 15 …Multiply
  • 10 – 5 = 5 …Division, 75 ÷ 15 = 5, before subtraction.

Example

| -2x2| – (3x/2)2 × 2

Solution

  • | -2x2 | – (3x/2)2 × 2
  • 2x2 – 9x2/4 × 2 …Do operations inside grouping symbols: absolute value and parentheses
  • 2x2 – 9x2/2 …Multiply
  • 4x2/2 – 9x2/2 …Use the common denominator
  • -5x2/2 …Subtract to get a final expression

Odd and Even Numbers

Even number

  • an integer that is divisible by 2 (…, -6, -4, -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 (…, -5, -3, -1, 1, 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 a prime number since it has only one positive divisor.

Only integers can be odd, even, or prime numbers.

Adding, Subtracting, and Multiplying Odd and Even

On the GMAT, you may be asked to add, subtract, or multiply very large numbers in order to answer a relatively simple question. These rules for odd and even can provide a shortcut through questions like that:

RuleExample
even + even = even4 + 4 = 8 (even)
odd + even = odd3 + 4 = 7(odd)
odd + odd = even3 + 3 = 6 (even)
even – even = even16 – 8 = 8 (even)
even – odd = odd16 – 5 = 11 (odd)
odd – odd = even9 – 5 = 4 (even)
even × even = even2 × 4 = 8 (even)
even × odd = even2 × 3 = 6 (even)
odd × odd = odd3 × 5 = 15 (odd)

Example

If k is an odd integer, are the following expressions even or odd?

  • 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.

Dividing Even and Odd

examples:

Dividing even by even can be even, odd, or not an integer.

            40 ÷ 4 = 10      40 ÷ 8 = 5      8 ÷ 40 = 1/5

Dividing even by odd can be even, odd, or not an integer.

            30 ÷ 5 = 6      30 ÷ 6 = 5      15 ÷ 30 = 1/2

Dividing odd by odd cannot be even. 15 ÷ 3 = 5
Dividing odd by even cannot be an integer. 15 ÷ 2 = 7.5

Positive and Negative Numbers

All numbers, except zero, are positive or negative.

Addition and Subtraction

  • Adding two numbers can give a positive or a negative number.
  • positive + positive = positive
    negative + negative = negative

Positive + negative can give a positive, a negative, or zero.

examples:   -7 + 9 = 2  4 + (-5) = 4 – 5 = -1  2 + (-2) = 0

Subtracting two numbers can give a positive, a negative, or zero.

positive – positive   9 – 7 = 2  7 – 9 = -2  7 – 7 = 0
negative – negative  -9 – (-7) = -2  -7 – 9 = -16  -9 – (-9) = 0
positive – negative   9 – (-7) = 9 + 7 = 16
negative – positive   -9 – 7 = -16

Be sure to distribute the negative sign to all terms inside parentheses.

9 – (14 – 8) = 9 – 14 + 8 = 3

10 – (5 + 2) = 10 – 5 – 2 = 3

You can use the order of operations to check:

9 – (14 – 8) = 9 – 6 = 3

10 – (7) = 3

  • 9 – (14 – 8) = 9 – 6 = 3
  • 10 – (7) = 3

Multiplication and Division

When multiplying or dividing: if the two numbers have the same sign, then the product is positive. If the two numbers have different signs, the product is negative. A way to remember this is Same Signs product PoSitive.

positive × positive = positive

6 × 2 = 12

positive × negative = negative

6 × (-2) = -12

(-6) × 2 = -12

negative × negative = positive

(-6) × (-2) = 12

positive ÷ positive = positive

6 ÷ 2 = 3

positive ÷ negative = negative

6 ÷ (-2) = -3

(-6) ÷ 2 = -3

negative ÷ negative = positive

(-6) ÷ (-2) = 3

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? Substitute -2 for x and you can see that (-2)2 is positive. This is also very useful for double checking.

Make a list. 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 make a list of all the number possibilities rather than try to solve the question using algebra.

Absolute Value

The absolute value of a number, written as | -5 |, is the distance of a number from zero on the number line: | +5 | = | -5 | = 5.

The value of an expression inside the absolute value signs could be positive or negative.
If | x | = 7, then x could be 7 or -7.

This means that absolute value equations usually have two possible solutions.

Example

Solve |x + 3| = 8

Solution

x + 3 = 8  or  x + 3 = -8 …Rewrite as two equations

x = 5  or x = -11 …Two solutions

Powers

Note: Powers are covered more completely in Section 6, Exponents.

Writing a number raised to a power is a way of expressing multiplication of a number by itself. A power has two parts, the exponent and the base. The exponent says the number of times the base is used as a factor.

Example

  • 64 = 6 × 6 × 6 × 6 = 1,296 …base=6, exponent=4
  • 52 = 5 × 5 = 25 …base=5, exponent=2
  • 43 = 4 ×4 ×4 = 64 …base=4, exponent=3

The expression 64 is read as “six to the fourth.”

Two powers have special names. Powers with the exponent 2 are called “squared,” since it can imply area. The expression 52 can be read as “five to the second power” or “five squared.”

Powers with the exponent 3 are called “cubed,” since it can imply volume. The expression 43 can be read as “four to the third power” or “four cubed.”

Powers of negative numbers

When using exponents with negative numbers, make sure your answer has the right sign. Pay careful attention to whether a negative sign is inside or outside of parentheses.

Example

  • (-4)3 = (-4) × (-4) × (-4) = -64
  • (-5)2 = (-5) × (-5) = 25  but  -52 = -(5 × 5) = -25

Roots

Just like addition and subtraction “undo” each other, so do exponents and roots.

You take a root to answer questions like:
If 3√64 = x, what is the value of x?
You know 4 × 4 × 4 = 43 = 64, so 3√64 = 4.

The expression √a is called a root or radical. The symbol √ is the radical symbol.
Radical symbols can have a number in front of the radical. The 3 in 3√64 is asking for the cube root. For square roots, the 2 is left off.

Example

  • 3√8 = 2 since 23 = 8
  • 3√(-1) = -1 since (-1)3 = -1

Odd and Even Roots

When taking an odd root, there is only one answer.

Example

  • 3√125 = 3√(5 × 5 × 5) = 5
  • 3√-125 = 3√((-5) × (-5) × (-5)) = -5
  • 5√32 = 5√(2 × 2 × 2 × 2 × 2) = 2

When taking an even root, there will be two answers. But the sign √ denotes only the non-negative answer. For example, there are two square roots of 25: 5 and -5, since 52 = 25 and (-5)2 = 25. But √25 has only one value: 5, √16 = 4, √1 = 1, etc.

Example

  • √16 = √(4 × 4) = 4
  • 3√1 = 3√(1 × 1 × 1) = 1
  • 4√16 = 4√(2 × 2 × 2 × 2) = 2

Be careful when dealing with equations – remember that an even root is undefined for negative numbers.
For example, √(-4) is undefined because there is no number squared that equals -4.

Example

If x2 = 4, what is x?

Solution

Since 22 = 4 and (-2)2 = 4, this answer is x = 2 and x = -2.

Alternative solution

  • x2 = 4
  • √(x2) = √4
  • |x| = 2
  • x = 2 and x = -2 are the solutions of the equation.
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