### Number Definitions

#### Integers

- a number such as 0, 1, 2, and 3 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

- neither positive nor negative, zero is an integer and an even number

#### Rational number

- a number that can be expressed as an integer, decimal, or fraction

#### 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

#### Integers

- a letter used to represent one or more numbers. A variable
*expression*has numbers, variables, and operations

If you forget a rule (which happens all the time under test day pressure) consider using the strategies in The 7 Techniques chapter, like Experiments or Backsolving/Plug-In.

### Order of Operations (PEMDAS)

Take a look at the equation 5 + 2 × 3. Is the correct answer **11** or **21?**

- (5 + 2) × 3 = (7) × 3 = 21 (add first, then multiply?)
- 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 **P**arentheses – **E**xponents – **M**ultiplication – **D**ivision – **A**ddition – **S**ubtraction. An easy way to remember the order is by the initials **PEMDAS** or the mnemonic phrase **P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally.

#### Rule 1: Please Excuse My Dear Aunt Sally

**P**lease

**P**arentheses

**(-5 + 2)**(-3) = **(-3)**(-3) = 9

**E**xcuse

**E**xponents

**2 ^{2}** + 4 =

**4**+ 4 = 8

**M**y

**M**ultiplication

5 + **2 × 3** = 5 + **6** = 11

**D**ear

**D**ivision

**8 ÷ 2** + 2 = **4** + 2 = 6

**A**unt

**A**ddition

**4 + 6** – 10 = **10** – 10 = 0

**S**ally

**S**ubtraction

**7 – 5** – 2 = **2** – 2 = 0

Parentheses can be used for notating fractions and negative numbers.

One-half can be written as ½ or (1 / 2).

- 25 × (1/5) = 5
- 5 + (-2) = 3

#### Rule 2: Combine all like terms in an expression.

The expression *2x + x – y + 4y *can be simplified by combining the like terms* (x*‘s and *y*‘s*)* into *3x + 3y.*

#### Rule 3: Distribute numbers to eliminate parentheses.

**Note**: Since subtraction is the same as addition of a negative number, e.g. 5 – 3 = 5 + (-3) = -2, these operations can be calculated interchangeably. Left to right is not always the best order.

Example: 15 + 8 – 6 – 5 is easier when you do 15 – 5 + 8 – 6 = 10 + 8 – 6 = 10 + 2 = 12.

**Note**: Multiplication and division are also interchangeable.

Example: 18 × 2 ÷ 3 ÷ 2 is easier when you do 18 × 2 ÷ 2 ÷ 3 = 18 × 1 ÷ 3 = 18 ÷ 3 = 6

- 2(
*x-y*) = 2*x*– 2*y* - -3(
*x-y*) = -3*x*+ 3*y*

Be sure to distribute the negative to each term!

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

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

Example

16 ÷ (2(8 – 3(4 – 2))) + 3 =

### Solution

**16 ÷ 2(8 – 3(4 – 2)) + 3 = ?**

**Combine the inner parentheses: (4 – 2) = 2**

**16 ÷ 2(8 – 3(2)) + 3**

**Combine the next parentheses: 8 – 3(2) = 2**

**16 ÷ 2(2) + 3**

**Multiplication before division: multiply out 2(2) = 4**

**16 ÷ 4 + 3**

**Division, 16 ÷ 4 = 4 before addition**

**4 + 3 = 7**

Example

10 – 3(8 – 3)^{2}÷ 15 =

### Solution

**10 – 3(8 – 3) ^{2} ÷ 15 …Combine inside the parentheses**

**10 – 3(5) ^{2} ÷ 15 …Calculate the exponent**

**10 – 3(25) ÷ 15 …Multiply**

**10 – 75 ÷ 15 …Divide**

**10 – 5 = 5**

Example

|-2x^{2}| – (3x/2)^{2}× 2

### Solution

**| -2x ^{2} | – (3x/2)^{2} × 2 …Do operations inside grouping symbols: absolute value and parentheses**

**2x ^{2} – 9x^{2}/4 × 2 …Multiply**

**2x ^{2} – 9x^{2}/2 …Use the common denominator**

**4x ^{2}/2 – 9x^{2}/2 …Subtract to get a final expression**

**-5x ^{2}/2**

### Odd / 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, Even, and Prime Numbers

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 can provide a shortcut through questions like that:

- even + even = even (
*4 + 4 = 8*) - odd + even = odd (
*3 + 4 = 7*) - odd + odd = even (
*3 + 3 = 6*) - even – even = even (
*16 – 8 = 8*) - even – odd = odd (
*16 – 5 = 11)* - odd – odd = even (
*9 – 5 = 4*) - even × even = even (
*2 × 4 = 8*) - even × odd = even (
*2 × 3 = 6*) - odd × odd = odd (
*3 × 5 = 15*)

Example

Ifkis an odd integer, are the following expressions even or odd?

k + k + k … (k + k) is even. Thus (k + k) + k is an even plus an odd, which is odd.k × k × k … (k × k) is odd. Thus, (k × k) × k is an odd times an odd, which is odd.k + 2k … k + 2k is an odd plus an even, which is odd.2k × k … 2k is even. An even times an odd is even.

#### Dividing Even and Odd

Dividing even by even can be even, odd, or not an integer. (40 ÷ 8 = 5)

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.

- -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

### 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 **S**ame **S**igns product Po**S**itive.

positive × positive = positive (*6 × 2 = 12*)

positive × negative = negative (*6 × -2 = -12*)

negative × negative = positive (*-6 × -2 = 12*)

positive ÷ positive = positive (*6 ÷2 = 3*)

positive ÷ negative = negative (*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 x? Substitute -2 for^{2}positivexand 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

**and the**

*exponent***. The exponent says the number of times the base is used as a factor.**

*base*

Example

6^{4}= 6 × 6 × 6 × 6 = 1,296 (base 6, exponent 4)5^{2}= 5 × 5 = 25 (base 5, exponent 2)4^{3}= 4 ×4 ×4 = 64 (base 4, exponent 3)

The expression 6^{4} 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 5^{2} 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 4^{3} 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 -5^{2}= -(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 ofx?

You know 4 × 4 × 4 = 4^{3} = 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 in 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 2^{3}= 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 5^{2} = 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. A root must be positive, so √(x^{2}) = |x|.

Example

If x^{2}= 4, what is x?

### Solution

**Since 2 ^{2} = 4 and (-2)^{2} = 4, this answer is x = 2 and x = -2.**

**Alternative solution**

**x**^{2}= 4**√(x**^{2}) = √4**|x| = 2****x = 2 and x = -2 are the solutions of the equation.**

Note: this section is designed as a math review for students scoring in the low to middle ranges.

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