Almost all of mathematics involves the use of the four basic mathematical operations--addition, subtraction, multiplication, and division. Understanding these basic mathematical operations is crucial to everything covered both in Pre-Algebra and in more complicated mathematics. It is impossible to master the complex principles of Pre-Algebra without first mastering the operations and their properties.

You are probably used to working with the four basic operations, but there are some things about these operations that you may not know. In particular, these operations have certain properties that can make evaluating complex expressions a lot easier.

The first section will explain how to correctly evaluate a complicated expression using the Order of Operations, which specifies the order in which to carry out operations when evaluating an expression. The Order of Operations is important to know; if you do not follow it correctly and instead carry out the operations in the incorrect order, your answer will also be incorrect.

Section two will teach some properties of addition that will make it easier to evaluate an expression without depending on a calculator. These properties are the Commutative Property, the Associative Property, and the Identity Property.

The third section will teach some properties of multiplication. Like addition, multiplication has its own version of the Commutative Property, the Associative Property, and the Identity Property. Multiplication has two additional properties--the Zero Product Property and the Distributive Property.

The fourth and final section will discuss inverse operations, which "reverse" other operations. These will be especially useful for future algebra.

Each section will teach something about basic operations that will help you evaluate expressions correctly and easily. These properties will also be useful when you approach more difficult topics in pre-algebra, such as solving an algebraic equation for a variable.

Associative Property of Addition - For any numbers a , b , and c , it is true that (a+b)+c = a+(b+c) .

Associative Property of Multiplication - For any numbers a , b , and c , it is always true that (a ×b) ×c = a ×(b ×c) .

Commutative Property of Addition - For any numbers a and b , it is true that a+b = b+a .

Commutative Property of Multiplication - For any numbers a and b , it is true that a ×b = b ×a .

Commutative Property of Multiplication

For any numbers a and b , it is true that a ×b = b ×a .

Distributive Property of Multiplication - For any numbers a , b , and c , it is true that a ×(b+c) = (a ×b)+(a ×c) .

Expression - A representation of a number. 5+3 and 10-2 are both expressions that represent the number 8 .

Identity Property of Addition - A number does not change when 0 is added: for any number a , it is true that a+0 = a . It is also true that 0+a = a .

Identity Property of Multiplication - A number does not change when it is multiplied by 1: for any number a , it is true that a ×1 = a . It is also true that 1 ×a = a .

Inverse Operation - An operation that "reverses" another operation. Addition and subtraction are inverses of each other, as are multiplication and division.

Order of Operations - The order in which to carry out operations when evaluating an expression-- parenthesis, multiplication, division, addition, subtraction.

Zero Product Property - Of multiplication. For any number a , it is true that a ×0 = 0 and 0 ×a = 0 .

An expression represents a number. For example, 6-2 is an expression that represents the number 4, and 3 ×5 is an expression that represents the number 15. This section will discuss how to find the unique number that each expression represents.

Consider the expression 2+4 ×3 . How might one search for the answer? One way is to start by adding 2+4 = 6 and then multiply 6 ×3 = 18 . Another way is to first multiply 4 ×3 = 12 and then add 2+12 = 14 . Only one of these answers can be correct. So which is it?

The solution lies in following the Order of Operations. This rule specifies an order in which to add, subtract, multiply and divide so that everyone can look at an expression and get the same correct answer.

There are three steps to finding the answer, or to evaluating the expression, as specified by the order of operations:

Step 1. Carry out the operations within parentheses.
Step 2. Multiply and divide (it does not matter which comes first).
Step 3. Add and subtract (it does not matter which comes first).

For example, to evaluate (3+2) ×5+(7-3) , go through the steps:

Step 1 (Parentheses). (3+2) ×5+ (7-3) = 5 ×5+ 4
Step 2 (Multiplication and Division). 5 ×5 +4 = 25+4
Step 3 (Addition and Subtraction). 25 + 4 = 29

In the example at the beginning of this section, 2+4 ×3 , the steps are:

Step 1. 2+4 ×3 = 2+4 ×3 (There are no parentheses)
Step 2. 2+ 4 ×3 = 2+ 12
Step 3. 2+12 = 14

Sometimes the expression within the parentheses might contain a combination of multiplication, division, addition, and subtraction, or even more parentheses. If this is the case, go through the steps in the order of operations for the expression within the parentheses.

For example, to evaluate (12-6/(3-1))/(7-4)+2 :

Step 1.1 (12-6/(3-1))/(7-4)+2 = (12-6/2)/(7-4)+2
Step 1.2 (12-6/2) / (7-4)+2 = (12-3) / (7-4)+2
Step 1.3 (12-3) / (7-4)+2 = 9/3+2
Step 2. 9/3+2 = 3+2
Step 3. 3+2 = 5

Here are some more examples:

Example 1. 12/(6-2)-(3 ×1) = ?

Step 1. 12/(6-2) - (3 ×1) = 12/4 - 3
Step 2. 12/4-3 = 3-3
Step 3. 3-3 = 0

Step 1. 1. (11-3 ×2)+2 ×3 ×4-(3+2) = (11-3 ×2)+2 ×3 ×4-(3+2)
2. (11-3 ×2)+2 ×3 ×4-(3+2) = (11- 6)+2 ×3 ×4-(3+2)
3. (11-6)+2 ×3 ×4-(3+2) = 5+2 ×3 ×4- 5
Step 2. 5+ 2 ×3 ×4-5 = 5+ 24-5
Step 3. 5+24-5 = 24

1.1. 5-2 ×2 = ? [Solution]

1.2. (8-3) ×4 = ? [Solution]

1.3. (8-7) ×6-10/5+4 = ? [Solution]

1.4. 7-(11-8)+14 = ? [Solution]

1.5. (2+8)/(6-1)+7 ×2 = ? [Solution]

1.6. (1+2 ×3)-7/(4-3)+2 = ? [Solution]

1.7. (12/(3 ×2)+4)/(13-(8+2)) = ? [Solution]

Sometimes it is necessary to add long strings of numbers without a calculator. For example, one might be asked to find 48+33+52+11+17 . This sum is difficult to compute without a calculator, but the task can be made a lot easier by knowing some simple properties of addition. In this section, we will focus on these properties, which will help make "mental math" easier and will be useful in later sections of Pre-Algebra.

2.1 Commutative Property

The Commutative Property states that for any numbers a and b , the following is always true:

a+b = b+a

For example, 3+5 = 5+3 . We can see that this is true because 3+5 = 8 and 5+3 = 8 , so 3+5 and 5+3 are equal to each other. Another way to think of the commutative property is the following: if you have a quarter and a dime in your pocket, and you add them together, you will come up with the same amount of money whether you add the quarter to the dime or the dime to the quarter.

By the commutative property, if we add two or more numbers, we can always add them in any order. This is useful because it might be easier to add numbers in a different order than the order given. In our example above, it takes a long time to add the numbers from left to right (try it). However, because addition has the commutative property, we can switch the order of the numbers in the expression:

48+33+52+11+17 = 48+52+33+17+11

This new expression is easier to evaluate, because 48+52 = 100 and 100+33+17 = 150 . It is easier to add numbers to numbers which end in "0". This expression can be made even easier to evaluate with the associative property:

2.2 Associative Property

The Associative Property states that for any numbers a , b , and c , the following is always true:

(a+b)+c = a+(b+c)

For example, (2+4)+7 = 2+(4+7) . We can see that this is true because (2+4)+7 = 6+7 = 13 and 2+(4+7) = 2+11 = 13 , so (2+4)+7 and 2+(4+7) are equal to each other. Or we can once again think about it using the example of coins: if I have a nickel and a dime in my left pocket and a quarter in my right pocket, I will have the same amount of money if I take the dime out of my left pocket and put it in my right pocket with the quarter.

Not only can we add numbers in any order, we can also add pairs of numbers within the expression before adding them all together. In other words, we can put parenthesis around any two (or more) numbers and add those numbers separately. Using our example above, we can rearrange the numbers using the commutative property and then use the associative property to add them in pairs:

48+52+33+17+11 = (48+52)+(33+17)+11 = 100+50+11

It's a lot easier to add these three numbers in one's head than to add the original five numbers one by one, and both methods yield the same answer--161.

The Commutative Property of Addition can be remembered by remembering that when only addition is involved, numbers can move ("commute") to anywhere in the expression. The Associative Property of Addition can be remembered by remembering that any numbers that are being added together can "associate" with each other. Another good rule of thumb is, when trying to decide which properties to use, look for numbers that add up to multiples of 10; these should be added first because they are easy to add to other numbers.

2.3 Identity Property

One final property of addition that will be very useful in algebra is the Identity Property, which says that for any number a , the following are always true:

a+0 = a   0+a = a

The Identity Property of Addition says that a number does not change its identity when 0 is added. For example, 12+0 = 12 . 0+17 = 17 . Or, if someone is given zero dollars, the amount of money he has does not change.

2.4 Using the Properties of Addition

These properties can be used in any order. Right now, they are useful because they make it easier to add long strings of numbers. Later, they will help us to solve algebraic equations, which we will discuss in Inverse Operations.

2.5 Examples

Here are some examples to show how these properties can make mental math easier:

Example 1. 12+67+98 = ?
Commutative Property: 12+67+98 = 12+98+67
12+98+67 = 110+67 = 177

Example 2. (13+21)+(9+5)+5 = ?

Associative Property: (13+21)+(9+5)+5 = 13+(21+9)+(5+5)
13+(21+9)+(5+5) = 13+30+10 = 53
Example 3. 54+17+6+12+3+18 = ?

Commutative Property: 54+17+6+12+3+18 = 54+6+17+3+12+18
Associative Property: 54+6+17+3+12+18 = (54+6)+(17+3)+(12+18)
(54+6)+(17+3)+(12+18) = 60+20+30 = 110

Do the following problems without your calculator, using the properties of addition to help you.

2.1. 46+78+54 = ? [Solution]

2.2. (56+7)+13 = ? [Solution]

2.3. 67+11+21+68 = ? [Solution]

2.4. (52+85)+(5+3) = ? [Solution]

2.5. 34+0+54+26+18+6 = ? [Solution]

Which properties of addition (Associative, Commutative, Identity) are used in the following?

2.6. (5+64)+6 = 5+(64+6) [Solution]

2.7. (5+64)+6 = (64+5)+6 [Solution]

2.8. 117+0+65 = 117+65 [Solution]

2.9. 54+2+7+1 = 1+2+7+54 [Solution]

2.10. 2+(3+4)+5 = (2+3)+(4+5) [Solution]

In the last section, we learned how to add long strings of numbers using the properties of addition. Similarly, it is sometimes necessary to multiply long strings of numbers without a calculator; this task is made easier by learning some of the properties of multiplication.

Multiplication and addition have some similar properties. Like addition, multiplication has a Commutative Property and an Associative Property.

3.1 Commutative Property

The commutative property for multiplication states that for any numbers a and b , the following is always true:

a ×b = b ×a

For example, 3 ×4 = 4 ×3 . We can see that this is true because 3 ×4 = 12 and 4 ×3 = 12 , so 3 ×4 = 4 ×3 . Just as in addition, we can multiply a long string of numbers in any order. This can make multiplication without a calculator easier. For example:

4 ×6 ×5 = 4 ×5 ×6 = 20 ×6 = 120

3.2 Associative Property

The associative property for multiplication states that for any numbers a , b , and c , the following is always true:

(a ×b) ×c = a ×(b ×c)

For example, (2 ×5) ×6 = 2 ×(5 ×6) . We can see that this is true because (2 ×5) ×6 = 10 ×6 = 60 , and 2 ×(5 ×6) = 2 ×30 = 60 . Thus, (2 ×5) ×6 = 2 ×(5 ×6)

3.3 Identity Property

Multiplication also has its own Identity Property. This property states that when any number is multiplied by 1, it does not change its identity. For any number a, the following is always true:

a ×1 = a 1 ×a = a

For example, 45 ×1 = 45 . 1 ×123 = 123 .

We can remember these three properties of multiplication just as we can remember the corresponding properties of addition. With the Commutative Property of Multiplication, when only multiplication is involved, numbers can move ("commute") to anywhere in the expression. With the Associative Property of Multiplication, any numbers that are being multiplied together can "associate" with each other. Also, multiplying by 1 does not change the Identity of a number.

3.4 Zero Product Property

Multiplication has two additional properties. The first is the Zero Product Property. This says that any number multiplied by 0 is equal to 0. For any number a, the following are always true:

a ×0 = 0 0 ×a = 0

For example, 3 ×0 = 0 . 4,567,892,435 ×0 = 0 .
Because multiplication commutes, if you are multiplying a long string of numbers that contains 0, you can move 0 to the beginning of the expression:

4 ×234 ×7 ×9 ×16 ×0 ×54 = 0 ×4 ×234 ×7 ×9 ×16 ×54

Because multiplication associates, this expression is equal to:

0 ×(4 ×234 ×7 ×9 ×16 ×54) = 0.

Thus, when multiplying any string of numbers, if 0 is one of the numbers, then the answer is always 0.

3.5 Distributive Property of Multiplication over Addition

The final property of multiplication is the Distributive Property of Multiplication over Addition. This property says that for any numbers a , b , and c , the following is always true:

a ×(b+c) = (a ×b)+(a ×c).

For example, 3 ×(5+1) = (3 ×5)+(3 ×1) . We can see that this is true because 3 ×(5+1) = 3 ×6 = 18 and (3 ×5)+(3 ×1) = 15+3 = 18 .

3.6 Examples

Just like the properties of addition, these properties of multiplication can be used in any order. Here are some examples to make the properties more familiar:

Example 1. 2 ×13 ×5 = ?
Commutative Property: 2 ×13 ×5 = 2 ×5 ×13
2 ×5 ×13 = 10 ×13 = 130

Example 2. 8 ×(5 ×9) = ?
Associative Property: 8 ×(5 ×9) = (8 ×5) ×9
(8 ×5) ×9 = 40 ×9 = 360

Example 3. 43 ×9 ×0 ×7 = ?
Zero Product Property: 43 ×9 ×0 ×7 = 0

Example 4. 1 ×591 = ?
Identity Property: 1 ×591 = 591

Example 5. 6 ×(2+20)
Distributive Property: 6 ×(2+20) = (6 ×2)+(6 ×20)
(6 ×2)+(6 ×20) = 12+120 = 132

Do the following problems without your calculator, using the properties of multiplication to help you.

3.1. 6 ×7 ×5 = ? [Solution]

3.2. 5 ×(27 ×20) [Solution]

3.3. 6 ×(2+10) [Solution]

3.4. 5 ×876 ×156 ×7 ×4 ×0 = ? [Solution]

3.5. 5 ×1 ×1 ×4 ×1 = ? [Solution]

3.6. (5+4) ×1 = ? [Solution]

Which properties of multiplication (Associative, Commutative, Distributive, Identity, Zero Product Property) are used in the following?

3.7. 5 ×78 ×0 = 0 [Solution]

3.8. 65 ×(9+10) = (65 ×9)+(65 ×10) [Solution]

3.9. (6 ×5) ×11 = (5 ×6) ×11 [Solution]

3.10. (9 ×8) ×3 = 9 ×(8 ×3) [Solution]

3.11. 52 ×1 = 52 [Solution]

An inverse operation "reverses" another operation. Addition and subtraction are inverses of each other because adding and subtracting the same number does not change the original number. For example, 7-6+6 = 7 and 13+11-11 = 13 .

Similarly, multiplication and division are inverses of each other because multiplying and dividing by the same number does not change the original number. For example, 11 ×5/5 = 11 and 6/2 ×2 = 6 . Dividing by 2 and multiplying by 2 cancel each other out, and so 6 does not change.

4.1 Inverse Operations and the Commutative Property

Because addition and subtraction commute, two numbers need not be next to each other to cancel each other out. Observe the following example:

43 + 5 - 43 = 5

Be careful, however. Addition and subtraction do not commute if there is a multiplication or division sign between the two numbers being moved.

Because multiplication and division also commute, two numbers need not be next to each other to cancel each other out. For example:

6 ×4 ×5/4 = 6 ×5 ×4/4 = 30 ×4/4 = 30.

One must be careful here, too. Multiplication and division do not commute if there is an addition or subtraction sign between the two numbers being moved.

Do the following problems without a calculator:

4.1. 86+92-86 = ? [Solution]

4.2. 68/12 ×(8+4) = ? [Solution]

4.3. 17-(19 ×6)/6+(7+19-7) = ? [Solution]

Find the number that fits into the question mark.

4.4. 82-19+?  = 82 [Solution]

4.5. (73/?) ×14 = 73 [Solution]

4.6. 12/7 ×(3+?) = 12 [Solution]