BMCC: Math Tutorials: Numbers:
Some Rules of Operation
This section presents quick summaries of several rules of operation for basic mathematics. You can use these quick links to jump directly to a specific rule farther down the page.
Addition Property of Zero
Commutative Property of Addition
Associative Property of Addition
Multiplication Property of Zero
Multiplication Property of One
Multiplication Property of Reciprocals
Commutative Property of Multiplication
Associative Property of Multiplication
The Distributive Property
Order of Operations Agreement
Adding zero (0) to any number n does not change the number.
n + 0 is always equal to n.
The order in which two or more numbers is added makes no difference. The sum will always be the same no matter what order is used.
When numbers are grouped within parentheses, the operations within the parentheses should be performed before any operations outside the parentheses. The associative property of addition states that the same sum is obtained no matter how numbers are grouped before adding. For example, (9 + 3) + 7 = 9 + (3 + 7).
The product of any number n and zero (0) is zero; n x 0 = 0.
The product of any number n and the number 1 is always equal to the number n itself;
n x 1 = n.
The product of any non-zero number or term and its reciprocal is always 1 . Shown as the general case, 
.
Some examples:
| Original | Reciprocal | Product |
| 4 |  |
 |
 |
3 |  |
 |
 |
 |
The order in which two or more numbers are multiplied makes no difference. The product of the numbers will be the same for any order.
The same product is obtained no matter how numbers are grouped before multiplying. For example,
If a mathematical expression has parentheses and contains both multiplication and addition operations, the parentheses can be removed using the Distributive Property. The Distributive Property states that if a, b, and c are three numbers or variables, then
Note that this expression uses the short-hand notation for expressing multiplication, whereby two variables that are written immediately adjacent to one another are supposed to be multiplied-- e.g., ab is the same as a x b . This works for expressions in parentheses too. If we write out the Distributive Property in words, it is expressed like this:
a times the sum of b plus c is equal to a times b plus a times c .
As mentioned above, for some mathematical expressions the order in which operations are performed makes no difference in the outcome. An expression involving only addition, or only multiplication, is order independent as stated by the commutative property of addition and the commutative property of multiplication. However, many mathematical expressions involve a mix of operators, and the outcome will depend on the order in which the operations are performed.
For example, consider the expression 4 + 5 x 3. If we first add 4 + 5, we get 9. Then, multiplying 9 x 3 gives a final answer of 27. However, if we first multiply 5 x 3, we get 15. Adding 4 to 15 gives a final result of 19. Performing the operations in one order gives an answer of 27, while the other order gives an answer of 15. Obviously, the order makes a difference, and we have to have an accepted order in which to perform the operations if the results are not going to be ambiguous.
The order of operations agreement solves this problem. The order of operations agreement isn't really any type of mathematical law or property, it's just a standardized procedure that, if followed by everyone, will always yield unambiguous results for a mathematical expression or equation. If the order of operations agreement is followed, only one answer is possible for a particular expression.
For any expression, the mathematical operations should be performed in this order:
Let's look again at the expression 4 + 5 x 3, and this time apply the order of operations agreement by going through the steps in order. Step 1 says to perform any operations inside parentheses. Since this expression has no parentheses, we move on. Step 2 says to simplify any exponents. The expression has no exponents, so again we move on. Step 3 says to perform any multiplication and division operations. The expression contains the multiplication operation 5 x 3, so we perform that and obtain an answer of 15. Now the expression reads 4 + 15. Step 4 says to perform any addition and subtraction operations. We add 4 + 15 for a final answer of 19. This is regarded as the "correct" answer for the expression. The earlier possible answer of 27 is incorrect because we obtained 27 by performing the addition operation (4 + 5) before the multiplication operation (9 x 3), which violated the order of operations agreement.
Let's try a slightly more detailed example.
Step 1: Perform any operations inside parentheses. This expression contains the operation 11 - 5 within parentheses, so we perform that operation and obtain a value of 6.
Step 2: Simplify any expressions containing exponents. The equation contains the expression 33, which equals 3 x 3 x 3, or 27.
Step 3: Perform any multiplication and division operations as they occur from left to right. The equation contains the operation 4 x 6, which is equal to 24.
Step 4: Perform any addition and subtraction operations as they occur from left to right. The entire original expression has now been reduced to only 27 + 24, which is equal to 51. This is the final answer for simplifying the original expression.
One more example. . .
Step 1: Perform any operations inside parentheses. This expression contains the operation 3 + 1 within parentheses, so we perform that operation and obtain a value of 4.
Step 2: Simplify any expressions containing exponents. The equation contains the expression 42, which equals 4 x 4, or 16.
Step 3: Perform any multiplication and division operations as they occur from left to right. The equation contains the operation 
, which is equal to 4.
Step 4: Perform any addition and subtraction operations as they occur from left to right. The entire original expression has now been reduced to only 16 - 4, which is equal to 12. This is the final answer for simplifying the original expression.
End of Numbers Tutorial
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