Order of the arithmetic operations
The order of the arithmetic operations describes the order in which the operations of a mathematical expression has to be carried out.
The hierarchy of the arithmetic operations
Hierarchy means that something is more important than anything else.
That also true to the arithmetic operations. Some of the operations always have to be carried out before the other, if more than one type of operation appears in a mathematical expression. Here is the order in which the operations have to be carried out:
2. Exponents and roots
3. Multiplication / division
4. Addition / subtraction
That is, you must multiply before you do the adding, and expressions inside parentheses always have to be calculated first etc.
Here are some examples of mathematical problems, where the order of the arithmetic operation is essential.
Multiplication and division must be carried out before addition and subtraction.
|Multiplication must be carried out before addition. So we calculate|
2 × 2 = 4 first.
|Now we do the addition: 2 + 4 + 2 = 8|
|8||The result is 8.|
Expressions inside parentheses must be calculated before multiplication and division.
|The expression inside the parentheses must be calculated first: 2 + 2 = 4|
|Then we do the multiplying: 2 × 4 = 8|
|Then we do the adding 2 + 8 = 10|
|10||The result is 10.|
Exponents and roots are calculated before multiplication and division.
|The expression inside the parentheses must be calculated first:2 + 2 = 4|
|Then the number with exponent is calculated: 4^2 = 16|
|Then we do the multiplying: 2 * 16 = 32|
|Finally we do the adding: 2 + 32 = 34|
|34||The result is 34.|
Commutative property of addition and multiplication
In some cases you can change the order of the parts of a mathematical problem.
The Commutative property of addition: The order of addends doesn’t matter.
The Commutative property of multiplication: The order of factors doesn’t matter.
The commutative property doesn’t apply to subtraction, because you cannot change the order of the minuend and the subtrahend without changing the result. But it is possible, however, to transform a subtraction problem into an addition problem, if you consider the last number as a negative number that has to be added to the first number, instead of two positive number being subtracted from each other.
In order to show this we write a minus sign in front of the last number and then put it into parentheses.
You can transform a division problem into a multiplication problem by multiplying the dividend by the divisor’s reciprocal instead. (The reciprocal of a number is one divided by the number)