Arithmetic/Order of Operations

From Wikibooks, open books for an open world
Jump to: navigation, search

Order of operations[edit]

The order of operations is the order in which all algebraic expressions should be simplified. Oftentimes, the meaning of a complex expression changes depending upon the order in which it is calculated. The order of operations is:

Parentheses means brackets()
Exponents (and Roots) means power
Multiplication & Division
Addition & Subtraction

EXAMPLE: 2 + 2 × 5 is equal to 12

This means that expressions within parentheses are evaluated first, then exponents (including roots, i.e. radicals), then multiplication and division (at the same level), and finally addition and subtraction (at the same level). If there are multiple operations at the same level on the order of operations, move from left to right.

There is a number of different abbreviations for memorizing the order. PEMDAS, BEDMAS and BODMAS (B is Brackets) are common. Another common way to remember the order is the mnemonic

  • "Please Excuse My Dear Aunt Sally,"

with the beginning letters standing for each operation. Whichever mnemonic you use, be aware that multiplication does not always come before division, and addition does not always come before subtraction. For example:

If you have an expression like

  • 3 × 3 - 5 + 2

you work like this: First notice that, there are no Parentheses or Exponents, so we move to Multiplication and Division. There's only the one multiplication, so we do that first and end up with 9 - 5 + 2. Now we move to Addition and Subtraction, working left to right. So firstly we do the subtraction to get 4 + 2, and finally the addition to give 6. If we had blindly done the addition first, we would have got the answer 2, which is wrong!

The rationale for the grouping (apart from parentheses, which are obviously first) is that multiplication is repeated addition and exponentiation is repeated multiplication. Also, division is multiplication by the reciprocal and subtraction is addition of the negative, so these operations are equivalent. In fact PEMA would be a better phrase ("Please Excuse My Aunt"), but in lower arithmetic courses MDAS is often taught without explaining reciprocals.

Parentheses are curved symbols, ( and ), that are put around part of an expression in order to convey that the expressions inside them should be evaluated first. Within a set of parentheses, the order of operations should be followed. Square brackets, [ and ], are sometimes used around parentheses to avoid confusion: [(3+5)×2]2 means the same as ((3+5)×2)2. The fraction bar and radical bar (often called a vinculum) groups expressions like parentheses.

For example, the expression 2×(6+7)-82 should be solved in the following order:

2×(6+7)-82 {first compute the expression inside the parentheses (6+7)}
 = 2×(13)-82 {second, calculate the exponent 82}
 = 2×(13)-64 {third, calculate the multiplication 2×(13)
 = 26-64 {finally, calculate the subtraction}
 = -38 {our final answer}

If the desired order for solving the expression were different (based on the initial problem), parentheses would be positioned differently, or even omitted.

The meaning of the fraction and radical bars must be deciphered carefully. The part of the expression directly below or above the bar is to be treated as parenthesised. (Care must be taken in writing expressions with a bar.)

The expression \sqrt{a + b} \times c means c times the root of a + b, not the root of a + b × c or even the root of c times the sum a + b, since the bar is above the a and the b, but not the c.

The expression \frac{4 + 5}{1 + 2} could be written in one line as
(4+5)/(1+2) = 9/3 = 3, not as 4+5/1+2 = 4+5+2 = 11. As you see, the expression above the bar is evaluated, as is the expression below the bar. Finally we can divide.

Because of order of operations -22 = -(22) = -4, not (-2)2 = +4: the negative sign can be considered to have an implicit 0 in front, making the expression 0 - 22.

When it comes to distributing a power, use the raise a power to a power rule. Example: (xy^2)to the fourth power (^4) =(x)^4 (y^2)^4 =x^4×y^8 (originally it was y^6; this would only be true for y^2 * y^4, where you add the exponents)

The order of operations is very important, and you must remember the order when using simple calculators. Expressions such as 2+3 × 5 vary on the order used. Entering [2] [+] [3] [×] [5] on most calculators would result in adding three to two and then multiplying by five, resulting in 25; the proper evaluation sequence would be 2+(3×5), multiplying three and five and then adding that to two to get 17. Some scientific calculators and most graphing calculators use the proper order of operations, but four-function calculators typically use "left-to-right" evaluation, which can return incorrect results.