Order of Calculations

January 17, 2017
Introduction to Formulas Order

"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like...

7 + (6 × 52 + 3)

So, long ago people agreed to follow rules when doing calculations, and they are:

Order of Operations

Do things in Brackets First. Example:

6 × (5 + 3) 6 × 8

48

30 + 3

33

(wrong)

Multiply or Divide before you Add or Subtract. Example:

2 + 5 × 3 2 + 15

17

7 × 3

21

Otherwise just go left to right. Example:

30 ÷ 5 × 3 6 × 3

18

30 ÷ 15

How Do I Remember It All ... ? BODMAS !

Brackets first
Orders (i.e. Powers and Square Roots, etc.)

DM

Division and Multiplication (left-to-right)

AS

Addition and Subtraction (left-to-right)

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)

After you have done "B" and "O", just go from left to right doing any "D" or "M" as you find them.

Then go from left to right doing any "A" or "S" as you find them.

Note: the only strange name is "Orders". "Exponents" is used in Canada, and so you might prefer "BEDMAS". There is also "Indices" which makes it "BIDMAS". In the US they say "Parentheses" instead of Brackets, so it is "PEMDAS"

Examples

Example: How do you work out 3 + 6 × 2 ?

Multiplication before Addition:

First 6 × 2 = 12, then 3 + 12 = 15


Example: How do you work out (3 + 6) × 2 ?

Brackets first:

First (3 + 6) = 9, then 9 × 2 = 18


Example: How do you work out 12 / 6 × 3 / 2 ?

Multiplication and ivision rank equally, so just go left to right:

First 12 / 6 = 2, then 2 × 3 = 6, then 6 / 2 = 3


Exponents of Exponents ...

What about this example?

432

Exponents are special: they go right-to-left. So we calculate it this way:

Start with: 432
32 = 3×3: 49
49 = 4×4×4×4×4×4×4×4×4: 262144

Oh, yes, and what about 7 + (6 × 52 + 3) ?

7 + (6 × 52 + 3)
7 + (6 × 25 + 3) Start inside Brackets, and then use "Orders" First
7 + (150 + 3) Then Multiply
7 + (153) Then Add
7 + 153 Brackets completed, last operation is add
160
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