MATHEMATICS; ORDER OF OPERATIONS ( ALGEBRA CON'TD)

Order of Operations
BODMAS

Operations

"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)

... what part should you calculate first?

Start at the left and go to the right?
Or go from right to left?

Warning: Calculate them in the wrong order, and you can get a wrong answer !

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

Order of Operations

Do things in Brackets First

		4 × (5 + 3)	=	4 × 8	=	32
		4 × (5 + 3)	=	20 + 3	=	23	(wrong)

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract

		5 × 22	=	5 × 4	=	20
		5 × 22	=	102	=	100	(wrong)

Multiply or Divide before you Add or Subtract

		2 + 5 × 3	=	2 + 15	=	17
		2 + 5 × 3	=	7 × 3	=	21	(wrong)

Otherwise just go left to right

		30 ÷ 5 × 3	=	6 × 3	=	18
		30 ÷ 5 × 3	=	30 ÷ 15	=	2	(wrong)

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

B	Brackets first
O	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)

So do it this way:

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 Division rank equally, so just go left to right:

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

A practical example:

Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?

Sam uses this special formula that includes gravity:

height = velocity × time − (1/2) × 9.8 × time²

Sam puts in the velocity of 20 meters per second and time of 2 seconds:

height = 20 × 2 − (1/2) × 9.8 × 2²

Now for the calculations!

Start with:20 × 2 − (1/2) × 9.8 × 2²

Brackets first:20 × 2 − 0.5 × 9.8 × 2²

Then Orders (2²=4):20 × 2 − 0.5 × 9.8 × 4

Then the Multiplies:40 − 19.6

Subtract and DONE !20.4

The ball reaches 20.4 meters after 2 seconds

Exponents of Exponents ...

What about this example?

4³²

Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:

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

So 4³² = 4⁽³²⁾, not (4³)²

 

 

And finally, what about the example from the beginning?

Start with:7 + (6 × 5² + 3)

Brackets first and then "Orders":7 + (6 × 25 + 3)

Then Multiply:7 + (150 + 3)

Then Add:7 + (153)

Brackets completed:7 + 153

Last operation is
an Add

Or we could use;

Order of Operations
PEMDAS

Operations

"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)

... what part should you calculate first?

Start at the left and go to the right?
Or go from right to left?

Warning: Calculate them in the wrong order, and you can get a wrong answer !

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

Order of Operations

Do things in Parentheses First

		4 × (5 + 3)	=	4 × 8	=	32
		4 × (5 + 3)	=	20 + 3	=	23	(wrong)

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract

		5 × 22	=	5 × 4	=	20
		5 × 22	=	102	=	100	(wrong)

Multiply or Divide before you Add or Subtract

		2 + 5 × 3	=	2 + 15	=	17
		2 + 5 × 3	=	7 × 3	=	21	(wrong)

Otherwise just go left to right

		30 ÷ 5 × 3	=	6 × 3	=	18
		30 ÷ 5 × 3	=	30 ÷ 15	=	2	(wrong)

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

P	Parentheses first
E	Exponents (ie Powers and Square Roots, etc.)
MD	Multiplication and Division (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)

So do it this way:

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

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

 

	You can remember by saying "Please Excuse My Dear Aunt Sally".
Or ...	Pudgy Elves May Demand A Snack Popcorn Every Monday Donuts Always Sunday Please Eat Mom's Delicious Apple Strudels People Everywhere Made Decisions About Sums

Note: in the UK they say BODMAS (Brackets,Orders,Divide,Multiply,Add,Subtract), and in Canada they say BEDMAS (Brackets,Exponents,Divide,Multiply,Add,Subtract). It all means the same thing! It doesn't matter how you remember it, just so long as you get it right.

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 ?

Parentheses first:

First (3 + 6) = 9, then 9 × 2 = 18

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

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

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

A practical example:

Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?

Sam uses this special formula that includes the effects of gravity:

height = velocity × time − (1/2) × 9.8 × time²

Sam puts in the velocity of 20 meters per second and time of 2 seconds:

height = 20 × 2 − (1/2) × 9.8 × 2²

Now for the calculations!

Start with:20 × 2 − (1/2) × 9.8 × 2²

Parentheses first:20 × 2 − 0.5 × 9.8 × 2²

Then Exponents (2²=4):20 × 2 − 0.5 × 9.8 × 4

Then the Multiplies:40 − 19.6

Subtract and DONE !20.4

The ball reaches 20.4 meters after 2 seconds

Exponents of Exponents ...

What about this example?

4³²

Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:

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

So 4³² = 4⁽³²⁾, not (4³)²

 

 

And finally, what about the example from the beginning?

Start with:7 + (6 × 5² + 3)

Parentheses first and then Exponents:7 + (6 × 25 + 3)

Then Multiply:7 + (150 + 3)

Then Add:7 + (153)

Parentheses completed:7 + 153

Last operation is an Add:160

:160