Order of Operations
GEMS
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 × 5^{2} + 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
Inside the Grouping symbols () [] first
4 × (5 + 3) | = | 4 × 8 | = |
32 |
|||
4 × (5 + 3) | = | 20 + 3 | = |
23 |
(wrong) |
Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract
5 × 2^{2} | = | 5 × 4 | = |
20 |
|||
5 × 2^{2} | = | 10^{2} | = |
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 ... ? GEMS !
G |
Grouping () [] first |
E |
Exponents (ie Powers and Square Roots, etc.) |
M |
Multiplication and Division (left-to-right) |
S |
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 "G" 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.
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 ?
Grouping 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 × time^{2}
Sam puts in the velocity of 20 meters per second and time of 2 seconds:
height = 20 × 2 − (1/2) × 9.8 × 2^{2}
Now for the calculations!
The ball reaches 20.4 meters after 2 seconds
Exponents of Exponents ...
What about this example?
4^{32}
Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:
Start with: | 4^{32} | |
3^{2} = 3×3: | 4^{9} | |
4^{9} = 4×4×4×4×4×4×4×4×4: | 262144 |
So 4^{32} = 4^{(32)}, not (4^{3})^{2}
And finally, what about the example from the beginning?