Lesson ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS

Source code of 'ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS'

This Lesson (ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS) was created by by Theo(13342)

ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS

First will be a summary of the procedures to be followed.
After that will be some examples explaining how the procedure works.

SUMMARY OF ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS

Calculate the results of an expression within the most inner set of parentheses in the equation.

Replace the most inner set of parentheses and the expression within the set of parentheses with the results of the calculation.

Calculate the results of an expression within the next most inner set of parentheses in the equation.

Replace the next most inner set of parentheses and the expression within the set of parentheses with the results of the calculation.

Continue working your way out until there are no more sets of parentheses to process.

Within each set of parentheses the calculations are done in the following order:

Process all powers and roots first going from left to right.
Process all multiplications and divisions next going from left to right.
Process all addition and subtraction next going from left to right.

Once all sets of parentheses have been processed and you are left with an equation that has no more sets of parentheses, then make one last pass through the equation using the same order of algebraic and arithmetic operations that was used within each set of parentheses.

Parentheses enclosing a number without any operation being performed within the parentheses are either ignored or processed at your discretion.

Entries would be something like (59) or (-59) or (-x).

Parentheses containing expressions involving an operation are what you are looking for. Those would contain something like x - 5 or 5*(x^2+3x-2).

TERMINOLOGY

Multiplication is shown as x*y where x is multiplied by y.
Example:
x = 5
y = 3
5*3 = 15

Division is shown as x / y where x is divided by y.
Example:
x = 15
y = 5
x/y = 3

Powers are shown as x^y where x is raised to the power of y.
Example:
x = 2
y = 3
2^3 = 8
This means that 2 raised to the 3d power = 8.
This also means that 2 cubed = 8.
This also means that 2 raised to the power of 3 = 8.

Roots are shown as root(x,y) where y is taken to the root of x.
Example:
x = 3
y = 8
root(3,8) = 2
This means that 8 taken to the root of 3 is equal to 2.
This also means that the cube root of 8 equals 2.
This also means that the 3d root of 8 equals 2.

Square roots are shown as sqrt(x) where x is taken to the root of 2.
Example:
x = 16
sqrt(16) = 4
This means that the square root of 16 is equal to 4.
This also means that 16 taken to the root of 2 equals 4.
This also means that the 2d root of 16 equals 4.
Note that sqrt(x) is simply a special form of root(2,x).
They mean the same thing.

AN EXAMPLE WITHOUT PARENTHESES EXCEPT FOR THE SQUARE ROOT FUNCTION

Your equation is:
2-5*2+3*3^2+sqrt(4)

First you scan from left to right processing powers and roots as you encounter them until all have been processed.

(3)^2 becomes 9.
sqrt(4) becomes 2.
Your equation becomes:
2-5*2+3*9+2

Next you scan from left to right processing multiplications and divisions as you encounter them until all have been processed.

5*2 becomes 10.
3*9 becomes 27.
Your equation becomes:
2-10+27+2

Next you scan from left to right processing additions and subtractions as you encounter them until all have been processed.

2-10 becomes -8
-8+27 becomes 19
19+2 becomes 21

Your answer becomes:
21

AN EXAMPLE INVOLVING A STRING OF MULTIPLICATIONS AND DIVISIONS TOGETHER WITHOUT PARENTHESES.

Your equation is:
5*3*7/3/5*10/2*9

Since there are no powers and roots to be processed, you scan from left to right processing multiplications and divisions until all have been processed.

5*3 becomes 15
15*7 becomes 105
105/3 becomes 35
35/5 becomes 7
7*10 becomes 70
70/2 becomes 35
35*9 becomes 315

Since there are no multiplications and divisions to be processed, you are finished.

Your answer becomes:
315

AN EXAMPLE INVOLVING ONE ADDITION AND ONE SUBTRACTION AND ONE MULTIPLICATION AND ONE DIVISION AND ONE ROOT AND ONE POWER WITHOUT PARENTHESES EXCEPT FOR THE SQUARE ROOT FUNCTION

Your equation is:
2^3-4/2+6*sqrt(16)

First you scan from left to right processing powers and roots until they are completely processed.

2^3 = 8
Sqrt(16) = 4
Your equation becomes:
8-4/2+6*4

Next you scan from left to right processing multiplications and divisions until they are completely processed.

4/2 = 2
6*4 = 24
Your equation becomes:
8-2+24

Next you scan from left to right processing additions and subtractions until they are completely processed.

8-2 = 6
6 + 24 = 30
Your answer becomes:
30

AN EXAMPLE INVOLVING NESTED POWERS

Your equation is:
2^3^3^3

You can leave this equation as is and process, or you can add parentheses to make the operation clearer.
If you add parentheses, the equation would look like this:
(((2^3)^3)^3)

You scan from left to right processing powers and roots as you encounter them until they have all been processed.

2^3 = 8
8^3 = 512
512^3 = 134217728

Since there are no multiplications and divisions to be processed and there are no additions and subtractions to be processed, you are finished.
Your answer is:
134217728

AN EXAMPLE INVOLVING NESTED ROOTS

Root(3,root(3,root(3,134217728)))

Here parentheses are necessary because the root function requires them.

If you could use the root symbol instead of the root function as shown, the equation would look like this:
{{{root(3,(root(3,(root(3,134217728)))))}}}

The order of processing with parentheses requires the inner parentheses be processed first.

The inner set of parentheses is found by scanning from left to right looking for an open parentheses followed by a closed parentheses.

We process first set of inner parentheses which is root(3,134217728) resulting in 512.
Our equation becomes:
Root(3,root(3,512))

We process the next set of parentheses which is root(3,512)) resulting in 8.
Our equation becomes:
Root(3,8)

We process the last set of parentheses which is root(3,8) resulting in 2.
Our equation becomes:
2
Our answer is:
2

AN EXAMPLE INVOLVING TWO SETS OF PARENTHESES NOT NESTED WITHIN EACH OTHER

Our equation is:
(5^2+3) - (root(3,512))

We scan left to right looking for inner sets of parentheses containing operators.
Our first one is (5^2+3) which we then process as (25+3) becoming (28).

Our second set of inner parentheses is (root(3,512)) which we process to become (8)

Our equation becomes:
(28) - (8)

Since there are no more parentheses to be processed and there are no roots and powers to be processed and there are no multiplications and divisions to be processed, we process additions and subtractions until they are completely processed.

Our equation becomes:
28 - 8 = 20
Our answer is:
20

AN EXAMPLE INVOLVING TWO SETS OF PARENTHESES NESTED WITHIN EACH OTHER

Our equation is:
(9+3-4*(8+(-2)^2))

Our first set of inner parentheses is (-2) which we ignore because no operation is indicated within it.

Our second set of inner parentheses is (8+(-2)^2) which we process because there are operations indicated in there.

We process from left to right looking for powers and roots.
(-2)^2 = 4
Our equation within the set of parentheses becomes:
(8+4)
There are no multiplications and divisions so we process additions and subtractions.
Our equation within the set of parentheses becomes:
(12)
Our equation becomes:
(9+3-4*(12))

Since there are no more sets of parentheses to process and there are no roots and powers to be processed, we process multiplications and divisions until they are completely processed.

4*12 equals 48
Our equation becomes:
(9+3-48)

Next we process additions and subtraction until they are completely processed.
(9+3-48) = 36
Our answer is:
36

More information regarding algebraic and arithmetic operations can be found in the lesson on:

RULES OF ALGEBRAIC AND ARITHMETIC OPERATIONS