pemdas - the proper order of mathematical operations

PEMDASThe Proper Order of Mathematical Operations

Prepared by:Mong Montances

www.mongmontances.wordress.com

www.mongmontances.wordpress.com 1

Learning OutcomesBy the end of this tutorial, you will be able to:

• Take the absolute value of a real number.

• Perform addition, subtraction, multiplication and division on signed real numbers.

• Use PEMDAS to simplify mathematical expressions.


Table of Contents• Absolute Values

• Addition of Signed Numbers

• Subtraction of Signed Numbers

• Multiplication of Signed Numbers

• Division of Signed Numbers

• Order of Operations

• PEMDAS


Absolute Values


Absolute ValuesIn a number line, an absolute value means the distance of a number from zero. The direction is not of importance. This means that -3 and +3 have the same absolute value which is 3!

To put it simply, you can think of taking the absolute value of a number by just removing any negative sign in front it.

Examples:

The absolute value of -12 is 12.

The absolute value of -49 is 49.

The absolute value of 17 is of course, 17.www.mongmontances.wordpress.com 5

Absolute ValuesIn taking the absolute value of an expression or any number, we use the symbol “| |” (which looks like a pair of bars).

For illustration, we can rewrite the previous example:

The absolute value of -12 is 12 as | -12 | = 12

Other examples:| -521 | = 521

| 12 -18 | = |-6 | = 6

| 5-2 | = | 3 | = 3


Addition of Signed Numbers


Addition of Signed NumbersWhen you add real numbers with like signs

1. Add their absolute values, and

2. Prefix the common sign.

Examples:

16 + 23 = +39 = 39

(-4) + (-13) + (-5) = - (4+13+5) = -22


Addition of Signed NumbersWhen you add two real numbers with unlike signs

1. Find the difference of their absolute values, and

2. Prefix the sign of the number with bigger magnitude.

Examples:

5 + (-7) = -|5 -7| = -|-2| = -2

-12 + 8 = -4

24 + (-3) = 21


Subtraction of Signed Numbers


Subtraction of Signed NumbersWhen you subtract two signed numbers

1. Change the sign of the subtrahend, and

2. Proceed as in addition of signed numbers.

Examples:

(-7) – (-4) = (-7) + (4) = -3

(13) – (-2) = (13) + (2) = 15

(-18) – ( 12) = -18 + (-12) = -30

(9) – (12) = 9 + (-12) = -3


Tips and Tricks


Tips and TricksHere are some tips and tricks which you can use to speed up your work with addition and subtraction. Let’s take the following examples:

60 − −15 can be simplified right away as 60 + 15.

9 + −3 can be viewed as 9 − 3.

You can look at this as if multiplying the difference operator with the negative number sign.

It then follows that, 5 + 3 can be written as 5 − −3

And 7 − 12 as 7 + (−12)

I hope this helps!www.mongmontances.wordpress.com 13

Multiplication of Signed Numbers


Multiplication of Signed NumbersWhen you multiply two signed numbers

1. Multiply their absolute values.

2. Prefix the correct sign:

The product is positive if the two numbers have the same sign(both + or -). Otherwise, the product is negative.

Examples:

(-7)*(-4) = +(7)*(4) = 28

(2)*(-3) = -(2*3) = -6

(-5)( 13) = -(5)(13) = -65

(8)(7) = 56


Division of Signed Numbers


Division of Signed NumbersWhen you divide two signed numbers

1. Divide their absolute values.

2. Prefix the correct sign:

The quotient is positive if the two numbers have the same sign(both + or -). Otherwise, the quotient is negative.

Examples:

(-36)/(6) = -(36/6) = -6

(72)/(-8) = -(72/8) = -9

(-15)/(-3) = +(15/3) = 5

(8)/(2) = 4


Order of Operations


Order of OperationsWhen simplifying expressions like this:

60 − 15 ÷ (2 − 5) + 7 × 23

The common mistake is to evaluate it right away from left to right.

We need to be aware that expressions like these are handled differently.

We need to understand that there is a proper way of solving or simplifying mathematical expressions. The correct order of operations can be easily remembered using the acronym PEMDAS.


PEMDAS


PEMDASPEMDAS is an acronym commonly used to remember the order of operations. It stands for:

Parenthesis | Exponents | Multiplication/Division | Addition/Subtraction

This simply means that the correct order starts by dealing with the Parenthesis. Then evaluating the exponents. Then the Multiplication/Division. Lastly is the Addition/Subtraction.

Note: The Multiplication/Division are of the same priority. Meaning, we will solve whichever appears first (from left to right) in the equation. The same is true for Addition/Subtraction

1 2 3 4


PEMDASGoing back to the previous example, 60 − 15 ÷ (2 − 5) + 7 × 23

Applying the PEMDAS rule step-by-step:

60 − 15 ÷ (2 − 5) + 7 × 23 Parenthesis

60 − 15 ÷ (−3) + 7 × 23 Exponent

60 − 15 ÷ (−3) + 7 × 8 Division

60 − (−5) + 7 × 8 Multiplication

60 − (−5) + 56 Subtraction

65 + 56 Addition

121


Exercises


ExercisesI have provided some exercises for you to test your understanding.

Answers are also given so you can check your work.

42 ÷14 + 25 ∙ 3 ÷1530 + (−28) ÷2 ∙ 3 + 12 ÷2 + 6125 ÷25 ∙ 3 + 16 ÷8 –33 ÷11 + 10 ÷53 ∙ (−4 + 5) + ( 12 ∙ 3 –6) ÷530 − 12 ÷ (6 − 2) + 6 × 23

5 + 12 ÷(8 –3 ∙ 2) –18 ÷3 + 6

Answers: 8, 0, 16, 9, 75, 11

1

2

3

4

5

6


Good to Know


Good to KnowWhen in doubt with your solution or answer, you can use a calculator to verify your work. These calculators are programmed to solve using the correct order of operation. Just make sure that you use the proper syntax and symbols.


Conclusion


Conclusion• Simplifying Mathematical Expressions relies on your ability to

perform fundamental arithmetic operations.

• While these concepts are straightforward, they are a key factor in handling more complex mathematical equations.

• Once these concepts are grasped with great depth, you will realize the simplicity of mathematical operations.


I hope you learned something from this short tutorial.

If you have questions or clarifications, feel free to send me an email.

Also, please let me know how if how can I improve this tutorial.

Thank you and God bless on your studies!

-Mong Montances

www.mongmontances.wordpress.com


pemdas - the proper order of mathematical operations

Education