PEMDAS - The Proper Order of Mathematical Operations
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The document discusses the proper order of mathematical operations known as PEMDAS. It covers absolute values, addition and subtraction of signed numbers, multiplication and division of signed numbers, and the order of operations using PEMDAS. Examples are provided to illustrate how to use PEMDAS to simplify expressions involving multiple operations. Exercises with answers are included to help readers practice applying these concepts.
PEMDAS - The Proper Order of Mathematical Operations
- 1. PEMDAS The Proper Order of Mathematical Operations Prepared by: Mong Montances www.mongmontances.wordress.com www.mongmontances.wordpress.com 1
- 2. Learning Outcomes By 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. www.mongmontances.wordpress.com 2
- 3. 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 www.mongmontances.wordpress.com 3
- 5. Absolute Values In 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
- 6. Absolute Values In 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 www.mongmontances.wordpress.com 6
- 8. Addition of Signed Numbers When 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 www.mongmontances.wordpress.com 8
- 9. Addition of Signed Numbers When 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 www.mongmontances.wordpress.com 9
- 11. Subtraction of Signed Numbers When 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 www.mongmontances.wordpress.com 11
- 13. Tips and Tricks Here 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
- 15. Multiplication of Signed Numbers When 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 www.mongmontances.wordpress.com 15
- 17. Division of Signed Numbers When 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 www.mongmontances.wordpress.com 17
- 19. Order of Operations When 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. www.mongmontances.wordpress.com 19
- 21. PEMDAS PEMDAS 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 www.mongmontances.wordpress.com 21
- 22. PEMDAS Going 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 www.mongmontances.wordpress.com 22
- 24. Exercises I have provided some exercises for you to test your understanding. Answers are also given so you can check your work. 42 ÷14 + 25 ∙ 3 ÷15 30 + (−28) ÷2 ∙ 3 + 12 ÷2 + 6 125 ÷25 ∙ 3 + 16 ÷8 –33 ÷11 + 10 ÷5 3 ∙ (−4 + 5) + ( 12 ∙ 3 –6) ÷5 30 − 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 www.mongmontances.wordpress.com 24
- 26. Good to Know When 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. www.mongmontances.wordpress.com 26
- 28. 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. www.mongmontances.wordpress.com 28
- 29. 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 www.mongmontances.wordpress.com 29