Last week, I mentioned having a chat with Jim in the US about Order of Operations and algebra mathematics problems. He sent me a link to a video on Order of Operations title “The Order of Operations is Wrong”. The video description does go on to explain that Order of Operations is “Morally Wrong, that is…”.
I actually agree which much of what the video says and wanted to clarify what I see as the issue with blindly (and sometimes incorrectly) following the Order of Operations mnemonic that you were taught at primary/elementary school.
The Order of Operations is purely a method to help simplify the basic rules of mathematics into a formula that can be followed to get you the “correct” answer. The problem is that many people now apply that formula without an understanding of what the formula means or how mathematics works.
Exactly what abbreviation you learnt varies depending on country and school (See Trending: Only for genius ?? 3 – 3 x 6 + 2 = ?? for 12 variations), but the regardless of terminology the concept is the same.
For example: If we take PEMDAS (Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction); those who following this blindly will always perform Multiplication before Division and Addition before Subtraction. And this can produce the wrong answer.
The truth is that the mnemonic should be PE[MD][AS]LTR. This is because Multiplication and Division should be performed at the same time and Addition and Subtraction should be performed at the same time. And both should be calculated Left To Right. I tried to explain this in my article Answer: Only for genius ?? 3 – 3 x 6 + 2 = ??.
Remember:
Division is just Multiplication of the inverse number:
3 / 2 (three divided two) is exactly same as 3 x 1/2 (three multiplied by a half)
Subtraction is just Addition of a negative number:
4 – 3 (four subtract three) is exactly the same as 4 + -3 (four add minus three)
Therefore:
Multiplication and Division can be performed at the same time and always give the same answer:
- 3 x 4 / 2 = (3 x 4) / 2 = 12 / 2 = 6
- 3 x 4 / 2 = 3 x (4 / 2) = 3 x 2 = 6
Addition and Subtraction can be performed at the same time and always give the same answer:
- 3 – 4 + 2 = (3 – 4) + 2 = -1 + 2 = 1
- 3 – 4 + 2 = 3 – (4 – 2) = 3 – 2 = 1
- 3 – 4 + 2 = 3 + -4 + 2 = 3 + (-4 + 2) = 3 – 2 = 1
It does not equal
- 3 – 4 + 2 = 3 – (4 + 2) = 3 – 6 = -3 This is WRONG!!
This is a very common mistake and is caused by ignoring the minus sign. The rules of maths state that if you add parenthesis that are proceeded by a minus sign you must invert the addition and subtraction signs inside the parenthesis. Which is why addition sign before the 2 in the second of the three correct examples above is changed to a subtraction sign.
Also once you have calculated everything as per the order of operations, the rest must be calculated from left to right.
- Correct: 12 / 4 / 4 = (12 / 4) / 4 = 3 / 4 = ¾
- Wrong: 12 / 4 / 4 = 12 / (4 / 4) = 12 / 1 = 12
So don’t remember PEMDAS, remember PE[MD][AS]LTR.
Another concept to help remember order of operations is the Boss Triangle. Where the boss is the higher levels and the boss always goes first. Powers (Exponents), then Multiplication and Division, then Addition and Subtraction. It does not include Parenthesis or Brackets as these are not operations themselves, they are just containers for operations. It is really the same, but makes it obvious that Multiplication and Division should be performed together and that Additional and Subtraction should be performed together.
A final point is to look at the spacing as this could imply parenthesis. Now spacing should not affect an equation and in these situations whoever wrote the equation is being ambiguous on purpose or lazy. For example:
- 8 / 1 / 2 = (8 / 1 ) / 2 = 4
- 8 / 1/2 = 8 / (1/2) = 8 / (½) = 8 x 2 = 16
A computer or calculator, which will ignore the spacing, will return 4 for both equations.
Anyhow, check out the video as I think it makes a number of good points, but watch closely I believe there is a mistake in it.
The Order of Operations is Wrong (direct link)
Note: It is worth clicking on the direct link and reading some of the comments on the video itself.
So, Did you find the mistake in the video? Let me know in the comments what the mistake is and what the correct answer should be.
Enjoy.
David
This article was originally posted on http://www.winthropdc.com/blog.
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