For this week’s Friday Funny we are going to see how people interpret the Order of Operations rules that they should have been taught at school. Most people know a mnemonic to help remember the basic rules, the actual mnemonic depends on whether you say Brackets or Parentheses, whether you say Indices, Exponents or Orders, and whether you put Multiplication or Division next.
Regardless of which terminology and mnemonic you use, they do not cover all situations as the following example demonstrates:
So here is the equation for you to solve:
Below is a poll for you to record your answer (please answer the poll before reading on):
Disclaimer: The following points are written in my own words, but seem to be ignored by many who learnt the mnemonic without understanding the underlying principles.
Now, that you have answered, here are some points to consider with the Order of Operations (Precedence) rules:
- The Multiplication and Division steps happen together, not one before the other.
- Hence Multiplication and Division can be swapped in the mnemonic.
- Division is Multiplication by the inverse number. For example: A ÷ 2 is the same as A x ½.
- The Addition and Subtraction steps happen together, not one before the other.
- Hence Addition and Subtraction can be swapped in the mnemonic (but AS is easier to remember than SA).
- Subtraction is Addition by the negated number: For example: A – 2 is the same as A + -2.
- When operations have the same priority, they should be calculated from left to right.
But here some other points that are less well known:
- A number before or after brackets/parentheses without an explicit operation has an implied multiplication (or multiplication denoted by juxtaposition)
Reference: Wikipedia (“In algebra, multiplication involving variables is often written as a juxtaposition (e.g., xy for x times y or 5x for five times x), also called implied multiplication. The notation can also be used for quantities that are surrounded by parentheses (e.g., 5(2) or (5)(2) for five times two).”).
- When there is an implied multiplication caused by brackets/parentheses it should be calculated before just after the brackets/parentheses step to ensure that the operand stays together.
Reference: Wikipedia (“multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division.”).
Reference: Quora (“implicit multiplication has a higher precedence than either division or explicit multiplication”).
- An implied multiplication cannot be changed to an explicit multiplication as this can change the order of operations and affect the result.
- You should not add your own brackets/parentheses to an equation just to force a change the order of operations and affect the result. However, to remove ambiguity from implicit multiplication, I believe an exception can be made to add brackets/parenthesis around the entire operand to keep it together.
If you correctly follow these guidelines, there is only ever one answer to the equation above.
That said, the equation above has been written to be deliberately ambiguous and confusing on purpose, which is why people disagree on the answer.
Here is the original question:
- 12 ÷ 2 (5 – 2)
We can use / instead of ÷ as this is easier on a computer keyboard:
- 12 / 2 (5 – 2)
This also helps separate the numerator from the denominator when expressed as a fraction:
- 12 12 12
——— = —– = —- = 2
2 (5 – 2) 2(3) 6
The operand 2 (5 – 2) can be expanded during the calculations (if desired), the result is the same:
- 2 (5 – 2) = 2 x 5 – 2 x 2 = 10 – 4 = 6 or 2 (5 – 2) = 2 (3) = 6
Adding an explicit multiplication operator changes the order of operations for this equation:
- 12 / 2 (5 – 2) = 12 / 6 = 2 is not the same as 12 / 2 x (5 – 2) = 12 / 2 x 3 = 6 x 3 = 18
Adding extra brackets/parentheses can change the order of operations for this equation:
- 12 / 2 (5 – 2) = 12 / 6 = 2 is not the same as (12 / 2) x (5 – 2) = 6 x 3 = 18
- (12 / 2) * (5 – 2) which is NOT the same as 12 / 2 (5 – 2).
- 12 / (2 (5 – 2)) = 12 / (2(3)) = 12 / 6 = 2
So according to the search engines I am wrong, but I firmly believe that the “bond” for implicit multiplication is stronger than explicit multiplication and division and that if adding the brackets/parentheses to “educate” the search engines is required, then so be it.
This might also come down to your age and when you learnt your maths. It seems people who learnt more recently do not differentiate between an implied multiplication and an explicit multiplication.
- Order of Operations: Implicit Multiplication?
- “implied multiplication” operator precedence? (see image of two calculators 🙂)
The best solution to this problem is not to write ambiguous equations and to add brackets/parentheses where required to avoid any possible confusion.
While we are discussing the topic of Order of Operations, which mnemonic did you learn at school? This probably depends on which country you lived in when you went to school. I know, for example, that my British background means I say Brackets, not Parentheses and Maths not Math.
Thanks for your involvement in this ad-hoc research into basic mathematics skills.
This article was originally posted on http://www.winthropdc.com/blog.