Order of Operations: The Pesky Math Problem from the Internet
Perhaps the World’s Most Problematic Math Problem
You’ve seen this, we all seen this, it’s the math problems that’s put all calculators in jeopardy, it’s the dreaded:
6 ÷ 2(1 + 2)
Is it 1 or is it 9? No matter what we answer, the debate still rages on and on. Why? Because not everyone agrees with the order of operations. And the problem lies with implied multiplication.
Seeing that this issues comes up multiple times of the HP Museum of Calculators (latest can be seen at [Source 5]), it’s time to input my two dollars into the subject. (It was two cents but the price went up due to inflation.)
What is Implied Multiplication?
Implied multiplication occurs when two or more quantities, often with or without a numeric coefficient, are combined with one or more variables without explicit multiplication symbols (*, ⋅, or ×). It is implied that when this notation is used, the quantities are to be added.
Examples:
xy is implied to mean x * y
2x is implied to mean 2 * x
3rs is implied to mean 3 * r * s
If there are two or more quantities and at least one of them is surrounded by parenthesis, this is also implied multiplication.
Examples:
The following are implied to mean 2 * 6:
2(6)
(2)6
(2)(6)
4(x)y is implied to mean 4 * x * y.
Implied multiplication is also known as juxtaposition. [Source 2]
Infix multiplication is multiplication with symbols (*, ⋅, or ×).
Where is implied multiplication in the order of operations?
PEMDAS vs PEJMDAS
PEMDAS and PEJMDAS are the two major order of operations.
In PEMDAS, also known as BODMAS in some parts of the world, put implied multiplication (juxtaposition) on the same level as infix multiplication and division. The video by The How and Why of Mathematics serves an excellent video to explain the difference and the potential problems that can be encountered. [Source 1]
PEMDAS
PEMDAS stands for:
P: parenthesis
E: exponents (and roots) (one-argument functions such as sin, int, abs and log are included)
MD: all multiplication and division, left to right
AS: all addition and subtraction, left to right
Similarly, BODMAS stands for:
B: brackets (serves as the same function as parenthesis)
O: operations (exponents, roots, one-argument functions such as sin, int, abs, and log)
DM: all division and multiplication, left to right
AS: all addition and subtraction, left to right
For this blog post, I will refer to this sequence as PEMDAS.
PEJMDAS
PEJMDAS moves implied multiplication to a higher priority:
P: parenthesis
E: exponents (and roots) (one-argument functions such as sin, int, abs and log are included)
J: multiplication by juxtaposition (implied multiplication), left to right
MD: infix multiplication and division, left to right
AS: all addition and subtraction, left to right
Examples
Problem 1: 48 / 2 * 3
PEMDAS:
48 / 2 * 3
24 * 3
72
PEJMDAS:
48 / 2 * 3
24 * 3
72
Here there is no difference because the multiplication is infix.
Problem 2: 48 / 2 (3)
PEMDAS:
48 / 2 (3)
24 (3)
72
PEJMDAS:
48 / 2 (3)
48 / 6
8
Notice in PEJMDAS the implicit multiplication, as marked by two numbers juxtaposed next to each other separated by parenthesis, takes priority.
Problem 3: 100 – 5 (2 + 3)
PEMDAS:
100 – 5 (2 + 3)
100 – 5 (5)
100 – 25
75
PEJMDAS:
100 – 5 (2 + 3)
100 – 5 (5)
100 – 25
75
Here we arrive at the same problem because subtraction has the lower priority than implication in both cases.
Problem 4: 100 / 5 (2 + 3)
PEMDAS:
100 / 5 (2 + 3)
100 / 5 (5)
20 (5)
100
PEJMDAS:
100 / 5 (2 + 3)
100 / 5 (5)
100 / 25
4
In PEJDMAS, implied multiplication has priority over division.
Conclusion – Which Method Reigns Supreme? Should we Even use Implied Multiplication?
I’m not usually a fan of the phrase “pick a lane”, but for the sake of consistency, I’ll make an exception. I would prefer the world to pick either PEMDAS or PEJMDAS and stick to it as a universal rule. It seems like PEJMDAS might get the favor since more mathematicians, scientists, and professionals prefer it. I view the PEMDAS vs PEMJDAS as similar to the way the world views which are the standard scientific units:
PEMDAS: ft, sec lbs (United States, especially in the education field)
PEJMDAS: m, sec, kg
Having grown up in the United States, it will take me a bit to adjust to PEJMDAS from PEDMAS.
I’m also in favor of just using additional multiplication, parenthesis, and perhaps using a fraction bar to make problems much clearer. To borrow a suggestion from Tony Barlow, a mathematician who tested the ill-fated never-released TI-88:
“Kill Implied Multiplication. Kill Implied Multiplication. Kill Implied Multiplication.” [2]
Disallowing implied multiplication allows us to avoid two potential problems:
1. In calculators with CAS capabilities and in Python, we are allowed variable names with more than one character (rate, chg, pts). If implied multiplication is allowed with variable names with more than one character, and “ratepts” is typed, do I mean “rate * pts”, the entire variable “ratepts”, or something else like “r * a * t * e * p * t * s”?
2. This makes it clear when we mean function calls, which thankfully is usually understood. But to someone who is unfamiliar with functions, something like “sin(x)” may mean “sin * x” or “ s * I * n * x” to them. “f(x)” may be misinterpreted as “f * x”.
Calculators with the classic AOS (algebraic operating system, postfix system), RPN (Reverse Polish Notation), or Chain operation do not deal with implied multiplication:
AOS: In all cases, the multiplication and parenthesis keys must be explicitly pressed. (Examples: TI-30 series, Casio fx-260 series)
RPN: RPN is a parenthesis-less operating system. (Examples: Almost all HP calculators, all Swiss Micros calculators)
Chain: Everything entered in the Chain operating system is done how you enter the keys. Therefore, these calculators don’t even deal with the order of operations, accuracy is completely up to you. (Examples: all basic four-function calculators)
I hope you find this blog entry helpful, and hopefully one day, we have a universal solution.
This is not the only issue with the order of operations, another issue is where to place the unary operator of negation, but that’s for another day.
Eddie
Sources
[1]
The How and Why of Mathematics. “The Problem with PEMDAS: Why Calculators Disagree” August 5, 2019. Accessed April 15, 2024. Video. https://www.youtube.com/watch?v=4x-BcYCiKCk
[2]
Wright, Gene. “HHC 2022: TI-88 Part 1: History and Go / No Go Decision” September 14, 2022. Accessed April 15, 2024. https://www.youtube.com/watch?v=wl16wzmn3wA Video. (Refer to time stamp 13:24).
[3]
“Multiplication” Wikipedia. Edited April 17, 2024. Accessed April 21, 2024. https://en.wikipedia.org/wiki/Multiplication
[4]
“Order of Operations” Wikipedia. Edited April 19, 2024. Accessed April 19, 2024. https://en.wikipedia.org/wiki/Order_of_operations
[5]
“What is the correct result?” Museum of HP Calculators. Thread started on March 18, 2024. https://www.hpmuseum.org/forum/thread-21474.html
Next Post: June 8, 2024
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