The Origin of the Order of Operations

Have you ever taken a math class?

And then you’re probably familiar with the order of operations, right?

It’s commonly used in algebra. Basically it says that…

• Parentheses come first. Use them to give a higher order of precedence to a part of your equation.
• Multiplication & Division come next (in order of left to right).
• Addition & Subtraction come last (also left to right).

So 2*27+19-78 will start by multiplying 2 by 27. But if you use parentheses like this…

2*((27+19)-78)

History Lesson

So where did this order of operations come from? Whose fault is it? =^)

1. 1646 - In Van Schooten's 1646 edition of Vieta, B in D quad. + B in D is used to represent B(D^2 + BD).
2. 1800s - The term "order of operations" was starting to get used in textbooks. It was used more by textbooks than mathematicians. The mathematicians mostly just agreed without feeling the need to state anything official.
3. 1920s - In this time period, the mathematicians were debating about whether or not multiplication should take precedence over division. Although they'd still argue over who won this argument, today it's become most common (and taught predominantly) that multiplication and division are equal, read from left to right. The reasoning is to keep it simple and to let the parentheses do it's thing!
4. 1960s - As mathematicians began writing books about algebraic notation, they basically agreed on the idea that multiplication would take precedence over addition. It's a natural hierarchy that lends itself well to writing polynomials with as few parentheses as possible. So at a time when the authors of these books on mathematics had to begin their book with a list of conventions... it wasn't needed on the basic order of operations... they all seemed to have agreed.

You'll still find textbooks that don't fully agree with each other, but the basics are commonly set now, and the world is full of order and peace, thanks to the Order of Operations!

- User Ed

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2. Ellis Godard says:

At least some of those dates are wrong. Van Shooten’s 1646 edition didn’t come out in 1964, and is evidence that mathematicians did not begin writing books about algebraic notation in the 1960s.

1. Ellis, good catch! I fixed the typo and reordered those bullets. Thanks!

3. Joshua Walsh says:

It seems the dates are still wrong. Point 3 says 1960s, but that actually happened in the 1600s. Point 4 says 1964, but that happened in 1646.

1. Thanks Joshua! Yes, it looks like my change never came through. I just updated it. Thanks!

4. Colin Etherington says:

“multiplication and division are equal, read from left to right.”
Is the reading of expressions left to right a generally accepted convention? I have read a suggestion that the convention for addition/subtraction has been inappropriately applied to division/multiplication. Expressions involving division/multiplication are more likely to be ambiguous and need parentheses.

1. It depends on the context. Using parentheses is a best practice for clarity for your reader or program. Thanks, Colin!