Why Is BODMAS Or PEMDAS In That Order?

BODMAS and PEMDAS are the same order-of-operations rule, just named differently (BODMAS in the UK, PEMDAS in the US). The order is brackets, then orders/exponents, then multiplication and division, then addition and subtraction. It is a convention, not a law of math, adopted so that calculations are written without ambiguity. Crucially, multiplication and division rank equally and are done left to right, as do addition and subtraction.

If you think 2+2×2 is equal to 8, then congratulations! You’re wrong. The correct answer is 6. Why? Because multiplication takes precedence over addition, so the last two 2s are multiplied first, and the product is then added to the first 2, such that the result is 6. Now, with that in mind, what is 2×2-4÷2 equal to? Which operation takes precedence then?

High school taught us that when mired in such a problem, we must seek the sovereign BODMAS, which Americans call PEMDAS (more on why those two names mean the same thing in a moment). This method shows us the way by declaiming the cardinal order: Brackets-Orders-Division-Multiplication-Addition-Subtraction. Or, Parentheses-Exponents-Multiplication-Division-Addition-Subtraction. Read the acronym literally and you might divide first, so that 2×2-4÷2 becomes 4-2, which is 2. (Hold on to that 2; we will soon see that you land on it whether you divide or multiply first.) But are we to blindly believe this method? Why is the order so particular? Why is the order not BOSAMD or any of the other 240 permutations?

If mathematics is the language in which the universe speaks, and these operations are its conjunctions, surely, as one would expect, the order must be based on logic and not chosen arbitrarily, right? Astoundingly, this is not the case… but it also kind of is.

Why BODMAS?

BODMAS or PEMDAS is simply a convention, a standard, and like any standard, it is devised to make processes, here calculations, less messy and ambiguous. In the above examples, I deliberately refrained from using brackets, and notice how ambiguous, at least for a novice, the calculations become. Convenience is actually why we use brackets. The sole purpose of a pair of brackets is to condense what would typically be written as a very long and cumbersome calculation.

Addition and subtraction are the most elemental operations in the sense that their operation is unaffected by direction. Consider this calculation:

5+5+5+6+7+3+3+3+3+4+1

We get the same result, regardless of the direction in which we add the numbers. In fact, one doesn’t even have to add in a linear, particular path: add 6 to 1 and the result to 5, that is, one can add the numbers in any random order, and the result is the same, which is 45. However, it is obvious that the process is cumbersome.

We know that multiplication is just repeated addition, while division is just repeated subtraction. Therefore, we can condense our calculation to this:

(5×3)+6+7+(3×4)+4+1

When we assemble the units or execute the process in reverse, we realize the importance of brackets. Now, it seems incorrect to read the sum like a sentence in English, namely, from left to right. The 3 mustn’t be added to 6, because the product of 5 and 3 forms a separate sub-calculation and therefore becomes elemental, another constituent to be added to the rest of the constituents. To identify this sub-calculation and its boundaries, we use brackets. Let’s add some variables to our calculation:

4b+ac+ab+4d+ad+4c

This can be reduced to:

(4+a)(b+c+d)

One can argue that we have sacrificed our freedom to choose the order, but we have indeed gained more than we have lost. What we have achieved is undoubtedly simpler and easier to solve. What was before a sum of six products is now just a product of two sums. At some point in the history of mathematics, someone realized that rather than writing a long string of additions and subtractions and perhaps multiplications (the E in PEMDAS stands for exponents, which is just repeated multiplication), one could simply condense them in this manner with the help of brackets. Brackets must be opened first, or whatever lies inside the brackets must be computed first, because that is how the computation logically unspools. We have, however, proved this in reverse.

Next, the reason why multiplication and division must be performed before addition and subtraction is that the former two are evidently higher operations. Contemplate the meaning of calculations, rather than reading them like a sentence. If someone asks you to buy 6 apples and 3 dozen eggs, would the total objects you’d buy be 108 or 42? The sum, which looks like 6+(3×12), is equal to 42 and not 108. This is why BODMAS or PEMDAS might seem arbitrary, and therefore “unprovable”, but it sure is logical.

In fact, the notion that BODMAS or PEMDAS is a strict “rule” has made teaching mathematics worse. The rule simply dictates that brackets must be first opened to unfold the calculation, after which must be performed any of the two higher operations before any of the two lower operations. There is no compulsion to divide before multiplying or add before subtracting; one must only ensure that higher operations are performed first. This is why, even though the puzzle 2×2-4÷2 in the introduction was solved, in the explanation, by dividing the 4 first, it isn’t essential. Multiplication and division are equally ranked, so you simply work through them from left to right; one can perform the former first or perhaps both simultaneously, just not after the subtraction. Indeed, that is why our introductory puzzle 2×2-4÷2 lands on 2 either way: divide first (2×2-2) or multiply first (4-4÷2), and both routes give 4-2, which is 2.

Similarly, people often blindly believe that, in accordance with the rule, addition must be performed before subtraction. If this is the case, shouldn’t 8-5+3 be equal to 0? The correct answer is, of course, 6, which we obtain when we write the sum as 8+(-5)+3. Notice now how, because we are merely adding numbers, the result is unaffected by the order or direction of adding. By putting the -5 in brackets and implying that subtraction is just addition, but with a negative number, we obtain the correct answer. Simple and convenient, but not arbitrary.

Is It BODMAS Or PEMDAS? Are They The Same?

If you grew up in the United Kingdom, Australia, Canada or India, you were drilled in BODMAS; if you went to school in the United States, you chanted PEMDAS instead. Cue the eternal internet argument: which one is correct? The slightly anticlimactic answer is that they are the same rule wearing different costumes. Neither is more correct than the other, and both deliver identical results.

Line them up and the equivalence is obvious:

• Brackets = Parentheses
• Orders = Exponents (powers and roots; in BODMAS the O is sometimes read as “Of”)
• Division and Multiplication = Multiplication and Division
• Addition and Subtraction = Addition and Subtraction

You may also bump into BIDMAS (the I standing for Indices) and BEDMAS (E for Exponents), but these are just regional rebrands of the same idea. Notice that BODMAS lists Division before Multiplication while PEMDAS lists Multiplication before Division, and yet, as we have already seen, that swap changes nothing. Multiplication and division share a single rung on the ladder, so you work left to right; addition and subtraction share the rung below, and again you work left to right. The acronym is a memory aid, not a strict pecking order, which is precisely the trap that snares so many students. So the URL of this very article spells it “PEDMAS,” a common misspelling; the proper US acronym is PEMDAS, but whichever letters you reach for, the underlying convention is the same.

Who Invented BODMAS, And Why Was It Invented?

Here is the part that disappoints anyone hoping for a heroic origin story: nobody sat down one afternoon and invented the order of operations. It accreted, slowly, as algebraic notation matured. The convention that multiplication outranks addition was already baked into the symbolic notation that took shape in the 1600s, because the distributive property (the very idea that turned our 4b+ac+ab+4d+ad+4c into (4+a)(b+c+d)) makes that hierarchy feel natural.

The tidy acronyms came much later. The terms “order of operations” and the PEMDAS- and BODMAS-style mnemonics were formalized only in the late 19th and early 20th centuries, when the booming market for standardized school textbooks demanded one agreed-upon way to write things down. And the agreement was not instant: as recently as the 1920s, the mathematics historian Florian Cajori documented that authors still squabbled over whether multiplication should outrank division or merely tie with it. The modern “they tie, so go left to right” resolution is the truce that eventually won. In short, BODMAS was not invented so much as negotiated, and it was adopted for one practical reason: to let everyone read the same string of symbols and arrive at the same answer.

Does that truce settle every argument? Not quite. Every few years the internet detonates over an expression like 8÷2(2+2). Apply the convention rigidly and the answer is (8÷2)×(2+2) = 4×4 = 16. Yet many people, and a fair few calculators, treat the implied multiplication in “2(2+2)” as a single glued-together unit and compute 8÷[2×(2+2)] = 8÷8 = 1. Neither camp is being stupid; the expression is simply written ambiguously, and BODMAS alone does not legislate how tightly implied multiplication binds. The cure is the same one mathematicians reach for every time: add a pair of brackets and the ambiguity evaporates. Which loops us right back to where we began, the humble bracket being the most important character in the whole acronym.