Order of Operations Explained: The 150-Year History Behind PEMDAS and the Obelus Symbol

A Deceptively Simple Math Problem

A few weeks ago I posted a puzzle that turned out to be more divisive than expected. The question was: what does 6²÷2(3)+4 equal?

Some people instinctively answered 10 — but that only works if you treat multiplication as higher priority than division, even though they appear in left-to-right sequence. The parentheses around the 3 throw people off.

How the Order of Operations Actually Works

The correct answer is 58. Here is the full calculation:

36 ÷ 2 × 3 + 4 = 18 × 3 + 4 = 54 + 4 = 58

Today we all use a universally accepted hierarchy of operations: parentheses first, then exponents, then multiplication and division (left to right), and finally addition and subtraction. This is what most English-speaking students learn as PEMDAS.

The 150-Year History of Mathematical Notation

Here is the thing — the notation and order of operations we take for granted only solidified around 150 to 200 years ago. Before that, mathematical symbols carried different meanings depending on who was writing.

Johann Rahn, the mathematician who introduced the obelus symbol (÷) for division, actually used it differently than we do today. Take a look at page 76 of his Algebra and you will see what I mean.

Why the Obelus Symbol Once Gave a Different Answer

If you apply the obelus the way Rahn originally intended, the same expression could reasonably yield 3.6:

36 ÷ (2 × 3 + 4) = 36 ÷ (6 + 4) = 36 ÷ 10 = 3.6

Under that historical reading, the divisor encompasses everything that follows — not just the next number.

Order of Operations Is a Convention, Not a Law of Nature

None of this means you should start calculating that way. Modern convention is clear, and consistency is what makes mathematics a shared language. But it is a useful reminder that even something as fundamental as the order of operations is a human agreement that evolved over time — not an eternal mathematical truth handed down from above.