It is easy to take mathematics for a set of truths carved into the fabric of the universe — independent of humans and unchanging. As far as the results go, that is true: two apples added to three always make five, regardless of era, language, or whether anyone likes it. But the way we write those operations down — the symbols, the precedence rules, the left-to-right direction — is not a law of nature. It is a human convention. The grammar of a certain language, one that evolved over centuries to avoid misunderstanding and fit more meaning into less space.
It pays to keep the two layers apart. Part of arithmetic is convention (the "+" sign, the "⋅", "multiplication before addition", reading left to right) and part is strict (the result of merging two sets, the impossibility of dividing by zero). This piece is the narrative layer above our arithmetic branch: where our symbols came from, why some operations break intuition, and why a simple equation from the internet can divide even professional mathematicians.
Symbols nobody invented all at once
Before today's notation, mathematics was rhetorical — operations, numbers, and relationships were described in full sentences of ordinary language. It was excruciatingly slow. The move from words to symbols took several hundred years, driven by very down-to-earth forces: maritime trade, banking, and the Renaissance printing industry.
The earliest shorthands for addition and subtraction were the medieval Italian più (more) and Latin minus (less), eventually squeezed down to the letters p and m with a horizontal stroke of abbreviation. Today's + descends from a ligature of the Latin et ("and"), while the origin of − is still disputed — some trace it to that stroke over the "m", others to the marks merchants used for a shortfall in a barrel. Tellingly, the first print to carry "+" and "−" (a 1489 work by Johannes Widmann) used them not as operators but as marks of surplus and deficit of goods in ledgers. They only earned the rank of proper operation signs in the 16th century, through textbooks like Robert Recorde's The Whetstone of Witte (1557).
The slanted cross × for multiplication was introduced by the English mathematician William Oughtred in 1631. Gottfried Wilhelm Leibniz criticised it almost at once: in a 1698 letter to Johann Bernoulli he wrote that "×" was too easily confused with the letter x standing for an unknown, and proposed a dot (⋅) instead. In continental Europe — Poland included — the dot became the standard. The shortest way to write multiplication, though, is no sign at all: 2a or 2(1+2) is multiplication "by juxtaposition", already described in the 16th century. Remember this trick — in a moment it will start a war.
| Sign | Author / first use | Date | Where it came from |
|---|---|---|---|
| + − | print by Johannes Widmann | 1489 | at first marks of surplus and deficit in merchants' ledgers |
| × | William Oughtred | 1631 | the slanted "St Andrew's cross" |
| ⋅ | popularised by Leibniz | 1698 | a dot instead of "×", to avoid clashing with the unknown x |
| ÷ | Johann Rahn (John Pell) | 1659 | the obelus: a picture of a fraction — a bar between two dots |
| / | Augustus De Morgan | 1845 | a slash to ease setting text on a single line |
| : | William Oughtred | 17th c. | originally a notation for a ratio (proportion) |
Why subtraction and division break the symmetry
The four basic operations are not equals. Addition and multiplication are "well-behaved" — symmetric; subtraction and division lack that symmetry, and not by accident.
Addition is commutative (a + b = b + a) and associative ((a + b) + c = a + (b + c)); so is multiplication. Subtraction and division break both rules: 7 − 3 = 4, but 3 − 7 = −4; (12 − 5) − 3 = 4, but 12 − (5 − 3) = 10. That is why, with subtraction and division, parentheses stop being decoration — they become the only way to state unambiguously what is being combined with what.
The deeper reason is elegant: subtraction and division are not separate operations. They are disguises. To subtract b is to add its opposite: a − b = a + (−b). To divide by b (for b ≠ 0) is to multiply by its reciprocal: a ÷ b = a · b⁻¹. Reduce them back to addition and multiplication, and the symmetry returns in full. So the asymmetry is not a property of the world — it is the price of a shorthand.
| Operation | Commutative | Associative | Identity element | Distributes over + |
|---|---|---|---|---|
Addition + | yes | yes | 0 (a + 0 = a) | — |
Subtraction − | no | no | right side only (a − 0 = a) | — |
Multiplication ⋅ | yes | yes | 1 (a · 1 = a) | yes: a(b + c) = ab + ac |
Division ÷ | no | no | right side only (a ÷ 1 = a) | right side only: (a + b) ÷ c = a ÷ c + b ÷ c |
Dots before dashes — the highway code of notation
The question "why do we do multiplication before addition?" cuts to the core. The answer is: because we agreed to. Had humanity adopted the opposite rule, the physical world would not have moved a millimetre — we would only have had to rewrite all of mathematics differently.
The reason for this particular agreement is practical: brevity. Polynomials like ax² + bx + c are the single most common thing anyone writes. If multiplication had no priority, you would have to write it as ((a·(x²)) + (b·x)) + c — a forest of brackets in every line. Giving multiplication priority makes brackets the exception rather than the rule. The split into precedence levels took shape informally in the 17th century along with algebraic notation; acronyms like PEMDAS or BODMAS are only mnemonics from the age of mass textbooks around the turn of the 20th century. The German tradition captures it without any acronym, in a single phrase: Punktrechnung vor Strichrechnung — "dot" operations (× ÷) before "dash" ones (+ −).
And here lurks the most common mistake: PEMDAS is often read as a rigid ladder in which "A" (addition) comes before "S" (subtraction). Not so — multiplication and division have equal priority, and so do addition and subtraction, and among equals you simply work left to right. That is why 10 − 4 + 2 = 8, not 4. We develop this on a dedicated page about the order of operations.
6 ÷ 2(1+2), or the war over notation
Every so often the internet splits into two warring camps over the expression 6 ÷ 2(1+2). One side defends the answer 9, the other 1, and — most interestingly — both are right, because the problem lies not in the people but in the notation itself.
The "9" camp sticks to the textbook rule: brackets first (1 + 2 = 3), then, since division and multiplication are equal, work left to right: 6 ÷ 2 = 3, then 3 · 3 = 9. The "1" camp invokes multiplication by juxtaposition: the sign-free 2(1+2) is treated as one fused term and computed before the explicit division — 2 · 3 = 6, so 6 ÷ 6 = 1. That second convention really is the default in some scientific literature and older academic textbooks.
The moral is not that one school is stupid. The moral is that a linear expression with the ÷ sign next to implicit multiplication is simply ambiguous — not, as mathematicians say, well-defined. Even scientific calculators (sometimes two models from the same company) can return different answers, depending on how their firmware treats juxtaposition. That is why professionals do not write this way at all — they replace ÷ with an unambiguous fraction bar, which shows at a glance what is in the numerator and what is in the denominator. The fight over 6 ÷ 2(1+2) is not a maths puzzle. It is proof that good notation is part of mathematics.
The ban that rescues everything else
There is one absolute boundary in arithmetic: you must not divide by zero. It is not about a lack of computing power — it is that allowing this operation would collapse the logical consistency of everything else.
Division is the inverse question to multiplication: a ÷ b = x means "which x satisfies x · b = a?". Substitute b = 0. For a non-zero a we want an x such that x · 0 = a. But x · 0 is always 0, so when a ≠ 0, no such x exists — the operation is impossible. And 0 ÷ 0? The reverse: the equation x · 0 = 0 is satisfied by every x, so the result is not unique — we call it an indeterminate form.
That this is no academic whim is shown by the classic "proof" that 2 = 1. Start from a = b:
a² = ab → a² − b² = ab − b² → (a − b)(a + b) = b(a − b)
It is now tempting to cancel (a − b) on both sides, getting a + b = b, and hence 2 = 1. The catch is that under the assumption a = b, the bracket (a − b) equals zero — and that hidden division by zero smuggles in the absurdity. The ban is therefore not arbitrary; it is a fuse that stops arithmetic from proving something false. You will find more on the operation itself, and on division with a remainder, on the page about division.
Two worked examples, step by step
Rules are clearest in action. First, an expression with brackets and a power:
18 − (3² − 5) · 2 + (12 ÷ 3)
Brackets first, and inside the first one the power before the subtraction: 3² = 9, so (9 − 5) = 4; the second bracket is 12 ÷ 3 = 4. That leaves 18 − 4 · 2 + 4. Multiplication has priority: 4 · 2 = 8, so 18 − 8 + 4. Finally, the equal-priority operations from the left: 18 − 8 = 10, 10 + 4 = 14. Result: 14.
The second example is the "equal priority" trap:
24 ÷ 4 · 3 ÷ 2
There is no hierarchy here — multiplication and division are computed strictly from the left. 24 ÷ 4 = 6, then 6 · 3 = 18, finally 18 ÷ 2 = 9. Result: 9. Anyone who "multiplied first" because multiplication sounds more important would get the wrong answer. Order among equals is not a ranking — it is simply reading the sentence from left to right.
What is convention and what is truth
Arithmetic has two layers, and it is worth not confusing them. The signs, the priorities, and the reading direction are convention — we could have agreed otherwise, and mathematics would still work the same, only look different. But that five apples are five apples, that subtraction is not commutative, and that you cannot divide by zero — that is truth, which no convention can touch. Next time you see an internet brawl over 6 ÷ 2(1+2), you will know they are arguing not about mathematics but about punctuation. And the real mathematics waits calmly in our arithmetic branch.
Further reading
- Wikipedia, Order of operations — the development of algebraic notation from Viète to today.
- Plus Magazine, The PEMDAS Paradox (plus.maths.org) — the linguistic and logical background to the internet fights over ambiguous equations.
- Wolfram MathWorld, Division by Zero — a formal treatment of division by zero in field theory.
- The Math Doctors, Order of Operations: Historical Caveats — how the notion of "order of operations" looked in 19th-century textbooks.
- MacTutor, Earliest Uses of Symbols of Operation — a chronicle of the first appearances of the operation signs.