Every script has rules nobody remembers learning. In English we read from left to right and from top to bottom; in Arabic, from right to left. Mathematical writing has rules of this kind too, and some are so strong that they look like mathematical truths. They are not: they are conventions, agreements that could have been different.

The order of operations

What is $2 + 3 \cdot 4$? Almost everyone answers $14$: first the multiplication, then the addition. But nothing in mathematics forces this order. If the convention were “from left to right”, the answer would be $20$. The rule we use (powers, then multiplications and divisions, then additions and subtractions) settled slowly in the algebra books of the sixteenth to nineteenth centuries, because it saves brackets in the expressions that appear most often: polynomials such as $3x^2 + 2x + 1$ would be unreadable otherwise.

In the United States students memorise it as PEMDAS; in the United Kingdom, as BODMAS. The names change, the convention is the same, and it has a gap that resurfaces from time to time on social media:

$$6 \div 2(1+2) = \ ?$$

Those who apply the rule to the letter, left to right among divisions and multiplications, do $6 \div 2 = 3$ and then $3 \cdot 3 = 9$. Those who read $2(1+2)$ as a block, because the multiplication is implicit, with no sign, do $6 \div 6 = 1$. Both readings have supporters, and some scientific style guides prefer the second. The point is not to decide who is right: it is to notice that the expression is badly written. A mathematician would never write it that way. They would write $\frac{6}{2}(1+2)$ or $\frac{6}{2(1+2)}$, and the doubt would vanish.

Signs that misbehave

  • $-3^2$. The power comes before the minus sign, so $-3^2 = -(3^2) = -9$. To get $9$, one writes $(-3)^2$. Many calculators and spreadsheets follow this convention; some do not.
  • $\sin^2 x$. It means $(\sin x)^2$, by an old custom that saves a pair of brackets. But $\sin^{-1} x$ means $\arcsin x$, the inverse function, and not $\frac{1}{\sin x}$. The same exponent on top of the same symbol has two meanings, as we saw in chapter 3.
  • $2a$ and $2\tfrac{1}{2}$. Placing a number next to a letter means multiplying: $2a = 2 \cdot a$. Placing a number next to a fraction, in books that use mixed numbers, means adding: $2\frac{1}{2} = 2 + \frac{1}{2}$. The same juxtaposition, two operations.
  • $f(x+1)$ and $a(x+1)$. The first is the value of the function $f$ at the point $x+1$; the second is a product. Only the context says whether the letter before the bracket is a function or a number.

The comma and the point

In Brazil, $1{,}5$ is one and a half and $1.500$ is one thousand five hundred. In the United States it is the other way round: $1.5$ is one and a half and $1,500$ is one thousand five hundred. The same sign, $1.000$, means a thousand in one country and one in the other. In 2003 the General Conference on Weights and Measures accepted both decimal separators and recommended separating thousands with a thin space, $1\,500$, precisely so that the number does not depend on the reader’s country.

Exercise: the sign and the power

Calculate $-2^2 + (-2)^2$ and $-2^3 + (-2)^3$.

$-2^2 + (-2)^2 = -4 + 4 = 0$. But $-2^3 + (-2)^3 = -8 + (-8) = -16$: with an odd exponent the two readings coincide, $-2^3 = (-2)^3 = -8$. That is why the problem only shows up with even exponents, and anyone who has practised only with cubes may not even notice it exists.

Exercise: write it unambiguously

Rewrite $6 \div 2(1+2)$ in two unambiguous ways, one for each reading, and give the value of each.

Left-to-right reading: $\dfrac{6}{2}\cdot(1+2) = 3 \cdot 3 = 9$. Block reading: $\dfrac{6}{2(1+2)} = \dfrac{6}{6} = 1$. The fraction bar, invented precisely to show what goes on top and what goes below, settles what the sign $\div$ leaves open.

Chapter index · ← The captain’s age: from word problem to equation · Errors that make sense: false rules →