What number goes in the box? $$8+4=\square+5$$In the late 1990s the American researchers Karen Falkner, Linda Levi and Thomas Carpenter asked this question of pupils from first to sixth grade. In every grade, fewer than one in ten answered $7$. The commonest answers were $12$ and $17$. Those who answered $12$ did the sum $8+4$ and wrote the result after the $=$, as they always did. Those who answered $17$ added up all the numbers they could see. Neither group got the sum wrong. Both got the reading of the equals sign wrong.
In 2019 a sum with four numbers spread across social media and provoked millions of arguments: $$8\div2(2+2)=\;?$$Half the people answered $16$; the other half, $1$. Calculators of different brands gave different answers. Mathematics teachers were divided. And the curious thing is that both sides were right, each according to a convention. Those who answer $16$ read from left to right: $8\div2=4$, and $4\cdot(2+2)=16$. Those who answer $1$ take the “glued” multiplication without a sign, $2(2+2)$, as a block that comes before the division: $8\div(2\cdot4)=1$. Both conventions exist and are used in serious books. The problem is not arithmetic. It is syntax: the expression was written in a way that does not make clear how the signs group together. A mathematician would never write it like that; they would write $\frac{8}{2}(2+2)$ or $\frac{8}{2(2+2)}$, and there would be no fight.
In Baghdad, around the year 820, Muhammad ibn Musa al-Khwarizmi worked at the House of Wisdom, the great library of the caliph al-Ma’mun. There he wrote a book with a long title, The Compendious Book on Calculation by Restoration and Balancing. “Restoration”, in Arabic, is al-jabr. From that word came algebra. Opening page of al-Khwarizmi’s book on al-jabr and al-muqābala, in a manuscript copy.Muhammad ibn Musa al-Khwarizmi · Wikimedia Commons · public domain The whole book contains not a single symbol. No $x$, no $+$, no $=$. Even the numbers are written out in words. One of the most famous problems is this: