The simplest signs in mathematics are also the youngest. Euclid never wrote a $+$. Al-Khwarizmi never wrote an $=$. Viète, who already used letters, still wrote aequatur in full. Almost all the operation signs a child learns in the first year of school were born in Europe between the end of the fifteenth century and the end of the seventeenth, in a span of about two hundred years. 1489Johannes Widmann prints $+$ and $-$ in a book of commercial arithmetic in Leipzig. 1525Christoff Rudolff uses $\surd$ for the square root in his algebra book, the Coss. 1557Robert Recorde invents $=$ in The Whetstone of Witte. 1591François Viète uses letters for known and unknown quantities. 1631William Oughtred uses $\times$ for multiplication; in the same year Thomas Harriot’s posthumous book appears with $<$ and $>$. 1637René Descartes writes powers as exponents: $x^3$, $x^4$. 1659Johann Rahn uses $\div$ for division. 1698Leibniz proposes the dot $\cdot$ for multiplication, so as not to confuse $\times$ with the letter $x$. Plus and minus: warehouse marks The signs $+$ and $-$ first appear in print in 1489, in an arithmetic manual for merchants written by Johannes Widmann. There they do not quite mean an operation yet: they mark excess and shortfall. A crate of goods that weighed more than it should was marked $+$; one that weighed less, $-$. The most accepted hypothesis is that $+$ is an abbreviation of et (“and”, in Latin), written quickly until it became a cross.
Try multiplying $\text{XLVII}$ by $\text{XII}$. Without converting to our digits. It is hard to know even where to start. The $\text{X}$ in $\text{XLVII}$ is worth ten, but it comes before the $\text{L}$, so it is subtracted: $\text{XL}=40$. The $\text{V}$ is worth five wherever it stands. There is no ones column and no tens column; each symbol carries its value hanging from its own shape. The same sum, written as $47\times 12$, is done in two lines:
Ask a student what the square root of 9 is and they will probably say “3”. Ask why it is called a root, and why square, and the conversation usually ends there. The name has become an empty label, and the symbol $\sqrt{\phantom{9}}$ a strange hook that “pulls” something out of the number. But the name once told the whole story. Greek geometers simply called the number they were looking for the side (pleurá): the side of a square of area 9 measures 3. Arab algebraists called it jidhr, the root from which the square “grows”, and medieval Latin translators wrote radix. A sentence such as