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