CHAPTER IX

What the Root of a Number and of Any Other Quantity Is

Folio 0026 (f. 5v) Initial

Because in the course of our treatise we shall frequently have occasion to speak of Roots (Radici), it seems fitting to explain briefly here what that term signifies, even though we have treated of it at length in every way in our great work [Summa].

I say, nevertheless, that the root (radice) of a quantity is likewise a quantity which, when multiplied by itself, produces that quantity of which it is said to be the root; and that multiplication made into itself is called the square (quadrato) of the said root.

Thus we say that the root of $9$ is $3$, the root of $16$ is $4$, and the root of $25$ is $5$, and so in other cases; while $9$, $16$, and $25$ are called squares (quadrati).

From this it must be known that there are certain quantities which do not have a root that can be named exactly by a number: for instance, $10$ has no number which, multiplied by itself, makes $10$ exactly; and the same is true of $11$, $12$, $13$, and others like them.

Wherefore there arise two kinds of roots:

  • One is called discrete (discreta), or let us say rational (rationale), which is a root that can be assigned exactly by a number, just as the root of $9$ is $3$;

  • The other is called surd (sorda), which is one that cannot be given exactly by a number, as we have said of the root of $10$ and others.

These latter are called by another name irrational (irrationali), because all quantities that cannot be assigned exactly by a number [or ratio of integers] are called irrational in the art, whereas those that can be given by a number are called rational (rationali). And let this suffice for our purpose concerning roots.