Muhammad ibn Musa alKhwārizmī (780–850 CE) was the first mathematician to write an algebra book: AlKitab almukhtasar fi hisab aljabr wa’lmuqabala (Compendium on Calculating by Completion and Reduction^{1}). In this treatise, alKhwārizmī first emphasized the notion of number via his statements [Rosen 1831, 5]:
When I considered what people generally want in calculating, I found that it always is a number.
I also observed that every number is composed of units, and that any number may be divided into units.
Moreover, I found that every number, which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled, just as before the units were: thus arise twenty, thirty, etc., until a hundred; then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; then the thousand can be thus repeated at any complex number; and so forth to the utmost limit of numeration.
He then proposed a classification of numbers into three types [Rosen 1831, 5–6]:
I observed that the numbers which are required in calculating by Completion and Reduction are of three kinds, namely, roots, squares, and simple numbers relative to neither root nor square.
A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions descending.
A square is the whole amount of the root multiplied by itself.
A simple number is any number which may be pronounced without reference to root or square.
Here, the phrase “in calculating by Completion and Reduction” is a reference to the practice of equationsolving. This, together with alKhwarizi’s descriptions of the kinds of numbers he had observed in that practice, suggests a natural correspondence between his categories and the three terms we associate with quadratic equations today: “roots” with the linear term; “squares” with the quadratic term; and “simple numbers” with the constant term.
We also notice the quantitative aspect of alKhwārizmī’s roots, squares, and simple numbers categories, which allows any two of the three to be used to generate a simple equation [Rosen 1831, 8]:
A number belonging to one of these three classes may be equal to a number of another class ; you may say, for instance, “squares are equal to roots," or “squares are equal to numbers,” or “roots are equal to numbers."
We further see the emergence of a new kind of quantity, which alKhwārizmi used to describe three “compound species” equations [Rosen 1831, 8]:
I found that these three kinds; namely, roots, squares, and numbers, may be combined together, and thus three compound species arise; that is, “squares and roots equal to numbers;” “squares and numbers equal to roots;” “roots and numbers equal to squares.”
alKhwārizmī’s treatise is thus generally considered a handbook for solving linear and quadratic equations. His text classified these, however, as six different equation types. The following table outlines this classification, along with the examples of each type that alKhwārizmī solved in the first part of his text. Note that the word “dirhem” that appears in the example for Type 4 refers to a type of coin that was historically used in Baghdad; it was often used by alKhwārizmī as the units for the simple number in an equation.
Equation Type (prose format)

Equation Type
(modern notation)

Type 1. Squares are equal to roots
Examples:
 a square is equal to five roots of the same
 one third of the square is equal to four roots
 five squares are equal to ten roots

\[ax^2 = bx\]

Type 2. Squares are equal to numbers
Examples:
 a square is equal to nine
 five squares are equal to eighty
 the half of the square is equal to eighteen

\[ax^2 = c\]

Type 3. Roots are equal to number
Examples:
 one root equals three in number
 four roots are equal to twenty
 half the root is equal to ten

\[bx = c\]

Type 4. Roots and squares are equal to numbers
Examples:
 one square, and ten roots of the same, amount to thirtynine dirhems
 two squares and ten roots are equal to fortyeight dirhems
 half of a square and five roots are equal to twentyeight dirhems

\[bx+ax^2 = c\]

Type 5. Squares and numbers are equal to roots
Example:
 a square and twentyone in numbers are equal to ten roots of the same square

\[ax^2+c = bx\] 
Type 6. Roots and numbers are equal to squares
Example:
 three roots and four of simple numbers are equal to a square

\[bx+c = ax^2\]

Notes
1. Aljabr = completion; almuqabala = reduction.