Vocabulary for Fractions

When you see $\frac{8}{3}$ (read as eight-thirds or the quotient of 8 divided by 3), what do you see? A quotient? A fraction? Or perhaps both at the same time.

I. Quotients

The quotient of the division of 8 by 3 cannot be written in terminating decimal form since 8 does not divide evenly by 3.

The quotient does, however, exist: It is the number that, when multiplied by 3, is equal to 8.

In general, if a and b are two numbers and b is not zero, the quotient of the division of a by b is the number obtained by dividing a by b. This is, therefore, the number that, when multiplied by b, is equal to a.
To present this number, we can use either fraction notation (which is exact) or decimal notation (which may be an approximation).

II. Fractions

To present the quotient of 8 by 3, we use the fraction $\frac{8}{3}$ (read as eight-thirds).


Generally, a fraction is a notation for a number in the form $\frac{a}{b}$ or a/b, where a is a whole number and b is a natural number.


Usually the definition given above is extended to include cases where the numerator is an integer; thus $\frac{-8}{3}$ is considered to be a fraction.

III. Fraction notation

Fraction notation of a number is the notation for the number in the form $\frac{x}{y}$ (sometimes read as x over y), where x represents a number and y represents a non-zero number.


Notes:
Let’s return to the question posed in the introduction: When we see $\frac{8}{3}$, we see a fraction. If we say that we “see” a quotient (that is, a number), we are misusing the terms somewhat; but this misuse, for reasons of convenience, is common practice.