Definition:Rational Number

Informal Definition

A number in the form $\displaystyle \frac p q$, where both $p$ and $q$ are integers ($q$ non-zero), is called a rational number.

The set of all rational numbers is usually denoted $\Q$.

Thus:

$\displaystyle \Q = \left\{{\frac p q: p \in \Z, q \in \Z^*}\right\}$

A rational number such that $q \ne 1$ is colloquially and popularly referred to as a fraction. The similarity between that word and the word fracture is no accident.

Variants on $\Q$ are often seen, for example $\mathbf Q$ and $\mathcal Q$, or even just $Q$.

A rational number is positive iff $p \cdot q > 0$. It is negative iff $p \cdot q < 0$ and it is zero iff $p \cdot q = 0$. Every non-zero rational number is either positive or negative.

Formal Definition

The field $\left({\Q, +, \times}\right)$ of rational numbers is the quotient field of the integral domain $\left({\Z, +, \times}\right)$ of integers.

This is shown to exist in Existence of Quotient Field.

In view of Quotient Field is Unique, we simply pick a quotient field of $\Z$, give it a label $\Q$ and call its elements rational numbers.

We note that $\left({\Z, +, \times, \le}\right)$ has a total ordering $\le$ on it.

From Total Ordering on Quotient Field is Unique, it follows that $\left({\Q, +, \times}\right)$ has a unique total ordering on it that is compatible with $\le$ on $\Z$.

Thus $\left({\Q, +, \times, \le}\right)$ is a totally ordered field.

Geometrical Definition

As Euclid defined it:

With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some on length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or square only, rational, but those which are incommensurable with it irrational.

And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

The Elements: Book X: Definitions $3$ and $4$

Comment

The name rational has two significances:

$(1): \quad$ The construct $\displaystyle \frac p q$ can be defined as the ratio between $p$ and $q$.

$(2): \quad$ In contrast to the concept irrational number, which can not be so defined. The ancient Greeks had such a term for an irrational number: alogon, which had a feeling of undesirably chaotic and unstructured. The proof that there exist such numbers was a shock to their collective national psyche.

Sources

• Seth Warner: Modern Algebra (1965): $\S 1$
• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.2$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.1$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $4$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 1$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(b)}$