Is a Whole Number A Rational Number

Question

Is a Whole Number part of A Rational Number or a whole number??

Answer 1

A whole number is a rational number. Write a whole number, $n$, as $\dfrac{n}{1}$.

Answer 2

Every whole number is a rational number: for example, $3 = \dfrac 31$. So it is rational.

Every whole number $n$ can be written as a fraction of integers: $n =\dfrac n1$. We aren't required to write it that way; we just need to know that it is possible to express every whole number as a fraction of integers, and hence it is rational.

Answer 3

The real answer, as usual, is "it depends". As the other answers have indicated, it is possible to identify whole numbers with certain rational numbers. On the other hand, it's also possible to identify rational numbers with certain ordered pairs of integers. So it really depends on your perspective/purpose. If you're doing something like number theory, you'll be thinking in terms of a whole number being a rational number. If you're thinking in terms of "mathematical foundations", you'll most likely be looking at it from the other direction.

Answer 4

It depends on how you set things up.

First, some notation. Write

• $\mathbb{N}$ for the natural numbers. (or 'whole' numbers, if you prefer).
• $\mathbb{Q}$ for the rational numbers.

Okay. So usually, we set things up such that if $x \in \mathbb{N}$, then $x \in \mathbb{Q}$. More tersely: $$\mathbb{N} \subseteq \mathbb{Q}.$$

However, we can also set things up so that natural numbers and rational numbers are different things that have a special relationship.

Specifically, they are different in that whenever $x \in \mathbb{N}$, it follows that $x \notin \mathbb{Q}$. (And vice versa). More tersely:

$$\mathbb{N} \cap \mathbb{Q} = \emptyset.$$

Whichever way you set it up, there's a function that maps each natural number $n$ to its corresponding rational number $\frac n 1$.

There is always a special natural number called $1_\mathbb{N}$, and a special rational number called $1_\mathbb{Q}$. This gives one (of possibly many ways) of defining this function (called the canonical embedding) of the natural numbers into the rationals. In particular, we can define this embedding as the unique $$f\colon \mathbb{N} \to \mathbb{Q}$$ such that $$f(1_\mathbb{N} + \cdots + 1_\mathbb{N}) = 1_\mathbb{Q} + \cdots + 1_\mathbb{Q},$$ where the number of terms we're summing on both sides are the same.

Okay, what's so special about this function $f$? Well, it preserves equality, order, addition, multiplication, exponentiation, etc. If this last comment is unclear, please say so and I will clarify.