set of natural numbers subset of the set of real numbers

Question

The natural numbers are said to be a subset of the real numbers but how is this possible since in the set of natural numbers division is not allowed.

Answer 1

Real numbers are basically all numbers, decimal, whole, negative, and positive except for imaginary numbers. They include both rational and irrational numbers. A more formal definition is any value that can represent a distance along a line (-ve and positive denoting direction).

Natural numbers are just whole positive numbers. Since whole positive numbers can represent a distance along a line, they are a subset of real numbers.

I really don't understand what division has to do with this.

Answer 2

That the set of natural numbers is a subset of the set of real numbers just means that all natural numbers are also real numbers. You may be thinking of the term subfield, which is a subset that is a field with respect to the same operations as the larger set. Since the set of natural numbers has no multiplicative inverses (or additive for that matter), it is indeed not a subfield of the set of real numbers.

Answer 3

A set $A$ is said to be a subset of a set $B$ if and only if every member of $A$ is a member of $B$.

Since every natural number is a real number, the set of natural numbers($\Bbb N$) is a subset of the set of real numbers($\Bbb R$). It should be clear from the definition that containment(subset) relationships have nothing to do with the possible binary operations defined on the set.

What you might be thinking of is a group, in which case the set of non-zero real numbers form a group under multiplication but the set of natural numbers do not form a subgroup because you cannot guarantee a multiplicative inverse for every element.

Answer 4

You mention that, "in the set of natural numbers, division is not allowed". That's not entirely true. We can do lots of division with natural numbers, e.g., $6\div 2=3$. I think you mean that the set of natural numbers is not "closed under division". That's true. There are lots of division problems involving natural numbers whose solution is not a natural number.

The set of real numbers is closed under division, with the usual provision that we don't divide by $0$. The natural numbers $6$ and $5$ are also real numbers, since the naturals are a subset of the reals. Their quotient, $\frac65$, is another real number, one which is not a natural number. There is no problem here.

Being a subset of the real numbers doesn't mean retaining all of the closure properties of real numbers. It's like this: the prime numbers are a subset of the natural numbers, and even though the natural numbers are closed under addition, the prime numbers are not: $3+5=8$. Here, the sum of two primes equals a natural that is not a prime, just like we saw the quotient of two naturals can equal a real that is not a natural.