Natural numbers are probably the most common numbers that you use in everyday life. Natural numbers are in general used for counting such as “there will be seven people at the party” and for ordering such as “look, Usain Bolt is first again”.
The set of natural numbers contains numbers 1, 2, 3, 4, 5, etc. We denote the set of natural numbers as ℕ. Sometimes we include the number zero to a set of natural numbers. You can also see the symbol ℕ0. It explicitly mark that we’re talking about set of natural numbers, zero included. On the other hand you can also see symbol ℕ + and this explicitly says that we are talking about positive numbers, i. e. without zero.
Properties of Natural Numbers
- The set of all natural numbers is infinite. The proof is quite easy. Think of any natural number. Now you add one – bam! You have a new natural number. Thus we can find for any natural number x a number x + 1 which is different and greater than x.
- Natural numbers are closed under addition. It means that if you add any two natural numbers you will always get another natural number. Try it! E. g. 5 + 6 = 11 and all of them are natural numbers.
- The same goes for multiplication. We can multiply any two natural numbers and we always get another natural number. E. g. 2 · 7 = 14.
- Natural numbers are not closed under subtraction. E. g. 4 − 9 = − 5. The numbers 4 and 9 are both natural numbers but the result of subtraction is not. The number − 5 is not a natural number.
- The same goes for division: 5 / 2 = 2.5 and the number 2.5 is not natural.
Graphical Representation of Natural Numbers
We can show natural numbers using a simple graph with a single axis:
We see the x-axis with ten marked points. Every point represents a natural number. As you can see, it’s the set ℕ + since the zero point is not highlighted. Of course we cannot show all natural numbers since there infinite amount of them.