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What are Natural Numbers?

Updated on January 15, 2011

What are numbers exactly?

The Set of Real Numbers



You have been using numbers since Kindergarten. And you will continue to use numbers everyday, throughout your life.

But have you ever asked yourself: What are numbers exactly?

This question is not as simple as it seems.

Thinking Question: two candy bars added to three candy bars is five candy bars. But what is two candy bars added to three gums or three teddy bears? In most cases, it is simply not logical to add objects of different kinds. Yet we blithely write 2+3 = 5. What exactly are we adding?

Numbers have been used for thousands of years. But to define a number with the rigour expected in mathematics is not easy. In most cases, we do not need that much rigour. We still can develop a workable concept of “number”.


Counting Sheep
Counting Sheep

Keeping Track with Tally Marks


The concept of number probably arose from the need to keep track of objects.

Let us imagine a time when human beings did not know how to count.

 Folk may not have known how to count but they were human. And human beings like to acquire property of all kinds. Suppose a farmer wanted to know if his flock of sheep was increasing or not. He could have made tally marks on a stick (or stone or clay tablet) to represent each animal in his flock. (Thinking Question: how could the farmer use the tally marks to determine whether the flock was increasing or not?)

Eventually people realized that the tally marks did not have to represent the objects originally being counted but could represent any set with the same number of objects. Consider a stick with three tally marks. This stick can be used to count the objects in any set with exactly three objects: three kittens, three golden necklaces, three white elephants and so on.

Now consider the class of all sets with exactly three elements. All the elements of this class have one to one correspondences with each other.

The number 1

 We call this class the number 3.

Similarly the number 1 is the class of all sets with exactly one element. The number 2 is the class of all sets with two elements. 

Our most important number is 1. Why?

Building the Set of Natural Numbers

We intuitively understand that by adding 1 to 1, we obtain 2. (Though to “prove it” with the rigour demanded by Mathematics is another story). The action of “adding” is represented by the mathematical operation of addition or +.

By consecutively adding 1 we obtain the set of natural or counting numbers.

1+ 1 = 2

2+ 1 = 3

3 + 1= 4

And so on…

Notice we used 1 to build the Set of Natural Numbers or

N= {1, 2, 3, 4, …}. (The three dots at the end of the list mean that the list keeps going on forever.

Operations on Natural Numbers

We can add any two natural numbers to obtain another natural number.

Thinking question: Why is 2 + 3 = 5?

Now suppose the farmer had 6 sheep but a pack of wolves killed 3 of them. How many sheep does the farmer have now?

The action of “taking away” can be expressed by the operation of “subtracting” (represented by the symbol “_”

We also use the natural numbers to” multiply”.

2 + 2 + 2 = 2 x 3

Properties of Natural Numbers

We notice that the Natural numbers have certain properties

Properties of the Natural Numbers

Let a,b, and c be  natural numbers.

Commutative Property of addition

a + b = b + a

Commutative Property of Multiplication

a x b = b x a

Associative Property of Addition

(a + b) + c = a + (b + c)

Associative Property of Multiplication

(a x b ) x c = a x (b x c )

Identity under Multiplication

1 x a = a x 1 = a

Distributive Property

a x (b + c) = a x b + a x c

You can verify these properties with any combination of natural numbers.


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    • simeonvisser profile image

      simeonvisser 6 years ago

      The big question is always when someone refers to the natural numbers, do they include zero or not? Sometimes there is even special notation to denote natural numbers with the zero and natural numbers without the zero.

    • HABMATH profile image

      HABMATH 6 years ago

      Good question! Zero is not considered a natural number.

      The set of natural numbers plus zero is known as the set of whole numbers. Thus zero is considered to be a whole number.I will provide more clarification on this point in forthcoming hubs.

    • simeonvisser profile image

      simeonvisser 6 years ago

      @HABMATH: That would be very good, thanks for clarifying already!

    • munirahmadmughal profile image

      munirahmadmughal 6 years ago from Lahore, Pakistan.

      "What are real numbers?"

      The hub is informative and educative and at the same time interesting.

      Mathematicians are to be saluted for having explored and exploited the numbers and their classifications and the precision they have obtained is really marvellous and hi ghly commendable in all fields of knowledge.

      The food for thought is that the digits are only ten (0,1,2,3,4,5,6,7,8,and 9). All kinds known or yet to be known are cofined to these digits like the alphabets to make any word in the world. This indictes towards the Creator, the Lord of all the worlds, the Beneficent, the Merciful.

      All accuracy and all precision are to guide towards the truth and to serve the truth. All confidence comes when a thing is found true. A man becomes trustworthy when he is tested by all means as truthful.

      May God bless all.

    • simeonvisser profile image

      simeonvisser 6 years ago

      @munirahmadmughal: You are aware that there are an infinite number of numeral systems and that we have just decided to settle on a base 10 numeral system? I mean to say that our numeral system is convenient but just one of many numeral systems.

    • HABMATH profile image

      HABMATH 6 years ago


      The base 10 or decimal system is written using the set of digits {0,1,2,3,4,5,6,7,8,9}

      Actually we can write the entire set of real numbers using any number greater than 1 as a base.

      THe base 2 (or binary system) requires the digits 0 and 1

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