Natural Numbers

Natural numbers are the basic numbers used for counting. It is a set of numbers that contains numbers from 1 to infinity (countless). They are made up of only positive numbers.

Natural numbers were in use even before they were exclusively categorized. The Babylonians gave the identity to the numerals 1 (one) to 10 (ten) under the title place value system. This system was expanded by the ancient Egyptians to encompass all powers of ten up to one million. Pythagoras (582–500 BC) and Archimedes (287–212 BC) were among the first Greek philosophers and mathematicians to take this category of numbers seriously.

Symbol and representation of natural numbers

All counting numbers are considered natural numbers. It is represented by the symbol ‘ℕ’. We use these them in our daily lives such as counting objects; measuring the quantity; calculating the distance traveled or speed at which a vehicle is moving and many more. Refer to the image below.

Counting objects and Measurements are represented as Natural Numbers
Counting objects and Measurements are represented as Natural Numbers

Natural Numbers are represented using a set notation as

ℕ = {1, 2, 3, 4, … ∞}

It is read as ℕ and is a set of all natural numbers from 1 to infinity.
{ } is the curly brackets used to denote the set.
∞ is the symbol used to represent infinity.

Natural numbers representation on number line

We can represent natural numbers on the number line. In this number line, the numbers starting from 1 followed by 2, 3, 4, … are marked at equal intervals.

For example, a set of natural numbers from 1 to 10 is represented on the number line as pictured.

Number line with natural numbers
Number line with natural numbers

What are axioms in mathematics?

The word “axiom” originated from the Greek word “Axioma” which means “true without needing a proof”.

There are some axioms in natural numbers introduced by Italian mathematician Giuseppe Peano. The axioms are well known as “Peano axioms”. These are now considered as a foundation for understanding the concepts of number theory and set theory.

The five Peano axioms are

1. 0 is a natural number.
2. Every natural number has a successor in the natural numbers.
3. 0 is not the successor of any natural number.
4. If the successor of two natural numbers is the same, then the two original numbers are the same.
5. If a set contains 0 and the successor of every number is in the set, then the set contains the natural numbers.

Is 0 a natural number?

Natural numbers, by definition, are positive numbers that begin with 1. However, we can get a number by adding 0 to a natural number, such as 10, 40, 70, and so on. 0 is not a natural number, according to the definition of natural numbers. But, several mathematicians across the world consider 0 as a natural number. The perceptions and thoughts of each mathematician differ from one another.

What is the predecessor and successor of a number?

A predecessor is an immediate number before a particular number, whereas a successor is an immediate number after a particular number. For example, consider the number 5. The predecessor of 5 is 4, as the number 4 appears right before 5. The successor of 5 is 6, as the number 6 appears right after.

Try it yourself!
Represent A = Set of natural numbers from 1 to 15 on the number line. Identify the odd and even natural numbers.

Can you answer?

1. How many days are there in a week?
2. How many colors are there in a rainbow?
3. What is the heartbeat rate in adults?
4. How many candies do you eat in a month?

These types of questions in our daily lives can be answered using natural numbers. These category of numbers play a vital role in our day-to-day communication and money transactions.


Math Facts

  • The axiom “If a set contains 0 and the successor of every number is in the set, then the set contains the natural numbers” is well known as the principle of induction.
  • The addition or multiplication of two natural numbers will always be a natural number.


    Pythagoras of Samos was a legendary founder of Pythagoreanism and an ancient Ionian Greek philosopher. His contributions in many fields were remarkable; he is most well-known for his contribution in the field of mathematics.

    Pythagoras of Samos
    Pythagoras of Samos – The legendary founder of Pythagoreanism

    The Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the spherical shape of the Earth, and the identity of the morning and evening stars as the planet Venus are only a few of Pythagoras’ mathematical and scientific achievements. He was supposed to be the first man to refer to himself as a philosopher and the first to split the earth into five climate zones. Many of the accomplishments attributed to Pythagoras presumably originated earlier or were done by his colleagues or successors, as classical historians disagree whether he made these discoveries.

    Some stories claim that Pythagoras’ philosophy was based on mathematics and that numbers were essential. However, it is debatable to what amount, if any, he contributed to mathematics or natural philosophy.

    Know more about other concepts of Number Systems on Fibonacci Numbers, Fractions, Natural Numbers, Prime numbers and Whole Numbers.

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