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WHAT ARE NATURAL NUMBERS?

Natural numbers form the basic set in the classification of numbers, as explained in my article on the types of numbers.. They are the numbers that are countable; used in counting real objects. They form the set

N = {1,2,3,4,5,6…} to infinity.

Natural numbers are positive integers that are obviously and tangibly used for everyday counting. They also fall under whole numbers when they include the number 0 (which does not represent any quantity). However, since 0 is not negative, it is still classified as a whole or non negative number just like the natural numbers.

Properties of natural numbers include divisibility and factorisation. Divisibility involves determining whether an integer can be evenly divided by a divisor without any remainder. For example, 16 is divisible by 4; 4 can divide it evenly without a remainder. Factorisation involves the expression of a positive integer as the product of other smaller factors or integers. 16 is a product of 1 and 16, 2 and 8, and 4 and 4. So 16 can be factored as 1, 2, 4, 8 and 16.

Since mathematicians saw the need to distinguish each and every number, there are different types of natural numbers.

On the basis of divisibility, natural numbers are classified under Even and odd numbers. Integers that can be evenly divided by 2 without leaving any remainders are called even numbers. For example 2 divided by 2 = 1; 4 divided by 2 = 2 : They are all multiples of 2, forming the set

E = { 2, 4, 6, 8, 10, 12…}.

On the other hand, numbers that cannot be equally divided by 2 without leaving a remainder are called odd numbers. Counting from 1 we have the odd numbers as O = {1, 3, 5, 7,9, 1…} Any of these numbers will leave remainders when divided by 2. We can see from the set of natural numbers

N = {1, 2, 3, 4, 5,...} that starting from 1, every other number is an odd number. This continues to infinity. It is the same with even numbers starting from 2. Thus both even and odd numbers interchange in the process of natural counting of numbers.

FACTS ABOUT ODD AND EVEN NUMBERS

  • To check if a two or more digit number is an odd or even number, we check the last digit; If it is even, then the whole big number is an even number. E.g 312 has 2 (an even number) as its last digit. So 312 is an even number. 170 is also an even number. This is because 0 ( a non-negative number) comes before 1 (an odd number) and this makes it an even number. 2917 has 7 (an odd number) as the last digit. So this 4 digit number is an odd number.

  • The sum of two even numbers is also an even number. 2 + 4 = 6, 16 + 18 = 34 ;

  • However, the sum of an odd number and an even number is an odd number: 7 + 8 = 15; 13 + 12 = 35

  • The product of two even numbers is an even number:

  • 4 x 6 = 24; 12 x 4 = 48

  • The product of two odd numbers is an odd number:

  • 3 x 5 = 15; 11 x 15 = 165

  • The product of an even number and odd number is an even number. 4 x 5 = 20, 16 x 13 = 208

On the basis of factorisation, natural numbers are grouped into Prime numbers and Composite numbers. A prime number has only 1 and itself as its factors. 2 is a prime number since its factors (numbers multiplying to make it) are 1 and 2 itself. 1 is not because it is already the constant prime factor for all numbers. 3 is a prime number since no other number apart from 1 and itself divide equally into it. This is the same for 5 ( 1 and 5 are the only factors). So the first four prime numbers are P = { 2, 3, 5 and 7}

No even number greater than 2 is a prime number because any other even number has three or more distinct divisors (factors)

A composite number is a natural number greater than 1 that has numbers apart from 1 and itself to divide it evenly. This means it has numbers apart from 1 and itself as its factors. For example 6 is a composite number because apart from 1 and itself it has 3 and 2 as numbers that can divide it evenly; 3 x 2 = 6 and 6 x 1 = 6.

To be able to tell whether a number is composite or not, you need to check whether any of the prime numbers {2, 3, 5, and 7} can divide them evenly. If they can divide then the number is a composite one. However, the number is a prime number if it is not divisible by any of the prime factors.

Shall we try with the following numbers:

33 = 3 x 11; it is composite (since a prime factor, 3, can divide)

23 = 1 x 23; it is a prime number ( none of the prime factors can divide evenly)

63 = 7 x 9; it is composite ( a prime factor, 7, can divide evenly)

51 = 3 x 17; it is composite (a prime factor, 3, can divide)

37 = 1 x 37; it is a prime number ( none of the prime factors can divide evenly)

PRIME NUMBERS BETWEEN 1 AND 100

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