by Kerry Kauffman

Created on: August 01, 2009   Last Updated: August 03, 2009

Understanding natural numbers is as easy as understanding how to count, because natural numbers can be thought of as the numbers used in counting. The natural numbers are 1,2,3,4,5,6,7,8,9,10 ...Interestingly, natural numbers can also be thought of as the positive integers and positive whole numbers.

The idea of counting numbers being considered "natural" makes sense, because when we think of counting, we don't start at 0 or negative 1, we start at 1, which comes NATURALLY to us.

Some properties of natural numbers are as follows. Adding any natural numbers together yields another natural number. Subtracting a smaller natural number from a larger natural number yields a natural number. Multiplying any two natural numbers together yields another natural number. If 2 divides evenly (meaning no remainder) into a natural number, then the result of the division is an even number. For example, 2 divides evenly into 48, 24 times 24 is an even number. Odd natural numbers cannot be divided by 2 evenly. All natural numbers ending in 5 are divisible by 5. All natural numbers ending in 0 are divisible by 2, 5 and 10. If the sum of the digits of a natural number is divisible by 3, then the natural number is divisible by 3. For example, take the natural number 621. The sum of the digits of 621 is 9 (6+2+1) and 9 is divisible by 3. Therefore 621 is also divisible by 3 (621 divided by 3 equals 207).

Some algebraic properties hold true with natural numbers. The associative property or addition and multiplication hold true. If a, b and c are natural numbers, then (a+b)+c = a+(b+c) and a times (b times c) = (a times b) times c. The commutative property holds true as well. a+b = b+a and a times b = b times a. The distributive property of multiplication holds true as well. a x (b+c) = a x b + a x c

Natural numbers can be used also to order things. Take a row of objects, sorted from smallest to largest. The smallest can be thought of 1st in line, then 2nd, 3rd, so on until we hit the nth object, the largest, where n is the number of objects.

A formal definition of natural numbers from the 19th century includes 0, but the traditional definition, one I've always seen in school and in my teaching, does not include 0 in the set.

Hope this clarifies what natural numbers are and shows the ease at which they can be used and the properties involved.

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