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The natural numbers are {1, 2, 3, 4, 5, 6, ...}. The ellipse (...) is to show that there are an infinite number of natural numbers that continue to get greater to infinity.
The natural numbers are not enough to represent all quantities we observe in life because they do not include zero or fractions. Negative numbers and Irrational Numbers also serve as additional sets which we require. Finally the imaginary numbers are added to make our system complete as far as the four basic operations and powers are concerned.
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Why integers need an extension?
In the first stage, the set of all integers needs an extension - to the set of rational numbers - to get closure for division (which is the inverse operation to multiplication).
Why do whole numbers need an extension?
The first need arose when it was found that the set of whole numbers was not closed under division. That is, given whole numbers A and B (B non-zero), that, in general, A/B was not a whole number - but a fraction.
Are natural numbers real numbers?
Yes, all natural numbers are real numbers. Natural numbers are a subset of real numbers, so not all real numbers are natural numbers.
Why does every integer have an opposite?
The numbers 1,2,3,... etc are called natural numbers or counting numbers. Integers are the natural numbers plus zero plus the negative ( or opposite ) natural numbers. Why do we need negative natural numbers ? For one thing x + 1 = 0 is an equation whose solution is x = -1. We could not solve this equation if we did not have negative integers. So over history these negative numbers came about as a way to solve certain math problems. The numbers 1,2,3,... etc are called natural numbers or counting numbers. Integers are the natural numbers plus zero plus the negative ( or opposite ) natural numbers. Why do we need negative natural numbers ? For one thing x + 1 = 0 is an equation whose solution is x = -1. We could not solve this equation if we did not have negative integers. So over history these negative numbers came about as a way to solve certain math problems.
Do set of natural numbers contain sets of whole numbers?
No. Natural numbers are a subset of whole numbers. Negative numbers are whole numbers but not natural.
