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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.
Why do the natural numbers need an extension?
While natural numbers are closed with respect to addition and mulitplication, they are missing the additive identity (zero). Furthermore, they are not closed with respect to two of the fundamental operations of arithmetic: subtraction and division.
What is the need to discover aliens?
An extension of natural human curiosity.
Why integers need 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 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).
Are there infinitely many natural numbers that are not prime?
This can be an extension to the proof that there are infinitely many prime numbers. If there are infinitely many prime numbers, then there are also infinitely many PRODUCTS of prime numbers. Those numbers that are the product of 2 or more prime numbers are not prime numbers.
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.
Why do you need a extenstion of rational numbers?
You need an extension because rational numbers are a tiny subset of all real numbers. There are transcendental numbers such as pi and e which are key to geometry and calculus (respectively), the Golden ratio, as well as all the non-rational roots of rational numbers.
What prompted the need to extend the system of natural numbers to the system of integers?
Natural numbers are separate from integers. I can't believe this was asked 9 years ago . . .
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Why whole numbers require an extension?
as whole numbers does not contain negative numbers, so for denoting ( eg: deapth of the sea,etc )negative things they require an extension.
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.
What is the intersection between rational numbers and natural numbers?
It is the set of natural numbers.
