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Resources » Articles/Knowledge Sharing » Education »
Step by Step Today's REAL NUMBER SYSTEM
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Small children first learn 1,2,3,4... so that they can be able to count how many apples they have or how many fingers are there in their hands and so on. These numbers are very natural to them and thus called NATURAL NUMBERS, also called positive integers and collection of all these numbers is denoted by N.
Addition comes when they have to count how many marbles they will possess if they already possessed 3 of them and someone gives them 5 more. They see the addition process of two natural numbers gives another natural number different from the first two (it should be remembered that the concept of 0 comes far later).
Next they have come up to the situation when there is 10 birds in a branch and suddenly 4 of them fly away, or they have 8 marbles initially and give their little sister 3 of them, i.e., Subtraction enters into their life and they first learn the concept of 0 (when all the birds fly away of the branch). Next they are said to consider the situation when they possessed Rs.10 and go to market and buy a fruitcake of price Rs. 12, i.e., they now have no money left in their hand, instead the shopkeeper must be given 2 more Rs. later. They are taught that in this situation they have Rs. -2, thus NEGATIVE INTEGERS comes into act.
So now they have the entire collection of natural numbers, 0 and also the collection of negative integers. Together we call all this numbers INTEGERS, and denote the collection of all these numbers by Z.
Next they learn multiplication by arriving the situation where they have 4 buckets, each containing 5 apples and they have to calculate how many apples are in total. They see this operation works very well in the sense that the two numbers of Z when multiplied yields another number belongs to the same set Z.
Problem occurs again while sharing a cake between 5 students. They learn division but when they divide some integer by another one (other than 0), they sometimes get resulting number (as in this case 1/5) not belonging to Z, this forces them to consider a new type of numbers. They indeed start calling those numbers RATIONAL NUMBERS (comes from the word ratio), the collection of all such numbers is denoted by Q, and, as any number can be thought of as that number divided by 1, so any member of Z also belongs to Q. They are also told this time that division of any number by 0 is not defined.
Thus the four basic operations addition, subtraction, multiplication and division go on smoothly in the set Q. So do they now live happily ever after?
Nay, a strange thing occurs when they are told to buy a string of length equal to that of a hypotenuse of a right angled triangle whose other sides are each of length 1cm. They are very much sure that they can buy a string of that length, but when they calculate it mathematically by Pythagoras Law they find the resulting number as v2=1.414213562.... (not terminating), not a member of Q, and the problem is they can not ask the shopkeeper to give them a string of length 1.414213562...; thus they got another type of number called IRRATIONAL NUMBERS, the collection of all irrational numbers is denoted by I.
Finally they get solution to all their problems. They start calling the set of all rational numbers and irrational numbers by the common name REAL NUMBER (the collection of all real numbers is denoted by IR) may be because all the real life problems have solutions in this system. Thus the evolution of today’s Real Number System.
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