NUMBER SYSTEMS - NCERT

1 NUMBER SYSTEMS1 CHAPTER1 NUMBER IntroductionIn your earlier classes, you have learnt about the NUMBER line and how to representvarious types of numbers on it (see Fig. ).Fig. : The NUMBER lineJust imagine you start from zero and go on walking along this NUMBER line in thepositive direction. As far as your eyes can see, there are numbers, numbers andnumbers!Fig. suppose you start walking along the NUMBER line, and collecting some of thenumbers. Get a bag ready to store them!2022-232 MATHEMATICS3-4016622-752190Z340167452601 422580-3-757-66-21-403171340167452601295 80W94016745260146525803117110 NYou might begin with picking up only naturalnumbers like 1, 2, 3, and so on.

2 You know that this listgoes on for ever. (Why is this true?) So, now yourbag contains infinitely many natural numbers! Recallthat we denote this collection by the symbol turn and walk all the way back, pick upzero and put it into the bag. You now have thecollection of whole numbers which is denoted bythe symbol , stretching in front of you are many, many negative integers. Put all thenegative integers into your bag. What is your new collection? Recall that it is thecollection of all integers, and it is denoted by the symbol there some numbers still left on the line?

3 Of course! There are numbers like1 3,2 4, or even 20052006 . If you put all such numbers also into the bag, it will now be theZ comes from theGerman word zahlen , which means to count .Q 67211213 1981161420052006 1213914 6625-656019199990 672758200520063 516609994 8 6625580277117981 121389 6723914 Why Z ?2022-23 NUMBER SYSTEMS3collection of rational numbers. The collection of rational numbers is denoted by Q. rational comes from the word ratio , and Q comes from the word quotient .You may recall the definition of rational numbers:A NUMBER r is called a rational NUMBER , if it can be written in the form pq,where p and q are integers and q 0.

4 (Why do we insist that q 0?)Notice that all the numbers now in the bag can be written in the form pq, where pand q are integers and q 0. For example, 25 can be written as 25;1 here p = 25and q = 1. Therefore, the rational numbers also include the natural numbers, wholenumbers and also know that the rational numbers do not have a unique representation inthe form pq, where p and q are integers and q 0. For example, 12 = 24 = 1020 = 2550= 4794, and so on. These are equivalent rational numbers (or fractions).

5 However,when we say that pq is a rational NUMBER , or when we represent pq on the numberline, we assume that q 0 and that p and q have no common factors other than 1(that is, p and q are co-prime). So, on the NUMBER line, among the infinitely manyfractions equivalent to 12, we will choose 12 to represent all of , let us solve some examples about the different types of numbers, which youhave studied in earlier 1 : Are the following statements true or false? Give reasons for your answers.(i)Every whole NUMBER is a natural NUMBER .

6 (ii)Every integer is a rational NUMBER .(iii)Every rational NUMBER is an : (i)False, because zero is a whole NUMBER but not a natural NUMBER .(ii)True, because every integer m can be expressed in the form 1m, and so it is arational (iii)False, because 35 is not an 2 : Find five rational numbers between 1 and can approach this problem in at least two 1 : Recall that to find a rational NUMBER between r and s, you can add r ands and divide the sum by 2, that is 2rs+ lies between r and s.

7 So, 32 is a numberbetween 1 and 2. You can proceed in this manner to find four more rational numbersbetween 1 and 2. These four numbers are 5 11 137.,,and4 884 Solution 2 : The other option is to find all the five rational numbers in one step. Sincewe want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1, , 1 = 66 and 2 = 126. Then you can check that 76, 86, 96, 106 and 116 are all rationalnumbers between 1 and 2. So, the five numbers are 7 4 3 511,,,and6 3 2 : Notice that in Example 2, you were asked to find five rational numbersbetween 1 and 2.

8 But, you must have realised that in fact there are infinitely manyrational numbers between 1 and 2. In general, there are infinitely many rationalnumbers between any two given rational us take a look at the NUMBER line again. Have you picked up all the numbers?Not, yet. The fact is that there are infinitely many more numbers left on the numberline! There are gaps in between the places of the numbers you picked up, and not justone or two but infinitely many. The amazing thing is that there are infinitely manynumbers lying between any two of these gaps too!

9 So we are left with the following are the numbers, that are left on the numberline, called? do we recognise them? That is, how do wedistinguish them from the rationals (rationalnumbers)?These questions will be answered in the next SYSTEMS5 EXERCISE zero a rational NUMBER ? Can you write it in the form pq, where p and q are integersand q 0? six rational numbers between 3 and five rational numbers between 35 and whether the following statements are true or false. Give reasons for your answers.

10 (i)Every natural NUMBER is a whole NUMBER .(ii)Every integer is a whole NUMBER .(iii)Every rational NUMBER is a whole irrational NumbersWe saw, in the previous section, that there may be numbers on the NUMBER line thatare not rationals. In this section, we are going to investigate these numbers. So far, allthe numbers you have come across, are of the form pq, where p and q are integersand q 0. So, you may ask: are there numbers which are not of this form? There areindeed such Pythagoreans in Greece, followers of the famousmathematician and philosopher Pythagoras, were the firstto discover the numbers which were not rationals, around400 BC.