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Irrational Numbers), , Can't be represented as x = p/q,, , Where p & q are integers and q #0., Decimal representation is non terminatin:, & non-recurring., , Terminating, , Non-Terminating, , , , , , , , x= p/q | £, and qz0 i | x aa,, (q is in the , Seow x, form of 2"5" (qis not tnt e, where m, n are form of 25’, non-negative where m, n are, integers, then ee, it is terminating | iieaers, Ten, decimal) i itis, 3 non- terminating, eg. =0.15 decimal), 2x5 ie, eeeee ae eg. 2: 3.3, ooo Momence, , Some Important Results :, , 1, Let 'p' be a prime number and ‘a’ be a positive, integer. If 'p' divides a’, then 'p' divides ‘a’., , 2. HCF x LCM = Product of two numbers., , 3. LCM is always divisible by HCF., , fone vem eeem ove ecemeven:, , , , , Real Numbers “4, , *, , Euclid division Euclid division, , <— Start From Here, , Lemma Algorithm, y . een,, ———swwwmy — / Tf ‘a’ and 'b’ are positive }, , ( Given positive integers a |, and b, there exist unique |, integers q and r satisfying |, a=bq+r,O<rcb, |, , }, , eee TNT, , . Check whether 6" can end with the digit 0 for any natural, , Isol., , Bud, , P—remrrreeremes, , , , | then every common divisor, | of ‘a’ and 'b’ is a common, , | divisor of 'b' and ‘r’, and, , | vice-versa., , | e.g. HCF of 420 & 48, , | 420 = 48 x 8 + 36, , | 48 = 36x1+12, , | 36 =12x3+0, , \ = H.CF. of 420 & 48 is 12, , number ‘n’., , Any positive integer ending with the digit zero is divisible, , by 5 and so its prime factorization must contain the prime 5., CoCxsyerss, , The prime in the factorization of 6” is 2 and 3., , 5 does not occur in the prime factorization of 6° for any n., , 6" does not end with the digit zero for any natural number n., , | integers such that a = bq+r, |, , , , , , + BOARD, NDA, FOUNDATION, , 1,1/2,-7/5..., , Can be represented as, , x = p/q, Where p &q are, integers and q #0., , Decimal representation is, either terminating or, non-terminating & recurring, , Fundamental, theorem of, Arithmetic, , , , Every composite number, can be expressed or, factorized as a product of, primes., e.g.48=2x2x2x2x3, , 420=2x2x3x5x7, , , , , , Application », , / Cicm), , e.g.48=2x2x2x2x3, , 2x2x3x5xZ, 2x2x2x2x3x5xZ, , 680, , , , , H.C.F. (48, 420) x L.C.M. (48,420), = 48 x 420, (This is true for 2 numbers only), , NCERT / X / Real Number