Class 10th real number complete ppt, Study notes of Mathematics

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CLASS X

REAL NUMBERS

MADE BY:S N MISHRA

Euclid’s Division Lemma

Here , 𝑎 = 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑, 𝑏 = 𝑑𝑖𝑣𝑖𝑠𝑜𝑟, 𝑞 = 𝑞𝑢𝑜𝑡𝑒𝑖𝑛𝑡 𝑟 = 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟. Example 13 = 2 × 6 + 1

Example :- Using Euclid’s division algorithm find the HCF of 12576 and 4052. Ans. Since 12576 > 4052 we apply the division lemma to 12576 and 4052 to get 12576 = 4052 × 3 + 420 Since the remainder 420 ≠ 0 , we apply the division lemma to 4052 and 420 to get 4052 = 420 × 9 + 272 We consider the new divisor 420 and new remainder 272 apply the division lemma to get 420 = 272 × 1 + 148 Now we continue this process till remainder is zero. 272 = 148 × 1 + 124 148 = 124 × 1 + 24 124 = 24 × 5 + 4 24 = 4 × 6 + 0 The remainder has now become 0 , so our procedure stops. Since the divisor at this stage is 4 , the HCF of 12576 and 4052 is 4.

Fundamental Theorem of

Arithmetic



Now factorize a large number say

32760=2x2x2x3x3x5x7x13x

Substituting for 𝑎 we get 2𝑏 2 = 4c 2 that is 𝑏 2 = 2c 2 . Here 2 divides 𝑏 2 , so it also divides 𝑏 .This creates a contradiction that a and b have no common factors other than 1. This contradiction has arisen because of our wrong assumption. So we conclude that √2 is a irrational number.

Revisiting Rational numbers

and their decimal expansions

 Theorem: 𝐿𝑒𝑡 𝑥 𝑏𝑒 𝑎 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑤 ℎ 𝑜𝑠𝑒 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒. 𝑇ℎ 𝑒𝑛 𝑥 𝑐𝑎𝑛 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑖𝑛 𝑡 ℎ 𝑒 𝑓𝑜𝑟𝑚 of 𝑝 and q, 𝑤ℎ 𝑒𝑟𝑒 𝑝 𝑎𝑛𝑑 𝑞 𝑎𝑟𝑒 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 , 𝑎𝑛𝑑 𝑡 ℎ 𝑒 𝑝𝑟𝑖𝑚𝑒 𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑞 𝑖𝑠 𝑜𝑓 𝑡ℎ 𝑒 𝑓𝑜𝑟𝑚 2 n 5 m , where n and m 𝑎𝑟𝑒 𝑛𝑜𝑛 - 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠. Example:0.375= 375/ 3

Theorem

 Let x =p/q be a rational number, such that the prime factorisation of q is not of the form 2 n 5 m , where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating ( recurring).  Example: 1 / 7=0.1428571…

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