Class 10 Maths Real Numbers.pptx ppt ppt

ppt

1. B.J.P.S Samiti’s M.V.HERWADKAR ENGLISH MEDIUM HIGH SCHOOL CLASS: 10 th CHAPTER 1: REAL NUMBERS Program: Semester: Course: NAME OF THE COURSE Staff Name: VINAYAK PATIL 1

2. REAL NUMBERS Real numbers can be defined as the union of both rational and irrational numbers. They can be both positive or negative and are demoted by the symbol “R” Real numbers include integers, fractions and decimals

3. Fundamental Theorem of Arithmetic 𝐸𝑣𝑒𝑟𝑦 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑐𝑎𝑛 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑎𝑠 𝑎 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑝𝑟𝑖𝑚𝑒𝑠, 𝑎𝑛𝑑 𝑡ℎ𝑖𝑠 𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑢𝑛𝑖𝑞𝑢𝑒 , 𝑎𝑝𝑎𝑟𝑡 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑜𝑟𝑑𝑒𝑟 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒𝑦 𝑜𝑐𝑐𝑢𝑟. Now factorize a large number say 32760 . 32760=2x2x2x3x3x5x7x13x13

4. HCF and LCM of a number For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b. HCF (a, b) = LCM (a, b) =  

5. Revisiting Irrational Numbers A number which cannot be written in the form where p and q are integers and q ≠ 0 Example: Theorem : 𝐿𝑒𝑡 𝑝 𝑏𝑒 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑓 𝑝 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 2 , 𝑡ℎ𝑒𝑛 𝑝 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎, 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟.  

6. 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 Proof: Let us assume that √2 is rational number then we can write √2= a/b where a and b are co-prime. √2 = 𝑎 /𝑏 (𝑏 ≠ 0) squaring on both sides 2 = 𝑎 2 / 𝑏 2 2 𝑏 2 = 𝑎 2 . Here 2 divides 𝑎 2 , so it also divides 𝑎 . So we can write a=2c for some integer c. 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 contradicts to our wrong assumption. So we conclude that √2 is a irrational number.