Table Of Content

Mathematics Classes 9-10 Chapter One Real Number Mathematics is originated from the process of expressing quantities in symbols or numbers. The history of numbers is as ancient as the history of human civilization. Greek Philosopher Aristotle According to the formal inauguration of mathematics occurs in the practice of mathematics by the sect of priest in ancient Egypt. So, the number based mathematics is the creation of about two thousand years before the birth of Christ. After that, moving from many nations and civilization, numbers and principles of numbers have gained an universal form at present. The mathematicians in India first introduce zero (0) and 10 based place value system for counting natural numbers, which is considered a milestone in describing numbers. Chinese and Indian mathematicians extended the idea zero, real numbers, negative number, integer and fractional numbers which the Arabian mathematicians accepted in the middle age. But the credit of expressing number through decimal fraction is awarded to the Muslim Mathematicians. Again they introduce first the irrational numbers in square root form as a solution of the quadratic equation in algebra in the 11th century. According to the historians, very near to 50 BC the Greek Philosophers also felt the necessity of irrational number for drawing geometric figures, especially for the square root of 2. In the 19th century European Mathematicians gave the real numbers a complete shape by systematization. For daily necessity, a student must have a vivid knowledge about ‘Real Numbers’. In this chapter real numbers are discussed in detail. At the end of this chapter, the students will be able to – (cid:190) Classify real numbers (cid:190) Express real numbers into decimal and determine approximate value (cid:190) Explain the classification of decimal fractions (cid:190) Explain recurring decimal numbers and express fractions into recurring decimal numbers (cid:190) Transform recurring decimal fraction into simple fractions (cid:190) Explain non-terminating non-recurring decimal fraction (cid:190) Explain non-similar and similar decimal fraction (cid:190) Add, subtract multiply and divide the recuring decimal fraction and solve various problems related to them. 2 Math Natural Number 1,2,3,4......... etc. numbers are called natural number or positive whole numbers. 2,3,5,7......... etc. are prime numbers and 4,6,8,9,......... etc. are composite numbers. Integers All numbers (both positive and negative) with zero (0) are called integers i.e. ....... (cid:16)3,(cid:16)2,(cid:16)1,0,1,2,3......... etc. are integers. Fractional Number p If p,q are co-prime numbers ; q(cid:122)0 and q(cid:122)1, numbers expressed in form are q called fractional number. 1 3 (cid:16)5 Example : , , etc. are fractional numbers. 2 2 3 If p(cid:31)q, then it is a proper fraction and if p(cid:33)q then it is an improper fraction : 1 1 2 1 3 4 5 5 Example , , , ,......... etc. proper and , , , ,.... etc. improper fraction. 2 3 3 4 2 3 3 4 Rational Number p If p and q are integers and q(cid:122)0, number expressed in the form is called rational q 3 11 5 number. For example : (cid:32)3, (cid:32)5.5, (cid:32)1.666... etc. are rational numbers. 1 2 3 Rational numbers can be expressed as the ratio of two integers. So, all integers and all fractional numbers are rational numbers. Irrational Number p Numbers which cannot be expressed in form, where p, q are integers and q(cid:122)0 are q called Irrational Numbers. Square root of a number which is not perfect square, is an 5 irrational number. For example: 2 (cid:32)1.414213....., 3(cid:32)1.732....., (cid:32)1.58113.... 2 etc. are irrational numbers. Irrational number cannot be expressed as the ratio of two integers. Decimal Fractional Number If rational and irrational numbers are expressed in decimal, they are known as decimal fractional numbers. As for instance, 3(cid:32)3(cid:152)0,5(cid:32)2(cid:152)5,10(cid:32)3(cid:152)3333......., 3(cid:32)1(cid:152)732.........etc. 2 3 are decimal fractional numbers. After the decimal, if the number of digits are finite, it is terminating decimals and if it is infinite it is known as non-terminating decimal number. For example, 0.52, 3.4152 etc. are terminating decimals and 1(cid:152)333......., 2(cid:152)123512367...........etc. are non-terminating decimals. Again, if the digits Math 3 after the decimal of numbers are repeated among themselves, they are known recurring decimals and if they are not repeated, they are called non-recurring decimals. For example : 1(cid:152)2323........, 5(cid:152)6(cid:6)5(cid:6)4(cid:6) etc. are the the recurring decimals and 0(cid:152)523050056........, 2(cid:152)12340314........etc. are non-recurring decimals. Real Number All rational and irrational numbers are known as real numbers. For example : 1 3 4 0,(cid:114)1,(cid:114)2,(cid:114)3,.......... (cid:114) ,(cid:114) ,(cid:114) ,........ 2, 3, 5, 6...... 2 2 3 1(cid:152)23, 0(cid:152)415, 1(cid:152)3333......., 0(cid:152)6(cid:6)2(cid:6), 4(cid:152)120345061.......... etc. are real numbers. Positive Number All real numbers greater than zero are called positive numbers. As for instance 1 3 1, 2, , , 2, 0(cid:152)415, 0(cid:152)6(cid:6)2(cid:6), 4(cid:152)120345061.............. etc. are positive numbers. 2 2 Negative Number All real numbers less than zero are called negative numbers. For example, 1 3 (cid:16)1, (cid:16)2, (cid:16) , (cid:16) , (cid:16) 2, (cid:16)0(cid:152)415, (cid:16)0(cid:152)6(cid:6)2(cid:6), (cid:16)4(cid:152)120345061.............. etc. are 2 2 negative numbers. Non-Negative Number All positive numbers including zero are called non-negative numbers. For example, 1 0, 3, , 0(cid:152)612, 1(cid:152)3(cid:6), 2(cid:152)120345.............. etc. are non-negative numbers. 2 Classification of real Number. Real Rational Integer Fraction Positive 0 Negative Simple Decimal Irrational Fractional 1 Composite Proper Improper Mixed Terminating Recurring Non- recurring 4 Math 3 9 4 Activity : Show the position of the numbers , 5, (cid:16)7, 13, 0, 1, , 12, 2 , 4 7 5 . . 1(cid:152)1234......,.323 in the classification of real numbers. Example 1. eDtermine the two irrational numbers between 3 and 4. Solution : Here, 3 (cid:32)1.7320508...... Let, a(cid:32)2.030033000333..... and b(cid:32)2.505500555....... Clearly : both a and b are real numbers and both are greater than 3 and less than 4. i.e., 3(cid:31)2.03003300333.........(cid:31)4 and 3(cid:31)2.505500555................(cid:31)4 Again, a and b cannot be expressed into fractions. (cid:63) a and b are the two required irrational numbers. Basic characteristics of addition and multiplication over a real number : 1. If a, b are real numbers, (cid:11)i(cid:12)a(cid:14)b is real and (cid:11)ii(cid:12) ab is a real number 2. If a, b are real numbers, (cid:11)i(cid:12)a(cid:14)b(cid:32)b(cid:14)a and (cid:11)ii(cid:12)ab(cid:32)ba 3. If a, b, c are real numbers, (cid:11)i(cid:12)(cid:11)a(cid:14)b(cid:12)(cid:14)c(cid:32)a(cid:14)(cid:11)b(cid:14)c(cid:12) and (cid:11)ii(cid:12)(cid:11)ab(cid:12)c(cid:32)a(cid:11)bc(cid:12) 4. If a is a real number, in real numbers there exist only two number0 and 1 where (cid:11)i(cid:12)0(cid:122)1 (cid:11)ii(cid:12)a(cid:14)0(cid:32)a (cid:11)iii(cid:12)a.1(cid:32)1.a(cid:32)a 1 5. If a is a real number, (cid:11)i(cid:12)a(cid:14)((cid:16)a)(cid:32)0 (cid:11)ii(cid:12) If a(cid:122)0, a. (cid:32)1 a 6. If a, b, c are real numbers, a(b(cid:14)c)(cid:32)ab(cid:14)ac 7. If a, b are real numbers, a(cid:31)b or a(cid:32)b or a(cid:33)b 8. If a, b, c are real numbers and a(cid:31)b, a(cid:14)c(cid:31)b(cid:14)c 9. If a, b, c are real numbers and a(cid:31)b, (i) ac(cid:31)bc where c(cid:31)0 (ii) If ac(cid:33)bc, c(cid:31)0 Proposition : 2 is an irrational number. eW know, 1(cid:31)2(cid:31)4 (cid:63) 1(cid:31) 2(cid:31) 4 or, 1(cid:31) 2(cid:31)2 Proof :12 (cid:32)1, (cid:11) 2(cid:12)(cid:21) (cid:32)2, 22 (cid:32)4 (cid:63) Therefore, the value of 2 is greater than 1 and less than 2. Math 5 (cid:63) 2 is not an integer. (cid:63) 2 is either a rational number or a irrational number. If 2 is a rational number p let, 2(cid:32) ; where p and q are natural numbers and co-prime to each other and q(cid:33)1 q p2 or, 2(cid:32) ; squaring q2 p2 or, 2q(cid:32) ; multiplying both sides by q. q p2 Clearly2q is an integer but is not an integer because p and q are co-prime natural q numbers and q(cid:33)1 p2 p2 (cid:63) 2q and cannot be equal, i.e., 2q(cid:122) q q p p (cid:63) aVlue of 2 cannot be equal to any number with the form i.e., 2(cid:122) q q (cid:63) 2 is an irrational number. Example 2. Prove that, sum of adding of 1 with the product of four consecutive natural numbers becomes a perfect square number. Solution : Let four consecutive natural numbers be x, x(cid:14)1, x(cid:14)2, x(cid:14)3 respectively. By adding 1 with their product we get, x(cid:11)x(cid:14)1(cid:12)(cid:11)x(cid:14)2(cid:12)(cid:11)x(cid:14)3(cid:12)(cid:14)1(cid:32) x(cid:11)x(cid:14)3(cid:12)(cid:11)x(cid:14)1(cid:12)(cid:11)x(cid:14)2(cid:12)(cid:14)1 (cid:32)(cid:11)x2(cid:14)3x(cid:12)(cid:11)x2(cid:14)3x(cid:14)2(cid:12)(cid:14)1 (cid:32)a(cid:11)a(cid:14)2(cid:12)(cid:14)1; [x2(cid:14)3x(cid:32)a] (cid:32)a(a(cid:14)2)(cid:14)1 ; (cid:32)a2(cid:14)2a(cid:14)1 (cid:32)(cid:11)a(cid:14)1(cid:12)2 (cid:32)(cid:11)x2(cid:14)3x(cid:14)1(cid:12)2; which is a perfect square number. (cid:63) If we add 1 with the product of four consecutive numbers, we get a perfect square number. Activity : rPoof that, 3 is an irrational number Classification of Decimal Fractions Each real number can be expressed in the form of a decimal fraction.