ALGEBRA Std 9 th Subject Algebra Topic Real

ALGEBRA Std - 9 th Subject- Algebra Topic- Real Numbers

Topic : Real Numbers Std - 9 th Subject- Algebra Topic- Real Numbers

Sub-Topics : Std - 9 th Subject- Algebra Topic- Real Numbers

REAL NUMBERS *Classify the following numbers into five groups natural numbers, whole numbers, rational numbers and Irrational numbers. 2/3, 7, -4/5, 0, -10, 0. 3, √ 2, √ 3, 0. 10111213……. Std - 9 th Subject- Algebra Topic- Real Numbers

Classification Ameya Natural numbers 7 Whole numbers 0 Integers Rational numbers Irrational numbers -10 0. 3 √ 2 Bismilla 7 0, 7, -10 2/3, -0. 3 Chaitanya 7 0, 7, -10, 2/3, 4/5, 0. 6 √ 2, √ 3 0. 1011… Let us compare these answers : Here we observe that thinking process of Chaitanya about number systems is more clear than others. Std - 9 th Subject- Algebra Topic- Real Numbers

Natural numbers: -The counting numbers 1, 2, 3, 4, … are called natural numbers. We write them as a set N = {1, 2, 3, 4, …} Is the sum of two natural numbers a natural number ? Dolly says, "yes. ” Since 4 is N, 7 is N, 4 + 7= 11 is N. Is the product of two natural numbers a natural number ? Elizabeth says, ”yes. ” Since 5 is N and 8 is N, 5 x 8 = 40 is N Topic- Real Numbers Std - 9 th Subject- Algebra

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Whole Numbers : The union of set of natural numbers and zero is a set of WHOLE NUMBERS & the set is denoted by ‘W’. W = {0, 1, 2, 3, 4, . . . . } Integer : The set of natural numbers, zero and opposite of all natural numbers is called as set of integers and denoted by I. I = {. . . -2, -1, 0, 1, 2, . . . } Std - 9 th Subject- Algebra Topic- Real Numbers

Is the division of two integers a integer? Govind says, “No. ” Since 2 is I, 5 is I, but 2/5 is I. Rational Number : If p & q are integers (q‡ 0) then the number p/q is called a rational number and the set of rational number is denoted by Q. Q = {. . . -1/2, 0, 1/2. . . } Std - 9 th Subject- Algebra Topic- Real Numbers

Are all the natural numbers a rational number ? Hemant says, “yes. ” Since 2 is N and 2= 2/1 is I Are all the whole numbers a rational number ? Indu says, “yes. ” Since 0 is W and 0= 0/5 is I Are all the integers a rational number ? Jaheer says, ”yes. ” Since – 3 is I and -3= -3/1 is I Rational numbers include natural numbers, whole numbers and integers Std - 9 th Subject- Algebra Topic- Real Numbers

*Natural numbers: - The smallest natural number is 1, and the largest number cannot be defined. *Whole numbers: - The smallest whole number is 0. *Integers: -The German word “Zahlen” means to count and “zahl” means number. *Rational number: - The word rational is derived from the word “Ratio. ” Std - 9 th Subject- Algebra Topic- Real Numbers

*Equivalent numbers: - 1/3 = 2/6 = 3/9…… These are equivalent numbers. *1/2=0. 5, 1/3=0. 333…, 1/4=0. 25, 1/5=0. 2, 1/6=0. 161616…, 1/10=0. 1. NOTE: -Rational whose denominators factors are 2 , 5 or 2 and 5 are terminating decimals. *Terminating decimals and non terminating recurring decimals are both rational numbers. *Rational numbers include natural numbers, Whole numbers and integers. Note : Terminating decimals and non-terminating recurring decimals are both rational numbers. Std - 9 th Subject- Algebra Topic- Real Numbers

TO FIND RATIONAL NUMBERS BETWEEN TWO RATIONALS 1. If a & b are two rational numbers then (a+b)/2 is between a and b. For ex. Between 3 and 4 there is a number (3+4)/2, i. e. , 7/2. Between 3 and 7/2 there is a number (3+7/2)/2=13/4. Also, between 3/2 and 4 there is a number (3/2+4)/2=11/4. . CONCLUSION : There are infinite rational numbers between two rational numbers Std - 9 th Subject- Algebra Topic- Real Numbers

*Archimedes was a Greek genius who first calculated the digits in decimal expansion of π. He found the value of π lies between 3. 140845 and 3. 142857. 3. 140845 < π < 3. 142857. *Aryabhatta was the greatest Indian mathematician who found the value of π correct to four decimal places. π =3. 1416 (approximately) Std - 9 th Subject- Algebra Topic- Real Numbers

Conversion of terminating decimals into form m/n. (i) Remove the decimal point from the numerator and write 1 in the denominator and put as many zeros on the right of 1 as the numbers of digits after the decimal point. (ii) Convert the rational number obtained into its lowest term by dividing the numerator and the denominator by the common factor. For ex. Convert 0. 45 into form m/n. 0. 45 = 45/100 = 9 x 5/20 x 5 = 9/20. Std - 9 th Subject- Algebra Topic- Real Numbers

CONVERSION OF NON-TERMINATING RECURRING DECIMAL NUMBER i. Put the given decimal number equal to x. ii. Remove the bar if any and write the recurring digits at least twice. iii. If the recurring decimal has one place repetition multiply by 10, if there is two place repetition multiply by 100 and so on. iv. Subtract the number in (ii) from the number obtained in step (iii). v. Divide both sides of equation by the coefficient of x. Std - 9 th Subject- Algebra Topic- Real Numbers

RULE FOR CONVERSION OF A PURE RECURRING DECIMAL IN THE FORM m/n. i. Write the given number after removing decimal as the numerator. ii. In the denominator write down as many 9’s as the number of repeating digits. For ex. 1. Express 0. 7 in the form m/n. Solution: - Write 7 as the numerator and one nine in the denominator (as only 1 repeating digit) 0. 7 = 7/9. Represent 0. 2323 --- in the form m/n - 23/99 Std - 9 th Subject- Algebra Topic- Real Numbers

IRRATIONAL NUMBERS The numbers which cannot be expressed in decimal form either in terminating or nonterminating recurring decimals are known as irrational numbers. An irrational number cannot be written in the form m/n, where m and n are both integers and n ¹ 0. Std - 9 th Subject- Algebra Topic- Real Numbers

PYTHAGORAS In 400 B. C. Pythagoras was Greek mathematician who was the first to discover the numbers which were not rational. These numbers are called irrational numbers. REAL NUMBERS: Rational numbers and irrational numbers taken together are known as real numbers. Std - 9 th Subject- Algebra Topic- Real Numbers

REPRESENTATION OF IRRATIONAL NUMBERS ON THE NUMBER LINE. Std - 9 th Subject- Algebra Topic- Real Numbers

DIVISION ALGORITHM We know that division is not defined in set of integers. For ex. If we divide 22/3, we get 7 as the quotient and 1 as the remainder. Thus, 22=3 x 7 + 1. This is division algorithm. Std - 9 th Subject- Algebra Topic- Real Numbers

Symbolically we state division algorithm as : a and b are integers and b > 0. There exists a unique pair of integers q and r, such that a=bq +r where 0< r < b. In arithmetic the students have seen the truth of this theorem. Here we see that a is the dividend, b is the divisor, q is the quotient and r is the remainder. Std - 9 th Subject- Algebra Topic- Real Numbers

EXAMPLE Now let a = 283, b = 23. Clearly, the remainder 7 lies between 0 and 23, i. e. , 0 < 7 < 23 implies 0 < r < 23. Such types of 0 examples help us to have DIVISION ALGORITHM In the form a = bq + r for all 0 < r < b. 23 12=q 283 -23 53 -46 7=r This theorem is very useful to find the HCF of two numbers a and b which is written as HCF of (a , b) = c or simply we write (a, b)=c Note(a, b)=c means HCF of two numbers a and b is c. Std - 9 th Subject- Algebra Topic- Real Numbers

Example to find the HCF of 240, 128. Solution: - 240=128 x 1 +112 Here the divisor is 128 and remainder is 112. Apply the division algorithm again on 128 and 112. 128=112 x 1 + 16. Now the divisor is 112 and the remainder is 16. We again apply the division algorithm on 112 and 16. 112 = 16 x 7 + 0 Now remainder is 0 and HCF of 112 and 16 is 16. Therefore we can say that HCF of 240 and 128 is 16. Here HCF(240, 128)= HCF of (128, 112)= HCF of (112, 16)=16. Std - 9 th Subject- Algebra Topic- Real Numbers

FUNDAMENTAL THEOREM OF ARITHMETIC We know that any natural number can be expressed as a product of primes and this is unique apart from the order in which the prime factors occur. For ex. , 3 = 3 x 1, 4 = 2 x 2 = 22 , 8 = 2 x 2 = 2 3, 6 = 2 x 3, 36 = 2 x 3 x 3 = 22 x 32, 192 = 8 x 24 = 4 x 2 x 4 x 6 = 2 x 2 x 2 x 3 = 26 x 3 In general when a composite number decomposes into primes and are written in ascending order. Std - 9 th Subject- Algebra Topic- Real Numbers

FUNDAMENTALTHEOREM Every composite number can be expressed as a product of prime numbers and this factorization is unique apart from the order in which the prime factor occur. This fundamental theorem of arithmetic helps us to find H. C. F and L. C. M of numbers.

WORKING RULE TO FIND H. C. F 1. Select the lowest of the powers of common primes. 2. The product of powers of common prime is H. C. F. 3. Factorize the numbers into primes. WORKING RULE TO FIND L. C. M 1] Factorize the numbers into primes. 2] Select the highest power of a prime present in all or some of the numbers. 3] The product of powers of common and uncommon prime is L. C. M.

Determine H. C. F. and L. C. M. of 120, 90 Solution: 120 = 8 x 15 = 2 x 2 x 3 x 5 = 23 x 5 90 = 6 x 15 = 2 x 3 x 5 = 2 x 32 x 5 H. C. F. = 21 x 31 x 51 = 30

DETERMIN H. C. F AND L. C. M OF 8, 25 Solution: 8 = 2 x 4 = 2 x 2 = 23 25 = 5 x 5 = 52 H. C. F of (8, 25) = 1 **No common factor between 8 and 25 L. C. M of (8, 25) = 8 x 25 = 200 NOTE: 1. H. C. F of any two or more prime numbers is always 1. 2. L. C. M of any two or more prime numbers is equal to their products.

SURDS If n is a natural number (n > 1) a is a positive rational number, and n√a is an irrational number is called a surd. n is called order of surd, a is called the radicand the symbol √ is a radical sign.

LAWS OF SURDS If a and b are rational a, b > 0 and m, n, p are natural numbers then: 1. 2. 3. 4. 5. 6. (n√a)n = a (n√a)(n√b) = (n√ab) (n√a)/(n√b)=(n√(a/b)) (n√(m√a))= (m√(n √a)) = mn√a (m√an) = (mp√anp) (m√an) = (m√a)n

FORMS OF SURDS 1. Mixed Surds- A surd of the form k n√a where k is a rational number, k ¹ 0 and k ¹ ± 1 is called a mixed surd. For ex. 2 3√ 7, ¾ √ 5 etc are mixed surds. NOTE: A mixed surd can be expressed as a pure surd. For ex. 5 √ 2 = √ 25 x √ 2 = √ 50 2. Pure Surds- A surd of the form k n√a where k is a rational number, k = ± 1 is called a Pure surd. For ex. √ 3, √ 5, 3√ 2 etc are Pure surds.

SIMILAR SURDS The surds of the form pn√a and qn√a where p and q are rational numbers are called similar surds or like surds. For ex. 5√ 2, 7√ 2, 3√ 2/4, are similar surds. √ 12, √ 50 are also similar surds.

COMPARISON OF SURDS Two surds of the same order can be compared by comparing their radicands. For ex. If n√a and n√b are two surds then: 1. n√a = n√b, if a = b. 2. n√a > n√a, if a > b. 3. n√a < n√b, if a < b.

METHOD TO COMPARE SURDS 1. Convert each surd into the same order. 2. Compare the radicands obtained. 3. A surd with larger radicand is larger of the given surds. For ex. √ 3, 3√ 2 √ 3 = 6√ 33 = 6√ 27 and 3√ 2 = 6√ 22 = 6√ 4 But 6√ 27 > 6√ 4. Therefore √ 3 > 3√ 2

OPERATION ON SURDS ADDITION AND SUBTRACTION OF SURD We know that the operations of addition and subtraction are defined on real numbers. As surds are irrational numbers they can be added or subtracted but addition and subtraction can be done only on like surds. Is sum of two surds a surd ? § For ex. 5√ 2 + 3√ 2 = 8√ 2 (5+√ 2) +(2 - √ 2)=7 7√ 3 - 2√ 3 = 5√ 3

MULTIPLICATION AND DIVISION OF SURDS: Using laws of surds, surds can be multiplied or divided. Note that multiplication and division of surds be performed on the surds of same order. Is product /division of two surds a surd ? For ex. √ 7 x √ 3 = √ 21 , 2 √ 3 x √ 3 =6 , 5 √ 3 / 2 √ 3 = 5/2 , 5 √ 6/ √ 2=5 √ 3 …

RATIONALIZATION OF SURDS If the product of two surds is a rational number then each surd is called a rationalizing factor of the other surd. For ex. √ 24 x √ 6 = √ 144 = 12 √ 24 x √ 24 = 24 Therefore √ 6 is a rationalizing factor of √ 24. Also, √ 24 is a rationalizing factor of √ 24. NOTE: The rationalizing factor of a given surd is not unique but it is convenient to use the lowest form of rationalizing factor of a surd Std - 9 th Subject- Algebra Topic- Real Numbers

• The binomial expressions of quadratic surds a + y√b and a-y √b are said to be conjugate of each other. • For ex. 2 √ 2 - √ 3 and 2 √ 2 + √ 3 are conjugate to each other. • Note: Conjugate surds are useful for rationalization since the product of two conjugate surds is a rational number. • For ex. (√ 3 + √ 2)(√ 3 - √ 2) = (√ 3)2 – (√ 2)2 = 3 - 2 = 1 Std - 9 th Subject- Algebra

BINOMIAL EXPRESSION OF A QUADRATIC SURD • The sum of two numbers one of which is a quadratic surd and the other, either a non-zero rational number or a quadratic surd is called Binomial expression of quadratic surd. • The general binomial expression of quadratic surd is a+ y √b where a is a non-zero rational number or a quadratic surd, y is a non-zero rational number and √b is a quadratic surd. • For ex. √ 5 - √ 2, √ 7 + √ 11, 3 + √ 5, 5 - √ 7 are binomial expressions of quadratic surds.

Exercise • • Write down the rationalizing factor of 3√ 625. (2 mark) Find the rational number between 3 and 4. (1 mark) Express 0. 6666… in the form m/n. (1 mark) Express 0. 1616… in the form m/n. (2 marks) Explain which is greater √ 17 - √ 12 or √ 11 - √ 8. (3 mark). Solve 5 x - 3 = 7 (2 marks) Find square root of 10+√ 24 + √ 60 + √ 40. (4 mark) Represent 1/ √ 7 on a number line. (5 mark) Topic- Real Numbers

Exercise Que. 1) Express each of the following in the form of p. q 1. 0. 555… 2. 0. 4747… Que. 2) Write the following real numbers in decimal form. 1. 3 2. -5 2 6 Que. 3) Represent √ 10 on the number line. Que. 4) Show that 7+ √ 7‾ is an irrational number. Que. 5)Use Euclid’s division lemma to find the HCF of the following. 1. ) 16 and 176 2. ) 40 and 248

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