Pocket Book of Maths By: Nab$ Sir Number System 2. NUMBER SYSTEM In general, numbers can be classified in following manner: ❖ Real Numbers (R): Numbers which can be expressed on number line are called Real no. Examples: All these numbers represented below are real numbers. Number line is geometrical straight line with arbitrarily defined zero (origin). ❖ Imaginary Numbers: Numbers which can’t be represented on number line OR we can say All the numbers whose square is negative are called imaginary numbers. For example: √−𝟒 is not a real number as its square is a negative number. ✓Note: √−𝟏 = 𝒊 (it is read as iota), hence √−𝟒 = √𝟒 × √−𝟏 = 𝟐𝒊 ✓Note: √−𝟐 ≠ −√𝟐 =√𝟐 𝒊 (its square is a negative number) ✓Remember: 𝒊 = √−𝟏 , 𝒊𝟐 = −𝟏, 𝒊𝟑 = −𝒊 & 𝒊𝟒 = 𝟏 Hence, 𝒊 + 𝒊𝟐 + 𝒊𝟑 + 𝒊𝟒 = 𝟎 𝑨𝒍𝒔𝒐 , √−𝟒 × √−𝟗 ≠ √𝟑𝟔 (𝒊. 𝒆. 𝟔) it is −√𝟑𝟔 = −𝟔 AMU Booster Page No. 13 Pocket Book of Maths By: Nab$ Sir Number System ✓ Note: The number 0 is real as well as imaginary number. ❖ Complex Numbers(C/z): The combined form of real and imaginary numbers is known as complex numbers. It is denoted by: z = x + iy where x is real part and y is imaginary part of z and x, y R. ❖ Real number are of two types: 1. Rational number 2. Irrational number Note: This is why sometimes we can say that real number is a collection of rational as well as irrational number. RATIONAL NUMBER (Q): A real no. that can be written in the 𝒑 form of , 𝒒 ≠ 𝟎 & 𝒑, 𝒒 ∈ 𝑰. Ex:𝟏𝟎, −𝟏𝟓, 𝟑𝟐 , 𝟕, 𝟏. 𝟑, 𝟎, −𝟐𝟎 etc. 𝒒 𝟏𝟑 SPECIAL TYPES OF RATIONAL NUMBERS Natural Numbers (N): The counting numbers 1, 2, 3, 4...... are called natural numbers i.e. N= {1, 2, 3, 4, .... , } In other words, Set of all nonfractional numbers from 1 to . Properties of Natural Numbers: ▪ Addition of natural numbers are closed, associative, and commutative. ▪ Natural Number multiplication is closed, associative, and commutative. ▪ The identity element of a natural number under addition is zero. ▪ The identity element of a natural number under Multiplication is one. Whole numbers(W): Natural numbers including zero are called whole numbers i.e. W= {0, 1, 2, 3, 4, ..... , } In other words, Set of all nonfractional numbers from 0 to . Properties of Whole Numbers: ▪ Whole numbers are closed under addition and multiplication. AMU Booster Page No. 14 Pocket Book of Maths By: Nab$ Sir Number System ▪ It obeys the commutative and associative property of addition and multiplication. ▪ Zero is the additive identity element of the whole numbers. ▪ 1 is the multiplicative identity element. ▪ Every natural number is a whole number but vice versa is not true. For example 0. Integers (I/Z): All counting numbers along with their negative including zero is called integers i.e. I or Z = {– ........., –3, –2, –1,0,1,2,3,....... } Note : 0 is neither positive nor negative integer. (a) Positive Integer (I+) : {1,2,3,.......................} (b) Negative Integer (I–) : {................., –3, –2, –1} (c) Nonpositive integers : {................., –3, –2, –1,0} (d) Nonnegative integers : {0,1,2,3,4,.....................} = W Properties of Integers: ▪ Integers are closed under addition, subtraction, and multiplication. ▪ The commutative property is satisfied for addition and multiplication of integers. ▪ It obeys the associative property of addition and multiplication. ▪ It obeys the distributive property for addition and multiplication. ▪ Additive identity of integers is 0. ▪ Multiplicative identity of integers is 1. Even Integers: Integers or numbers which are divisible by 2. Example: 𝟎, ± 𝟐, ± 𝟒, ± 𝟔, . . . . . . . . . . . . . . . .. ✓ Note: Even numbers are denoted by the expression 2n, where n is any integer. Odd integers: Integers or numbers which are not divisible by 2. Example: ±𝟏, ±𝟑, ±𝟓, . . . . . . . . .. Note: Odd numbers are denoted by the general expression ✓ 2n – 1 where n is any integer. AMU Booster Page No. 15 Pocket Book of Maths By: Nab$ Sir Number System Prime numbers: A natural number is said to be prime if it has exactly two distinct factors, namely 1and itself. Example :2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ………… etc. Note: There are 15 prime numbers between 1 and 50. There are 25 prime numbers between1 to 100. Types of Prime numbers (a) CoPrime: Two natural numbers (not necessarily prime) are co prime, if there H.C.F. is one. Example: (1,2), (1,3), (3,4), (3,10), (3,8), (5,6), (7,8) etc. ✓ Note: These numbers are also called as relatively prime numbers. ✓ Note: Two prime numbers (s) are always coprime but converse need not be true. ✓ Note: Consecutive numbers are always coprime numbers. (b) Twin Prime: If the difference between two prime number is two, then the numbers are twin prime pair or prime pair. Example: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139) etc. ✓ Note: There are only eight twin prime numbers from 1 to 100. ✓ Note: Every prime pair other than (3,5) is in the form of (6n – 1, 6n+1), where n is any natural number. ✓ Note: The sum of each twin prime except (3,5) is divisible by 12. IDENTIFICATION OF A PRIME NUMBER Certain note about prime number will do the job easily. N1: Prime numbers never ends with 0, 2, 4, 6 & 8 except 2. N2: Prime numbers never ends with 5 except 5. Step 1: Find approximate square root of given number. Step 2: Divide the given number by prime numbers less than approximate square root of number. If given number is not divisible by any of this prime number then the number is prime otherwise not. AMU Booster Page No. 16 Pocket Book of Maths By: Nab$ Sir Number System Example: 571, is it a prime or not? Sol. Approximate square root of 571 = 24. Prime number < 24 are 2, 3, 5, 7, 11, 13, 17, 19, & 23. But 571 is not divisible by any of these prime numbers so 571 is a prime number . Composite Numbers: A number is said to be composite numbers if it has at least three distinct factors. Or in other words we can say No’s which are not prime are composite numbers except 1. Example: 4, 6, 8, 9, 10, 12…. Note: (i) ‘1’ is neither prime nor composite number. (ii) ‘2’ is the only even prime number. (iii) There are 74 composite numbers from 1 to 100. (iv) ‘4’ is the smallest composite number. (v) ‘4’ is smallest even composite number. (vi) ‘9’ is smallest odd composite number. SIMPLEST FORM OF A RATIONAL NUMBER 𝑝 A rational number 𝑞 is said to be in simplest form, if p and q are integers having no common factor other than 1(or p & q coprimes). Here q 0. They are also called as equivalent rational numbers. 2 3 4 5 6 1 Thus, the simplest form of each of 4, 6, 8, 10, 12 etc. is 2. 𝟔 𝟐 𝟕𝟔 𝟒 Similarly, the simplest form of 𝟗 is 𝟑 and that of 𝟏𝟑𝟑 is 𝟕. ❖ FRACTION: A fraction is a number which can be written in 𝑎 the form , where both a and b are natural numbers and the 𝑏 number 'a' is called numerator and 'b' is called the denominator 𝑎 of the fraction , b 0.hence every fraction is a rational 𝑏 2 1 0 7 number. For example: , , , , are fractions. 5 3 5 15 AMU Booster Page No. 17 Pocket Book of Maths By: Nab$ Sir Number System DIFFERENT TYPES OF FRACTIONS Types Definition Examples 2 3 12 Proper Numerator < Denominator , , , ... 9 7 29 7 29 17 Improper Numerator > Denominator , 5 17 13 , ,... 7 5 11 Like same denominator , , , ..... 12 12 12 2 4 11 7 Unlike different denominators , , , , .... 3 5 13 8 1 1 1 1 Unit numerator as 1 , , , , .... 2 4 3 7 combination of a whole number 1 1 1 Mixed 3 , 5 , 8 ,… and a proper fraction. 2 3 4 Numerator and Denominator 2/3 3/5 Compound , ,…. themselves are fractions 5/7 2/9 Fractions representing the same 4 2 6 Equivalent = = portion of the whole. 6 3 9 denominator is any of the number 8 11 17 Decimal , , , .. 10,100,1000 etc. 10 100 1000 denominator is a whole number, 2 3 11 Vulgar , , other than 10,100,1000 etc. 7 8 17 ✓ Note: Improper fraction can be written in the form of mixed fractions. ❖ DECIMAL: In algebra, a decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point. The dot in a decimal number is called a decimal point. The digits following the decimal point shows a value smaller than one. AMU Booster Page No. 18 Pocket Book of Maths By: Nab$ Sir Number System ❖ There are two types of decimal numbers, namely: 1. Terminating decimals: Decimals that terminates/ends. In other words, we can say a decimal which has finite number of digits after decimal place. Examples:𝟎. 𝟒𝟓, 𝟎. 𝟐, 𝟏𝟐. 𝟐𝟓, 𝟐𝟏𝟑. 𝟔𝟐𝟓 etc. ✓ Note: Every terminating decimal can easily be converted into p/q forms, hence are always rational numbers. 2. Nonterminating Decimals: Decimals that don’t terminates. It is of two types: a. Nonterminating repeating (Recurring) decimals: Decimals that repeats itself.it can be of two types: (i) Pure recurring decimals: A decimal in which all the digits after the decimal point are repeated. Ex.: 0. 6, 0. 16 , 0. 123 are pure recurring decimals. (ii) Mixed recurring decimals: A decimal in which at least one of the digits after the decimal point is not repeated and then some digit or digits are repeated. Ex: 2.16, 0.35, 0.785 are mixed recurring decimals. ✓ Note: Nonterminating repeating decimals can be converted into p/q form. Hence, they are always rational numbers. b. Nonterminating non repeating decimals: Decimals which neither terminates nor repeats itself. Ex: 567.142857142857….. , 0.10100100010000….etc. ✓ Note: These types of decimal can never be converted into p/q form. Hence, they are not rational numbers. 𝒑 Case I: In order to convert a pure recurring decimal to form 𝒒 We follow the following steps: Step I: Obtain the repeating decimal and put it equal to x (say). Step II: Write the number in decimal from by removing bar from the top of repeating digit and listing repeating digits at least twice. Ex: x = 0. 8 as x = 0.888 AMU Booster Page No. 19 Pocket Book of Maths By: Nab$ Sir Number System Step III: Determine the number of digits having bar on their heads. Step IV: If the repeating decimal has 1 place repetition, multiply by 10; a twoplace repetition. multiply by 100; a threeplace repetition, multiply by 1000 and so on. Step V: Subtract the number in step II from the number obtained in step IV. Step VI: Divide both sides of the equation by the coefficient of x. Step VII: Write the rational number in its simplest form. 𝑝 Example: Convert the 23. 43 number in the form . 𝑞 Sol. Let x = 23. 43 Then1 x = 23.434343..... ....(i) Here we have only two repeating digit after decimal so we multiply both sides by100. Multiplying both sides of (i) by 100, we get 100 x = 2343.4343.... ....(ii) Subtracting (i) from (ii), we get 100x – x = (2343.4343...) – (23.4343....) 2320 99x = 2320 𝑥 = 99 Alter method: We have, 23. 43 = 23 + 0. 43 43 23. 43 = 23 + 99 43 [Using the above rule, we have 0. 43 = 99] 23×99+43 23. 43 = 99 = 2277+43 99 = 2320 99 𝒑 Case II: To convert a mixed recurring decimal to the form 𝒒 We follow the following steps: Step I: Obtain the mixed recurring decimal and write it equal to x (say). AMU Booster Page No. 20 Pocket Book of Maths By: Nab$ Sir Number System Step II: Determine the number of digits after the decimal point which do not have bar on them. Let there be n digits without bar just after the decimal point. Step III: Multiply both sides of x by 10n so that only the repeating decimal is on the right side of the decimal point. Step IV: Use the method of converting pure recurring decimal to the 𝑝 form and obtain the value of x. 𝑞 𝑝 Example: Convert 0.2353535... = 0.235 in form? 𝑞 Sol. Let x = 0.235. Then 10x = 0.235 10x = 2 + 0. 35 35 35 10𝑥 = 2 + 99 [∴ 0. 35 = ] 99 2×99+35 10𝑥 = 99 198+35 233 233 10𝑥 = ⇒ 10x = ⇒ 𝑥 = 990 99 99 DETERMINING THE NATURE OF THE DECIMAL EXPANSIONS OF RATIONAL NUMBERS (a) Let x be a rational number whose decimal expansion terminates. 𝑝 Then we can express x in the form 𝑞, where p and q are coprimes, and the prime factorization of q is of the form 2m × 5n. where m, n are nonnegative integers(W). 𝑝 (b) Let x = 𝑞 be a rational number, such that the prime factorization of q is of the form 2m × 5n where m, n are nonnegative integers. Then, x has a decimal expansion which terminates. 𝑝 (c) Let 𝑥 = be rational number, such that the prime factorization 𝑞 of q is not of the form 2m × 5n. where m, n are nonnegative integers. Then, x has a decimal expansion which is nonterminating repeating (recurring decimal). AMU Booster Page No. 21 Pocket Book of Maths By: Nab$ Sir Number System 𝑝 ✓ Note: If a rational number 𝑞 terminates as q is of the form 2m × 5n, then it will terminate up to minimum (m, n) places of decimal. PROPERTIES OF RATIONAL NUMBERS 𝑎 𝑐 𝑒 If 𝑏 and 𝑑 𝑓 , are three rational numbers then: 𝒂 𝒄 𝒄 𝒂 1. 𝒃 + 𝒅 = 𝒅 + 𝒃 [commutative law of additional] 𝑎 𝑐 𝑒 𝑎 𝑐 𝑒 2. (𝑏 + 𝑑) + 𝑓 = 𝑏 + (𝑑 + 𝑓) [associative law of additional] 𝒂 −𝒂 −𝒂 𝒂 𝒂 𝑎 3. (𝒃 + ) = ( 𝒃 + 𝒃) = 𝟎, − 𝒃 is called the additive inverse of 𝑏 𝒃 𝒂 𝒄 𝒄 𝒂 4. (𝒃 × 𝒅) = (𝒅 × 𝒃) [commutative law of multiplication] 𝒂 𝒄 𝒆 𝒂 𝒄 𝒆 5. ( × ) × = ( × ) [associative law of multiplication] 𝒃 𝒅 𝒇 𝒃 𝒅 𝒇 𝒂 𝒄 𝒆 𝒂 𝒄 𝒂 𝒆 6. × (𝒅 + 𝒇) = (𝒃 × 𝒅) + (𝒃 × 𝒇) [distributive law] 𝒃 TO FIND RATIONAL NUMBER(S) BETWEEN GIVEN RATIONAL NUMBERS ✓ Note: There are infinitely many rational numbers between any two given rational numbers. Method 1: Let x and y be two rational numbers such that x < y. 1 Then, 2 (𝑥 + 𝑦) is always a rational number between x and y. Use above formula repeatedly to get desired rational numbers. Method 2: Let x and y be two rational numbers such that x < y. Suppose we want to find n rational numbers between x and y. 𝑦−𝑥 Let 𝑑= . 𝑛+1 Then, n rational numbers between x and y are: (𝑥 + 𝑑), (𝑥 + 2𝑑), (𝑥 + 3𝑑), . . . , (𝑥 + 𝑛𝑑) Method 3: Making Denominator same by taking LCM. Method 4: Converting given rational numbers into decimals. AMU Booster Page No. 22 Pocket Book of Maths By: Nab$ Sir Number System ❖ Irrational Numbers (Qc): The numbers whose decimal representation is neither terminating nor repeating. Also these numbers can never be written in p/q form. Ex: √𝟐, 𝟑√𝟐, 𝟐 + √𝟑, , 1.327185…, 0.10100100010000….etc. DIFFERENCE BETWEEN RATIONAL AND IRRATIONAL NUMBERS Rational Number (Q) Irrational Number (Qc) 𝑃 • They can be written in form • These numbers can’t be 𝑄 𝑃 where Q 0 and P, Q ∈ I written in form. 𝑄 Ex √2, √3, √5, etc. −1 Ex1, 3, 0, , 1. 3, 1.25 etc. 5 1.0100100010001… etc. • Even No, Odd no, Integers, • √𝑃𝑟𝑖𝑚𝑒 𝑁𝑢𝑚𝑏𝑒𝑟 is always Natural No, Whole Numbers, irrational number. Prime No, Composite No, Fractions are all examples of Ex √2, √3, √5, etc. rational no. • √𝐶𝑜𝑚𝑝. 𝑁𝑜 ≠ 𝑃𝑒𝑟. 𝑆𝑞𝑢𝑎𝑟𝑒 is always irrational No. • √𝐶𝑜𝑚𝑝. 𝑁𝑜 = 𝑃𝑒𝑟. 𝑆𝑞𝑢𝑎𝑟𝑒 is Ex.√6, √8, √10, etc. always rational number Ex √4, √9, √16, . . . . ., etc. • 𝑛√𝑎 if it is a surd then it is always irrational No. • √𝑎 if not a surd then it is 𝑛 3 4 3 4 Ex: √9, √16, 2√10, √32 etc. 3 4 rational. Ex: √27, √16 etc. • NonTerminating and non repeating decimals are always • Terminating and repeating irrational decimal are always rational Ex 𝜋 = 3.1415. . . . 𝑒 = 2.71. .. Ex 1.21, 1. 67, 1.231, 1.5 etc 4537.82572481923745…. AMU Booster Page No. 23 Pocket Book of Maths By: Nab$ Sir Number System 𝟐𝟐 ✓ Note: π is an irrational No., While 𝟕 is a rational No. π is 𝟐𝟐 approximately written equal to . 𝟕 ✓ Note: Between two real number (Q/Qc) there exists infinitely many real no(Q/Qc) on number line. OPERATIONS ON REAL NUMBERS 𝟒 𝟐 𝟑𝟖 Q for Rational numbers like 0,1, 7,3,4,10, , , etc. 𝟓 𝟕 𝟑𝟓 c 𝟑 Q for Irrational numbers like √𝟐, √𝟑, √𝟔, √𝟑, 𝟐√𝟑, 𝟐 + √𝟑 , 𝟐 − √𝟑 Operations Result Examples 𝟒 𝟐 𝟑𝟖 𝑸+𝑸 Always Q +𝟕= , 7 + 3 = 10 𝟓 𝟑𝟓 𝟒 𝟐 𝟏𝟖 𝑸−𝑸 Always Q 𝟓 −𝟕= 𝟑𝟓 ,7–3=4 𝒄 𝟑 𝑸+𝑸 Always Qc 𝟐 + √𝟑, 4 + √𝟑 𝑸 − 𝑸𝒄 Always Qc 𝟑 𝟐 − √𝟑, 4 −√𝟑 √𝟑 + 𝟐√𝟑 = 𝟑√𝟑 , √𝟑 + √𝟓 𝑸𝒄 + 𝑸 𝒄 May be Q or Qc (𝟐 + √𝟑) + (𝟐 − √𝟑) = 𝟒 𝑸𝒄 − 𝑸 𝒄 May be Q or Qc 𝟐√𝟑 − √𝟑 = √𝟑 , √𝟑 − √𝟑 = 𝟎 𝟒 𝟐 𝟖 𝑸×𝑸 Always Q ×𝟕= , 7 × 3 = 21 𝟓 𝟑𝟓 𝒄 c 𝑸×𝑸 May be Q or Q 2× √𝟑 = 𝟐√𝟑 , 0× √𝟑 = 𝟎 𝒄 𝒄 c 𝑸 ×𝑸 May be Q or Q √𝟐 × √𝟑 =√𝟔 , √𝟐 × √𝟐 = 𝟐 𝟒 𝟐 𝟏𝟒 𝑸÷𝑸 Q or Not defined ÷ = , 7 ÷ 0 = ND 𝟓 𝟕 𝟓 𝒄 𝑸÷𝑸 May be Q or Q c 2÷ √𝟐 = √𝟐 , 0÷ √𝟐 = 0 𝑸𝒄 ÷ 𝑸 Qc or Not defined 𝟐√𝟑 ÷ 𝟐 = √𝟑 , √𝟑 ÷ 𝟎 = ND 𝑸𝒄 ÷ 𝑸 𝒄 May be Q or Qc √𝟔 ÷ √𝟑 =√𝟐 , √𝟐 ÷ √𝟐 =1 AMU Booster Page No. 24 Pocket Book of Maths By: Nab$ Sir Number System SOME IMPORTANT NOTE N1: Negative of an irrational number is an irrational number. N2: The product of a nonzero rational number & an irrational number will always be an irrational number. N3: The division of a rational number with a nonzero rational number always results into a rational number. N4: The division of an irrational number with a nonzero rational number is always irrational number. N5: Sum, difference, product and division of two irrational numbers results into a rational or an irrational number. N6: Every rational as well as irrational number can be represented on number line (this we have studied in our class IX) N7: If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then √𝒂𝒃 is an irrational number lying between a and b. CLOSURE PROPERTY OF REAL NUMBERS A real number is closed (under an operation) if and only if the operation on any two real numbers produces real number. If the operation produces even one number not real, then that operation is said as not closed. For example: The set of real numbers is closed under addition. If you add two real numbers, you will get another real number. There is no possibility of ever getting anything other than another real number. 5 + 12 = 17, 3½ + 6 = 9½, √𝟑 + 𝟐√𝟑 = 𝟑√𝟑 etc. Another example: The set of integers {... 3, 2, 1, 0, 1, 2, 3 ...} is NOT closed under division. 5 ÷ 2 = 2.5 Since 2.5 is not an integer, closure fails. There are also other examples that fail. Next example: The set of real numbers is NOT closed under division. 3 ÷ 0 = undefined. Since "undefined" is not a real number, closure fails. Division by zero is the ONLY case where closure fails for real numbers. AMU Booster Page No. 25 Pocket Book of Maths By: Nab$ Sir Number System Note: Some textbooks state that " the real numbers are closed under nonzero division" which, of course, is true. This statement, however, is not equivalent to the general statement that "the real numbers are closed under division". Always read carefully! CLOSURE PROPERTY TABLE FOR BASIC MATHEMATICAL OPERATIONS Numbers Add (+) Subtract () Multiply ( ×) Division ( ÷) Real Closed Closed Closed Not Closed Rational Closed Closed Closed Not Closed Irrational Not Closed Not Closed Not Closed Not Closed Natural Closed Not Closed Closed Not Closed Whole Closed Not Closed Closed Not Closed Integers Closed Closed Closed Not Closed Fundamental theorem of Arithmetic: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. In other words, it can be stated as “The prime factorization of a natural number is unique, except for the order of its factors” Theorem: Let P be a prime number. If P divides a2, then P divides a, where a is a positive integer. HIGHEST COMMON FACTOR(HCF) Factors: A number is a factor of another, if the former exactly divides the latter without leaving a remainder (remainder is zero). They are also called as divisors. For example: Factors of 10 are 1, 2, 5 & 10. Similarly factors of 24 are 1, 2, 3, 4, 6, 8, 12 & 24. Note: 1 and the number itself is always one of the factors of a given number. Note: A prime number has only/exactly two factors. Ex: 2, 3, 5 etc. AMU Booster Page No. 26 Pocket Book of Maths By: Nab$ Sir Number System Definition: HCF also known as GCM (Greatest common Measure) or GCD (greatest common divisor). The HCF of two or more than two positive integers is the highest/greatest positive integers (that must be common Factor) that divides each of the given positive integer exactly (remainder is zero). For Example: HCF of 3 and 6 is 3 as three divides 3 as well as 6. (i) 14 is the largest positive integer that divides 28 and 70 completely; therefore H.C.F. of 28 and 70 is 14. (ii) H.C.F. of 75, 125 and 200 is 25 as 25 divides each of 75, 125 and 200 completely and so on. Note: HCF of two or more prime numbers is equal to the smallest prime numbers in the given prime numbers. Note: The HCF of positive integers is always less than or may be equal to the least/smallest of these integers. METHODS TO FIND HCF I. HCF by Listing Method: The listing method involves the process of listing the factors of the given numbers. For example, find the HCF of 20 and 35. • All possible factors of 36 are 1,2,3,4,6,12,18 and 36. • All possible factors of 48 are 1,2,3,4,6,8,12,16,24 and 48 The common factors of the given numbers are: 1,2,3,4,6 and 12. The greatest among all other numbers is 12, so it shall be the HCF of both the numbers. II. By applying the Fundamental theorem of Arithmetic method or Prime Factorization method: When we find HCF by prime factorization method, we are finding the greatest common factor among the prime factors or numbers. Steps to be followed for the method are: Let us use these steps in the example below: For Example: Find the HCF of 36 and 48? AMU Booster Page No. 27 Pocket Book of Maths By: Nab$ Sir Number System Step 1: Finding prime factors individually: All possible factors of 36 are: 2×2×3×3 All possible factors of 48 are: 2×2×2×2×3×1 Step 2: Choose out the common factors: 2×2×3 Step 3: Multiply all the common factors to get the HCF of the given numbers: Here the given numbers are 36 and 48. The product of the common factors: 2×2×3 = 12. So the HCF for the numbers 36 and 48 is 12. III. Long division method: Follow following steps to find HCF: Step 1: We divide the bigger number by smaller one. Step 2: Divide smaller number in step 1 with remainder obtained in step 1. Step 3: Divide divisor of second step with remainder obtained in step 2. Step 4: We will continue this process till we get remainder zero and divisor obtained in end is the required H.C.F. For Example: let us find HCF 36 ) 48 ( 1 of 36 and 48 36 by this method. 12) 36 (3 HCF 36 IV. Euclid’s Division Algorithm: ×× An algorithm is a series of welldefined steps which gives a procedure for solving a type of problem. Euclid’s Division lemma (EDL): For any two positive integers a and b, there exist unique integers q and r satisfying: a = bq + r, where 0 r < b also q or r can also be zero. Lemma: A lemma is a proven statement used for proving another statement. AMU Booster Page No. 28 Pocket Book of Maths By: Nab$ Sir Number System ✓ Note: In the relation a = bq + r, where 0 r < b is nothing but a statement of the long division of number a by number b in which q is the quotient obtained and r is the remainder. Thus, dividend = divisor × quotient + remainder a = bq + r SOME IMPORTANT NOTES ABOUT EDL N1: EDL helps in proving many divisibility properties of integers. N2: Euclid’s division lemma and division algorithm are so closely interlinked that people often call former as the division algorithm also. N3: Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0. N4: Euclid’s division algorithm can be used to compute the Highest Common Factor (HCF) of two given positive integers. For Example: Consider positive integers 48 and 36 Step 1: Taking bigger number (48) as a and smaller number (36) as b. Express the numbers as: a = bq + r i.e. 48 = 36 × 1 + 12 Step 2: Now taking the divisor 36 and remainder 12, apply the Euclid’s division lemma to get. 36 = 12 × 3 + 0 [Expressing as a = bq + r] We will repeat this step until we get remainder as 0. Step 3: Since, the remainder = 0 so we can’t proceed further. Step 4: The last divisor is 12 and we say H.C.F. of 48 and 36 is 12. ✓ Note: This method works because If a = bq + r, then in each step H.C.F. of (a, b) = H.C.F. of (q, r). ✓ Note: we can note down that all the above four methods will give same HCF. These methods can also be used to find HCF of more than two Positive integers. AMU Booster Page No. 29 Pocket Book of Maths By: Nab$ Sir Number System SOME RESULTS OF INTEGERS PROVED BY EDL R1: Every even positive integer is of the form 2q and that every positive odd integer is of the form 2q + 1, where q is any integer. R2: Square of any positive integer is either of the form 3m or 3m +1 for some integer m. R3: Any positive odd integer is of the form 6q + 1, 6q + 3 or 6q + 5, where q is some integer. R4: The cube of any positive integer is either of the form 9q, 9q + 1 or 9q + 8. R5: Only one of the numbers (n + 2), n and (n + 4) is divisible by 3. R6: Any positive even integer is of the form 4q, 4q + 2, where q is some integer. R7: Any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. R8: 𝑛 2 − 𝑛 is divisible by 2 for every +ve integer n. R9: Multiplication of two consecutive positive integers is divisible by 2. R10: One and only one out of 𝑛, 𝑛 + 3, 𝑛 + 6, 𝑛 + 9 is divisible by 4 R11: If 𝑎 = 𝑥𝑞 + 𝑎𝑟 , 𝑏 = 𝑦𝑞 + 𝑏𝑟 , 𝑐 = 𝑧𝑞 + 𝑐𝑟 , then the remainder 𝑎×𝑏×𝑐 𝑎𝑟×𝑏𝑟 ×𝑐𝑟 of 𝑞 is same as the remainder of 𝑞 . LOWEST COMMON MULTIPLE (LCM) Multiples: The multiple of a number are numbers obtained by multiplying the given number with same or other numbers. For example: Multiples of 6 are 6, 12, 18, 24, 30 etc. or indirectly we can all the numbers which lies in the table of 6. ✓ Note: 0 is a multiple of every number as the product of 0 multiplied by any number is 0. ✓ Note: Number of multiples of a number is infinite while number of its factor is always finite (how bigger may number is). ✓ Note: A number is always a multiple of all its factors. AMU Booster Page No. 30 Pocket Book of Maths By: Nab$ Sir Number System Definition: LCM of two or more than two positive integers is the lowest/least positive integer (that must be common multiple) which is divisible by each of the given positive integers completely (means it leaves no remainder). For example: LCM of 3 and 6 is 6. (i) LCM of 12 and 18 is 36 as 36 is the smallest positive integer that is divided by 12 and 18 completely. (ii) LCM of 3, 6, 9 and 24 is 72 as 72 is divisible by each of 3, 6, 9 and 24 completely. Note: LCM of two or more than two prime numbers is the product of all given prime numbers. Note: The LCM of positive integers is always greater than or may be equal to the greatest/highest of these integers. METHODS TO FIND LCM I. LCM by Listing Method: The listing method involves the process of listing the multiples of the given numbers. For example, To find the LCM of 12 and 18. • All possible multiples of 12 are 12, 24, 36, 48, 60, 72, 84, … • All possible multiples of 18 are 18, 36, 54, 72, 90, 108, …. The common multiples of the given numbers are: 36, 72,108, … The lowest among all other numbers is 36, so it shall be the LCM of 12 and 18. You can check 36 is exactly divisible by 12 and 18. II. By prime Factorization method: The steps are same as followed in finding HCF. In HCF we take minimum power of all prime factors involved while in LCM we take maximum powers of all the prime factors involved. For Example: to find LCM of 36 and 48 we prime factorize both numbers. 36 = 2×2×3×3 = 22×32, Similarly 48 = 24×3 Therefore: LCM (36, 48) = 24×32 = 16 × 9 = 144 III. Ladder or short cut method: We can also find the L.C.M. of the given numbers by dividing all the numbers at the same time by a number that divides at least two of the given numbers. AMU Booster Page No. 31 Pocket Book of Maths By: Nab$ Sir Number System a. When a number is not exactly divisible, we write the number itself below the line. b. When we cannot divide the numbers by a common factor exactly we discontinue dividing the number. L.C.M. = 2 × 2 × 2 × 3 × 2 = 48 NOTE: For any two positive integers: Their L.C.M. × their H.C.F. = Product of the number 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 (i) LCM = 𝐇𝐂𝐅 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 (ii) HCF = 𝐋𝐂𝐌 𝐋𝐂𝐌 × 𝐇𝐂𝐅 (iii) One number = 𝐎𝐭𝐡𝐞𝐫 𝐍𝐮𝐦𝐛𝐞𝐫 HCF AND LCM OF FRACTIONS 𝑯𝑪𝑭𝒐𝒇 𝑵𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓𝒔 HCF of fractions = 𝑳𝑪𝑴 𝒐𝒇 𝑫𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓𝒔 4 4 2 36 𝐻𝐶𝐹𝑜𝑓(4,4,2,36) 2 For example: The HCF of , , , = 𝐿𝐶𝑀𝑜𝑓 (3,9,15,21) = 315 3 9 15 21 𝑳𝑪𝑴 𝒐𝒇 𝑵𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓𝒔 LCM of fractions = 𝑯𝑪𝑭 𝒐𝒇 𝑫𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓𝒔 4 4 2 36 𝐿𝐶𝑀 𝑜𝑓 (4,4,2,36) 36 For example: The LCM of , , , = = = 12 3 9 15 21 𝐻𝐶𝐹 𝑜𝑓 (3,9,15,21) 3 Note: For any three positive integers p, q and r: HCF (P, q, r) × LCM (p, q, r) p × q × r AMU Booster Page No. 32 Pocket Book of Maths By: Nab$ Sir Number System However, the following results hold good for the three positive integers p, q and r: p×q×r×HCF (p, q, r) LCM (p, q, r) = HCF (p, q) ×HCF (q, r) × HCF (p, r) p×q×r×LCM (p, q, r) HCF (p, q, r) = LCM (p, q) × LCM (q, r) × LCM (p, r) Exponents or Powers For any real number ‘a’ and a positive integer ‘n’ we define a n as: an = a ×a × a × ....... × a (n times) a is called the n power of a. The real number ‘a’ is called the base n th and ‘n’ is called the exponent/power or index of the n th power of a. Ex. 34 = 3 × 3 × 3× 3 = 81 (It is read as 3 raised to the power 4) LAWS OF EXPONENTS There are certain laws that govern the operations in numbers which are expressed in the exponential notation. 1. am × an = am+n Ex: 24 × 23 = 24+3 = 27 2. am an = am – n Ex: 25 23 = 25 – 3 = 22 3. (𝒂𝒃) 𝒎 = 𝒂𝒎 𝒃𝒎 Ex: (2.3)4 = 24 34 𝒂 𝒎 𝒂𝒎 𝟔 𝟑 𝟔𝟑 4. (𝒃) = 𝒃𝒎 (𝒃 ≠ 𝟎) Ex: (𝟐) = 𝟐𝟑 = 33 5. (am)n = (an)m = amn Ex: (23)4 = (24)3 = 23.4 = 212 6. abn = ab+b+b….n times Ex: 23.6 = 23+3+….6 times 𝟏 1 7. 𝒂−𝒎 = 𝒂𝒎 , (𝒂 ≠ 𝟎) Ex: 2−3 = 23 8. am am = a0 = 1 (𝒂 ≠ 𝟎) Ex: 20 = 1 x y 9. If a = a ⇒ x = y (𝒂 ≠ 𝟎, 𝟏, −𝟏) For example: (1)6 = (1)8 ⇒ 6 = 8, but 6 ≠ 8 Where a and b are positive real numbers and m, n are rational numbers. AMU Booster Page No. 33 Pocket Book of Maths By: Nab$ Sir Number System ✓ Note: But if we have to solve the equations like [𝒇(𝒙)]𝒑(𝒙) = [𝒇(𝒙)]𝒈(𝒙) then we have to solve: (a) 𝑓(𝑥) = 1 (b) 𝑓(𝑥) = −1 (c) 𝑓(𝑥) = 0 (d) 𝑝(𝑥) = 𝑔(𝑥) Verification should be done in (b) and (c) cases 10. If 𝑎 𝑥 = 𝑏 𝑥 then consider the following cases: (i) If 𝑎 ≠ ±𝑏, then 𝑥 = 0 (ii) If 𝑎 = 𝑏 ≠ 0, then x may have any real value (iii) If 𝑎 = −𝑏, then 𝑥 is even. ✓ Note: If we have to solve the equation of the form [𝒇(𝒙)]𝒑(𝒙) = [𝒈(𝒙)]𝒑(𝒙) i.e., same index, different bases, then we have to solve (a) 𝑓(𝑥) = 𝑔(𝑥), (b) 𝑓(𝑥) = −𝑔(𝑥), (c) 𝑝(𝑥) = 0 Verification should be done in (b) and (c) cases. Surds (Radicals) and Indices ❖ Any root of a number which cannot be exactly found is called a surd i.e. in other words we can say ‘Any irrational number of the form 𝑛√𝑎 is given a special name surd’. Hence 𝑛√𝑎 is a surd if it is an irrational number where ‘a’ is a rational number. Here 𝑛 √ is called as radical sign and ‘a’ is called as radicand and n is called as index whose plural is indices. √𝒂 = 𝒂𝟏/𝒏 𝒉𝒆𝒓𝒆 𝒏 {2, 3, 4, 5, ….} 𝒏 3 For Ex: √4 is a surd as radicand ‘4’ is a rational number. 3 1 1 • √√3 is a surd as √√3 = √ 32 = 36 = √3 also √ √3, √ √6 …. Etc 3 3 6 3 3 4 5 • √7 − 4√3 is a surd as 7 – 4√3 is a perfect square of (2 – √3) • 3 3 3 8 is not a surd because √8 = √23 which is a rational number. • √2 + √3 is not a surd because 2 + √3 is not a perfect square. AMU Booster Page No. 34 Pocket Book of Maths By: Nab$ Sir Number System 3 • √1 + √3 is not a surd because radicand is an irrational number. • 2 + √3 is a surd (as surd + rational number will give a surd) 3 Similarly, these √3 − √2, √3 + 1, √3 + 1, . . .. are all surds ❖ Let a be a rational number and n is a positive integer. If the 𝑛 𝑡ℎ root of x i.e., 𝑥1/𝑛 is irrational, then it is called surd of order n. ❖ Order of a surd is indicated by the number denoting the root. 3 3 𝑛 For example:√7, √9, (11)5 , √3 are surds of second, third, fifth and nth order respectively. ➢ A second order surd is often called a quadratic surd, a surd of third order is called a cubic surd. ✓ Note: Every surd is an irrational number but converse is not true. For Example: 𝜋 is an irrational but not a surd. ❖ Simplest form of a surds: 3 3 Ex. (i) √135 it’s simplest form is 3√5 4 4 (ii) √1875 it’s simplest form is 5√3 4 6 (iii) √8 = √23 = √2 Simplest form TYPES OF SURDS 1. Quadratic surds: Surds of order 2. Ex: √2 , √3 𝑒𝑡𝑐 3 3 2. Cubic surds: Surd of order 3. Ex: √3, √15 𝑒𝑡𝑐. 4 4 3. Biquadratic surds: Surd of order 4. Ex: √8 , √9 etc. 4. Like surds: Two or more surds are called like if they have or can be reduced to have the same irrational or surds factor. Ex: √2 & 3√2 , √45 & √80 𝑎𝑠 𝑖𝑡 𝑠𝑎𝑚𝑒 𝑎𝑠 3√5 and 4√5 5. Unlike surds: Two or more surds are called unlike, if they are not similar, (i.e. radicand as well as index are different). 3 4 Ex: √5, √3, √6, √3, √9 etc. 6. Pure surds: A Surds which has unity only as its rational factor, the other being irrational, is called pure surd. 3 4 6 Ex: √5, √7, √𝟑, √15, √1875, √8 etc. AMU Booster Page No. 35 Pocket Book of Maths By: Nab$ Sir Number System 7. Mixed surds: A surd which as a rational factor other than unity, the other factor being irrational, is called a mixed surd. 4 Ex: 5√3 , 6√2, 2√3, 5√3 etc. 8. Simple surds: A surd consisting of a single term is called a 4 73 simple surd. Ex: √3, 3√5, √6 etc. 2 9. Compound surds: An algebraic sum of two or more surds is called as compound surd. 3 3 Ex: √3 + √5 − √4, √3 − √5 etc are compound surds. Types of compound surds: • Monomial surds: Single surd is called monomial surds. 3 43 Ex: √2, √2, 2 √3 𝑒𝑡𝑐. • Binomial surds: An algebraic sum of two simple surds or a rational number and a simple surd is known as a binomial surd. 3 3 3 Ex: 2 + √3, √3 + √2, 2 + √3, √2 + √3 etc. • Trinomial surds: An algebraic sum of three simple surds or the sum of a rational number and two simple surds is known as a 3 3 3 trinomial surd.Ex: 2 + √3 − √5, √9 − √6 + √4 etc. 10. Equiradical surds:  Surds of the same order are called equiradical surds. Ex: √2, √3, √5 etc. 11. Nonequiradical surds: Surds of the different orders are 3 4 known as nonequiradical surds. Ex: √2, √4, √5 etc. 12. Binomial quadratic surds: Binomial surds consisting of pure (or simple) surds of order two i.e., the surds of the form 𝑎√𝑏 ± 𝑐√𝑑 or 𝑎 ± 𝑏√𝑐 are called binomial quadratic surds. Note: Two binomial quadratic surds which differ only in the sign which connects their terms are said to be conjugate or complementary to each other. The product of a binomial quadratic surd and its conjugate is always rational. For example: The conjugate of the surd 2√7 + 5√3 is the surd 2√7 − 5√3. AMU Booster Page No. 36 Pocket Book of Maths By: Nab$ Sir Number System Note: If 𝑎 + √𝑏 = 𝑐 + √𝑑, where a and c are rational, and √𝑏, √𝑑 are irrational, then 𝑎 = 𝑐 and b= d. LAWS OF SURDS 𝟏 𝒏 𝟏/𝒏 𝟑 1. √𝒂 = 𝒂 Ex: √𝟐 = 𝟐𝟑 𝒏 𝒏 𝒏 𝟑 𝟑 𝟑 2. √𝒂 √𝒃 = √𝒂𝒃 Ex: √𝟐. √𝟑 = √𝟔 𝒏 𝟑 √𝒂 𝒏 𝒂 √𝟏𝟔 𝟑 𝟑𝟏𝟔 3. 𝒏 = √𝒃 Ex: 𝟑 = √ 𝟐 = √𝟖 = 𝟑 √𝒃 √𝟐 𝒎 𝒏 𝒏 𝟑 𝟐 𝟑 𝟔 𝟐 4. √ √𝒂 = √ 𝒎√𝒂 = 𝒏𝒎√𝒂 Ex: √ √𝟖 = √ √𝟖 = √𝟖 𝟏 𝟏 𝒎 𝟑 𝒏 𝒏 𝟒 𝟒 5. √𝒂𝒎 = (𝒂𝒎 ) = (𝒂 )𝒎 = ( √𝒂)𝒏 𝒏 Ex: √𝟐𝟑 = (√𝟐) 𝒏 𝒏 𝟑 𝟑 𝟑 6. ( 𝒏√𝒂) = (𝒂𝟏/𝒏 ) = 𝒂 Ex:(√𝟐) = (𝟐𝟏/𝟑 ) = 𝟐 𝑛 𝑛𝑝 𝑛 𝑛𝑝 3 3.2 7. √𝑎 = √𝑎𝑝 or √𝑎𝑚 = √(𝑎𝑚 )𝑝 Ex: √23 = √(23 )2 = 2 (This law is useful if we have to change order of the surd) 3 3 3.2 6 For Ex: To make √62 of order 6 then, √62 = √62.2 = √64 𝟑 3 3.5 15 And if to make √𝟔 of order 15 then √6 = √61.5 = √65 𝒏 𝒏 ✓ Note: 𝒏√𝒂 √𝒃 = √𝒂𝒃 is not always true. This is true when at least any of a and b is positive. ✓ Note: √𝒙 = 𝒙 when x = 0 or 1 ✓ Note: √𝒙 < 𝒙 when x > 1 ✓ Note: √𝒙 > 𝒙 when 0 < x < 1 OPERATION OF SURDS (a) Addition and subtraction of surds: Addition & subtraction of surds are possible only when order and radicand are same i.e. only for like surds. Ex: Simplify:15√16 − √216 + √96 = 15√6 − √62 × 6 + √16 × 6 = 15√6 − 6√6 + 4√6 = (15 − 6 + 4)√6 = 13√6 AMU Booster Page No. 37 Pocket Book of Maths By: Nab$ Sir Number System (b) Multiplication and division of surds: 3 3 3 3 3 Ex: (i) √4 × √22 = √4 × 22 = √23 × 11 = 2√11 3 4 3 4 12 12 12 12 (ii) √2 by √3 = √2 × √3 = √24 × √33 = √24 × 33 = √432 (c) Comparison of surds: • Case 1: When indices are same: It is clear that if x > y > 0 and n > 1 is a positive integer then 𝑛√𝑥 > 𝑛√𝑦 3 3 5 5 Ex: √16 > √12, √36 > √25 i.e. if indices are same then greater is the value of radicand, greater will be the given surd. • Case 2: When radicands are same: It is clear that if x > 0 and n > m > 1 and are positive integers then 𝑛√𝑥 < 𝑚√𝑥. 3 4 Ex: √1000 = 10 > √1000 = 5.623 i.e. if radicands are same then greater is the value of index, lesser will be the given surd. • Case 3: When neither index nor radicand are same: In this case we reduce our surds to either case 1 or case 2. The best way is to convert our given surds to case 1 by taking LCM of indices and then using law 7 to make indices same. 3 5 Ex: which is greater √6 or √8 ? Sol: L.C.M. of 3 and 5 is 15 3 3.5 15 Now making indices same: √6 = √65 = √7776 5 3.5 3 15 √8 = √8 = √512 15 ∴ √7776 > √512 15 √6 > 5√8 3 SQUARE ROOTS OF BINOMIAL QUADRATIC SURDS 2 Since (√𝑥 + √𝑦) = (𝑥 + 𝑦) + 2√𝑥𝑦 𝑎𝑛𝑑 2 (√𝑥 − √𝑦) = 𝑥 + 𝑦 − 2√𝑥𝑦 (a) Square root of x + y + 2√𝑥𝑦 = ±(√𝑥 + √𝑦) (b) Square root of (x + y) – 2√𝑥𝑦 = ±(√𝑥 − √𝑦) (c) Square root of a2 + b + 2a√𝑏 = ±(𝑎 + √𝑏) (d) Square root of a2 + b – 2a√𝑏 = ±(𝑎 − √𝑏) AMU Booster Page No. 38 Pocket Book of Maths By: Nab$ Sir Number System Square Roots of a +b and a − b Let √𝑎 + √𝑏 = √𝑥 + √𝑦, where 𝑥, 𝑦 > 0are rational numbers. Then squaring both sides, we have, 𝑎 + √𝑏 = 𝑥 + 𝑦 + 2√𝑥√𝑦 𝑎 = 𝑥 + 𝑦, √𝑏 = 2√𝑥𝑦 𝑏 = 4𝑥𝑦 So, (𝑥 − 𝑦)2 = (𝑥 + 𝑦)2 − 4𝑥𝑦 = 𝑎2 − 𝑏 After solving we can find x and y. Similarly square root of 𝑎 − √𝑏 can be found by taking √𝑎 − √𝑏 = √𝑥 − √𝑦, 𝑥 > 𝑦 Example:√𝟑 + √𝟓 is equal to Solution: Let √3 + √5 = √𝑥 + √𝑦 3 + √5 = 𝑥 + 𝑦 + 2√𝑥𝑦. Obviously x + y = 3 and 4 xy = 5 . So (𝑥 − 𝑦)2 = 9 − 5 = 4or (𝑥 − 𝑦) = 2 5 1 5 1 √5+1 After solving 𝑥 = 2 , 𝑦 = 2. Hence √3 + √5 = √2 + √2 = √2 Square Roots a + b + c + d Where b , c , d are Surds . Let √𝑎 + √𝑏 + √𝑐 + √𝑑 = √𝑥 + √𝑦 + √𝑧, (𝑥, 𝑦, 𝑧 > 0) and take √𝑎 + √𝑏 − √𝑐 − √𝑑 = √𝑥 + √𝑦 − √𝑧. Then by squaring and equating, we get equations in x, y, z. On solving these equations, we can find the required square roots. Example: [10 − (24 ) − (40 ) + (60 )] = ? Solution: Let 10 − 24 − 40 + 60 = ( a − b + c )2 10 − √24 − √40 + √60 = 𝑎 + 𝑏 + 𝑐 − 2√𝑎𝑏 − 2√𝑏𝑐 + 2√𝑐𝑎, 𝑎, 𝑏, 𝑐 > 0. Then a + b + c = 10, ab = 6 , bc = 10, ca = 15 AMU Booster Page No. 39 Pocket Book of Maths By: Nab$ Sir Number System a2b 2c 2 = 900 abc = 30 ( 30 ) . So a = 3, b = 2, c = 5 Therefore, (10 − 24 − 40 + 60 ) = ( 3 + 5 − 2 ) ✓ Note: If 𝑎2 − 𝑏 is not a perfect square, the square root of 𝑎 + √𝑏 is complicated i.e., we can't find the value of √𝑎 + √𝑏 in the form of a compound surd. Rationalization ❖ The process of converting a surd to a rational number by using an appropriate multiplier is known as rationalization. ❖ If the product of two surds is a rational number then each of them is called the rationalizing factor (R.F.) of the other. SOME COMMON RATIONALIZING FACTOR: R.F. of √𝑎 is √𝑎 and vice versa. (∴ √𝑎 × √𝑎 = 𝑎). 3 3 3 R.F. of 3√𝑎 𝑖𝑠 √𝑎2 (∴ 3√𝑎 × √𝑎2 = √𝑎3 = 𝑎) R.F of (𝑎 + √𝑏) is (𝑎 − √𝑏) [(𝑎 + √𝑏)(𝑎 − √𝑏) = (𝑎2 − 𝑏)] R.F of (𝑎 + 𝑏√𝑥) is (𝑎 − 𝑏√𝑥) and vice versa. [ ∴ (𝑎 + 𝑏√𝑥)(𝑎 − 𝑏√𝑥) = (𝑎2 − 𝑏2 𝑥)] R.f of (√𝑥 + √𝑦) is (√𝑥 − √𝑦) and vice versa. [∴ (√𝑥 + √𝑦)(√𝑥 − √𝑦) = (𝑥 − 𝑦)] R.F. of √𝑎 + √𝑏 𝑖𝑠 √𝑎 − √𝑏 & vice versa [∴ (√𝑎 + √𝑏)(√𝑎 − √𝑏) = 𝑎 − 𝑏]. R.F. of a +√𝑏 is 𝑎 − √𝑏 & vice versa [∴ (𝑎 + √𝑏)(𝑎 − √𝑏) = 𝑎2 − 𝑏] 3 3 3 3 R.F. of √𝑎 + √𝑏 is (√𝑎2 − √𝑎𝑏 + √𝑏2 ) 3 3 3 3 3 3 3 3 [∴ ( 3√𝑎 + √𝑏 )(√𝑎2 − √𝑎𝑏 + √𝑏 2 ) = ( 3√𝑎 ) + (√𝑏 ) = 𝑎 + 𝑏] 3 3 3 R.F. of √𝑎 − √𝑏 is (√𝑎2 + √𝑎𝑏 + √𝑏2 ) 3 3 3 3 3 3 3 3 3 [∴ ( 3√𝑎 − √𝑏 )(√𝑎2 + √𝑎𝑏 + √𝑏 2 ) = ( 3√𝑎 ) + (√𝑏 ) = 𝑎 + 𝑏] AMU Booster Page No. 40 Pocket Book of Maths By: Nab$ Sir Number System Nab$ Tricks Trick 1: Converting pure repeating decimal in p/q form 𝑎𝑏𝑐𝑚𝑛𝑜𝑝 −𝑎𝑏𝑐 If 𝑥 = 𝑎𝑏𝑐. 𝑚𝑛𝑜𝑝 then 𝑥 = 9999 Here total number of 9 in denominator of p/q form is equal to number of digits in repetition in decimal form (as mnop = four ‘9’) 12357 −123 12234 Example: 𝑥 = 123. 57 then 𝑥 = = 99 99 Trick 2: Converting mixed repeating decimal in p/q form 𝑎𝑏𝑐𝑘𝑙𝑚𝑛𝑜 −𝑎𝑏𝑐𝑘𝑙 If 𝑥 = 𝑎𝑏𝑐. 𝑘𝑙𝑚𝑛𝑜 then 𝑥 = 99900 Here total number of 9 in denominator is equal to total no of digits in repetition after decimal (as here mno = three ‘9’) and number of 0 in denominator is equal to total number of digits which are not in repetition (as here kl = two ‘0’). 536573 −536 536037 Example: If 𝑥 = 5.36573 then 𝑥 = = 99900 99900 𝒂+ √𝒂𝟐−𝒃 𝒂−√𝒂𝟐 −𝒃 Trick 3: √𝒂 + √𝒃 = √( ) + √( ) 𝟐 𝟐 √𝒂𝟐 −𝒃 𝒂−√𝒂𝟐−𝒃 √𝒂 − √𝒃 = √(𝒂+ ) − √( ) 𝟐 𝟐 Trick 4: If a is a rational number, √𝑏, √𝑐, √𝑑, are surds then 𝑏𝑑 𝑏𝑐 𝑐𝑑 (i) √𝑎 + √𝑏 + √𝑐 + √𝑑 = √ +√ +√ 4𝑐 4𝑑 4𝑏 𝑏𝑑 𝑏𝑐 𝑐𝑑 (ii) √𝑎 − √𝑏 − √𝑐 + √𝑑 = √ 4𝑐 + √4𝑏 + √4𝑑, 𝑏𝑐 𝑏𝑑 𝑐𝑑 (iii) √𝑎 − √𝑏 − √𝑐 + √𝑑 = √ −√ −√ 4𝑑 4𝑐 4𝑏 AMU Booster Page No. 41 Pocket Book of Maths By: Nab$ Sir Number System 𝒙 𝒙 𝒙 𝒙 Trick 5: + + +⋯+ = 𝒙(√𝒏 − √𝒂) √𝒂+√𝒃 √𝒃+√𝒄 √𝒄+√𝒅 √𝒎+√𝒏 Given: a, b, c, d, …., m, n are all consecutive positive integers 𝟏𝟎 𝟏𝟎 𝟏𝟎 𝟏𝟎 For Example: 𝟏+√𝟐 + + +⋯+ =? √𝟐+√𝟑 √𝟑+√𝟒 √𝟖+√𝟗 Sol: The answer is always: 10× (√9 − √1) = 10 × 2 =20 Trick 6: To get number of factors (or divisors) of a number N, express N as N = ap. bq. cr. ds……… (where a, b, c, d are prime numbers and p, q, r, s are indices) Then the number of total divisors or factors of N are (p + 1) (q + 1) (r + 1) (s + 1) …… Example: Total no of factors of 540 = ? Sol: We know 540 = 22 × 33 × 51 Total number of factors of 540 = (2 + 1) (3 + 1) (1 + 1) = 24 Trick 7: Nab$ way: Elimination Trick In this we eliminate the options which are not satisfying and thus reaches to final answer Example: The LCM of 6, 72 and 120 is 360, their HCF is: (a) 120 (b) 6 (c) 72 (d) None of these Sol: As we know HCF of positive integers is always less than or may be equal to the least/smallest of these integers., hence (a) and (c) will never be the answer. Now check for option (b). As it is dividing 6, 72 and 120 exactly hence (b) is the correct answer. Trick 8: Nab$ way: Time saving Trick In questions in which two or more values of variables are being asked, then on getting one variable helps in finding out the correct option. Example: If the GCD of x2 + ax – 3 and 2x2 + x + b is (x + 1), then the value of a and b are respectively? (a) {–1, 0} (b) {0, –1} (c) {2, 1} (d) {–2, 1} Sol: As (x + 1) is GCD or HCF of x2 + ax – 3, hence x = –1 will make this equal to 0. Therefore (–1)2 + a(–1) –3 = 0 implies a = – 2. Now we can see that a = – 2 is only given in option (d), hence (d) is our final answer. No need to calculate b. This saves our time. AMU Booster Page No. 42
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