NUMBERS( A group of digits, denoting a number, is called a numeral. Every digit in a numeral has two values:

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1 NUMBERS( A number is a mathematical object used to count and measure. A notational symbol that represents a number is called a numeral but in common use, the word number can mean the abstract object, the symbol, or the word for the number. Number is first discovered in India and ten symbols are used, namely o, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to represent any number. These symbols are called digits. A group of digits, denoting a number, is called a numeral. Every digit in a numeral has two values: a. Its face value is equal to it and never changes. b. Its place value depend on its position in numeral. 1. Natural Number : N = {1, 2, 3, 4, 5, 6, 7, 8, 9, }$(0$excluding) 2. Whole Numbers : W= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, } (0 including). All natural numbers are whole numbers. 0 is only a whole number and not a natural number. Aaryabhatt, who was an astronomer and lived in India around 500 AD, invented the symbol 0 for nothing. 3. Even Number : E= {2, 4, 6, 8, 10..} 4. Odd Numbers : O= {1, 3, 5, 7, 9, 11, } 5. Positive integers : I + = {+1, +2, +3,.} 6. Negative Integers : I - = {-1, -2, -3, -4,..$} 7. Integers : I= { 24,23,22,21,0,1,2,3,4..} 8. Zero : a. Zero is an integer and it is neither positive nor negative. b. Any number multiplied by 0 is equal to 0. c. Face value and place value of 0 is always 0 9. Imaginary Numbers: The square root of negative numbers are called imaginary numbers. Example :,, etc. 10. Real Numbers : When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Real numbers are Rationals + Irrationals All points on the number line Or all possible distances on the number line Real numbers include

Whole Numbers (like 1,2,3,4,5,6 etc) Rational Numbers (like 3/4, 0.1, 0.635..., 1.5, etc ) Irrational Numbers (like π, 3, etc ) What is not a Real number?

actual geometric line. A point is chosen on the line to be the "origin", points to the right will be positive, and points to the left will be negative.

2 Whole Numbers (like 1,2,3,4,5,6 etc) Rational Numbers (like 3/4, 0.1, , 1.5, etc ) Irrational Numbers (like π, 3, etc ) What is not a Real number? -4 (the square root of minus 4) is not a Real Number, it is an Imaginary Number Infinity is not a Real Number The Real Number Line The Real Number Line is like an actual geometric line. A point is chosen on the line to be the "origin", points to the right will be positive, and points to the left will be negative. A distance is chosen to be "1", and the whole numbers can then be marked off: {1,2,3,...), and also in the negative direction: {-1,-2,-3,...} Any point on the line is a Real Number:

3 The numbers could be rational (like 20/9) or irrational (like$!) Properties of Real Number : a. Closure Property : The sum and multiplication of the real numbers are always a real number. Example : x + y = z and a x b = c, here x, y, z, a, b and c are integers. b. Opposite : Two real numbers that are the same distance from the origin of the real number line are opposites of each other. c. Reciprocals :! Two numbers whose product is 1 are reciprocals of each other. d. Commutative Property : When adding or multiplying two numbers, the order of the numbers does not matter. Example : a + b = b + a or a x b = b x a, where a and b are rational numbers. e. Associative Property : When three numbers are added, it makes no difference which two numbers are added first. And at the same time when three numbers are multiplied, it makes no difference which two numbers are multiplied first. Example : (a+b)+c=a+(b+c) or (axb) x c= ax (bxc), where a, b and c are rational numbers. f. Distributive Property : Multiplication distributes over addition ie. ax(b+c)=(axb) + (axc), where a, b and c are integers. g. Additive Identity Property :!The additive identity property states that if 0 is added to a number, the result is that number. Example : 5+0=0+5=5 h. Multiplicative Identity Property: The multiplicative identity property states that if a number is multiplied by 1, the result is that number. Example : 5 x 0= 0 x 5=0 i. Additive Inverse : The additive inverse property states that opposites add to zero ie. if the sum of two rational numbers is zero the numbers are known as additive inverse of the other. Example : + ( )=0. Additive inverse of every rational number x is x. j. Multiplication Inverse : The multiplicative inverse property states that reciprocals multiply to 1 ie. if the multiplication of two numbers is 1, then each number is known as multiplication inverse of the other. Example : 3 x = 1 or x = 1. Multiplication inverse of every rational number x will be 11. Composite Numbers : A composite number is a positive integer which has a positive divisor other than one or itself. In other way a Composite number has more than two divisors. i.e. 10 is a composite number as it has 1,2,5 and 10 as divisors. Other examples : 4,6,9,16,21, etc. When a number can be divided up evenly it is a Composite Number

4 12. Prime Numbers : When a number can not be divided up evenly it is a Prime Number. A natural number other than 1, which has no other factor except 1 and itself, is called a Prime Number. Examples : 2,3,5,7,11,13,17, are prime numbers. When only two factors of a number are 1 and the number, then the number is called Prime Number and it must be greater than 1. A prime number is greater than 1 and a Positive integer. Any natural number greater than 1 is either composite or prime number. List of prime number : is neither Prime nor Composite number. The smallest prime number is 2. 2 is the only even Prime number, all other Prime Numbers are odd. 13. Co-Prime Numbers : Two numbers are said to be Coprimes when they have only 1 as a common factor. Two natural numbers are called co-prime numbers if they have no common factor other than 1 or two natural numbers are co-prime if their H.C.F is 1. Example 1 : Find out whether number 7 and 12 are Coprime?

5 Answer : Factors of 7 are 1 and 7 Factors of 12 are 1, 2,3,4 and 12 On Comparing the factors of number 7 and 12, we find that both have only 1 as a common factor. Hence, the numbers 7 and 12 are Coprimes. 14. Twin Prime Number : The pairs of Prime Numbers whose difference is 2 are called Twin Prime Numbers. Example : (5,7), (11,13),(17,19) etc. 15. Prime Triplet Number : The three natural numbers whose H.C.F. is 1, are called Prime triplet Numbers. Example; (8,9,25). In mathematics, a prime triplet is a set of three prime numbers of the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since every third odd number greater than 3 is divisible by 3, and hence not prime. The first prime triplets are (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887) A prime triplet contains a pair of twin primes (p and p + 2, or p + 4 and p + 6 ), a pair of cousin primes (p and p + 4, or p + 2 and p + 6), and a pair of sexy primes (p and p + 6). 16. Perfect number : A given number for which sum of all its factors is equal to it's twice, the number is known as A Perfect Number. Example : Examples 1 = Is 6 a Perfect number? Answer = Find the factors of 6 Factors of 6 = 1, 2, 3, 6. Add all the above factors. Sum = = 12 Twice of given number 6 = 12 Since the sum of all the factors of 6 is equal to its twice So, 6 is a perfect number. Examples 2 = Is 9 a Perfect number? Answer = Find the factors of 9.

6 Factors of 9 = 1, 3, 9. Add all the above factors. Sum = = 13 Twice of given number 9 = 18 Since the sum of the factors of 9 is not equal to its twice Hence, 9 is not a perfect number. Examples 3 = Is 28 a Perfect number? Answer = Find the factors of 28 Factors of 28 = 1, 2, 4, 7, 14, 28. Add all the above factors. Sum = = 56 Twice of given number 28 = 56 Since, the sum of all the factors of 28 is equal to its twice So, 28 is a perfect number. 17. Literal Numbers : An English Alphabet that is used to represent a variable is called a Literal Number. 18. Fraction : A fraction is a quantity that can not be expressed in whole number. A fraction is an ordered number of whole numbers, the first one is written on top of other, eg. ¼, 3/5,2/7 etc. The top number is called the Numerator, it is the number of parts one has. The bottom number is called the Denominator, it is the number of parts the whole is divided into. Numbers which can be expressed as where (a) x y, (b) x 0, y 0 (c) y 1 (d) there is no common factor. Example:,, etc. Equivalent Fraction : A fraction can have many different appearances these are called Equivalent Fraction. Example: and and. and We cannot tell whether two fractions are the same until we reduce them to their lowest terms. Improper Fraction : An improper fraction is a fraction with the numerator larger than or equal to the denominator. Example :,, etc. Any natural number can be converted into Improper Fraction. Mixed Number : A mixed number is natural number and a fraction together. An improper fraction can be converted to a mixed number and vice versa.

7 19. Inverse Numbers : Numbers who have negative power are called Inverse Numbers. Example : 3-2, 4-5 etc. 20. Even Number : Even numbers can be divided evenly into groups of two( or any natural number which is divisible by 2 is called even number. Example :2, 4, 8, 10,22 etc. 21. Odd Number : Odd numbers can not be divided evenly into groups of two or any natural number which is not divisible by 2 is called an odd number. Example : 1, 3, 5, 7, 111 etc. 22. Rational numbers : Numbers which can be represented as where x and y are integers and y 0. Eg. 1, 2, -1, -2,, -!$etc. Special characteristics of rational numbers: a. All natural numbers are rational. b. Rational number can be expressed either as a terminating decimal or a repeating decimal. c. Every terminating decimal is a rational number. d. Every repeating decimal is a rational number. 23. Irrational Number : Numbers which can not be represented as ratio of two natural numbers, are called irrational number.!example :, 5 etc. An irrational number is a non repeating and non terminating decimal.! and! are irrational numbers. History of Irrational Numbers Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and so it was irrational. However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!

8 24. Decimal form of a rational number : When we express a rational number in decimal, then either the decimal will be exact, as = 0.5 (called terminating decimal) or it will not, as = (called recurring decimal).

CHAPTER 1 REAL NUMBERS KEY POINTS

CHAPTER 1 REAL NUMBERS KEY POINTS CHAPTER 1 REAL NUMBERS 1. Euclid s division lemma : KEY POINTS For given positive integers a and b there exist unique whole numbers q and r satisfying the relation a = bq + r, 0 r < b. 2. Euclid s division