JMATHSLEARNING

1. Irrational numbers satisfy the commutative, associative and distributive laws for addition and multiplication. 2. (i) Sum of two irrationals need not be an irrational. Example: Each one of (2 +  √3 ) and (3 -  √3 ) is irrational.                   But,  (2 +  √3 ) + (3 -  √3 ) = 5, which is rational.        (ii). Difference of two irrationals need not be an irrational. Example: Each one of ( 4 +  √5 ) and ( 2 +  √5  ) is irrational.                    But,   ( 4 + √5 ) - ( 2 +  √5  ) = 2, which is rational.            (iii). Product of two irrationals need not be an irrational. Example:   √5  is irrational.                     But,  √5  x  √5  = 5, which is rational.            (iv). Quotient of two irrationals need not be an irrational. Example: Each one of 3 √5  and  √5  is irrational.                      But, 3 √5 / √5  = 3, which is rational.   3 (i). Sum of a rational and an irrational is irrational. Example: 2 is rational and 

Real Numbers A number whose square is non-negative, is called a real number. Example: All rational and irrational numbers are real  numbers. Addition Properties of real numbers:   1.       Closure Property : The sum of two real numbers is always a real number. Example : Two real numbers are 1 and  √ 2 , then (1 +  √ 2)  is a real number.    2.        Commutative Property : a and b are two real numbers, then                                           a + b = b + a Example : two real numbers are (2 +  √3)  and (2 –  √3) , then                   (2 +  √3)  + (2 –  √3)  = 4                   (2 –  √3)  + ( 2 +  √3)  = 4 Then                    left hand side = Right hand side Hence Commutative Property is verified.   3.        Associative Property : a, b  and c are three real numbers, then                                   a + ( b + c ) = ( a + b ) + c Example : Three real numbers are  2 , (5 +  √7)  and (5 -  √7)                   2 + [ (5 + 

Rational Numbers The numbers of the form a/b, where a and b are integers and b ≠ 0 are known as rational numbers. Example: 1/2, -4/5, 0, -1, 2 etc. or A number which can be expressed as a terminating decimal or repeating decimal , is called a rational numbers. Example:   5/3 = 1.666666....... =  1. 6               16/45 = 0.355555 ...... = 0.3 5  etc. Important result all natural numbers, whole numbers and integers are rational numbers. We can understand above terms with a picture diagram. Irrational Numbers A number which can neither be expressed as a terminating decimal nor a repeating decimal , is called an irrational number. Thus, non-terminating, non-repeating decimals are irrational numbers. Examples : Type 1. 0.2020020002....... , 0.4040040004...... etc. are a non-terminating and non-repeating decimal, therefore these are irrational number. Type 2 : If n is positive which is not a perfect square, then √ n   is irrational, e.g.  √2,  √3,  √5,

Decimal Number Decimal numbers are numbers that are written according to their place value. The decimal point is used to separate whole part from number value less than one.A symbol is used " . ", this is called decimal point. Example: 65.894 where 65 is a whole part and 894 is a fractional part. Type of Decimals 1. Terminating Decimals  2. Non-terminating Decimals 1. Terminating Decimals Every fraction a/b can be expressed as a decimal. If the decimal expression of a/b terminates i.e. comes to end, then the decimal so obtained is called a terminating decimal. Example: 1/2 = 0.5                   5/4 = 1.25                 13/5= 2.6 Thus, each of the numbers 1/2, 5/4 and 13/5 can be expressed in the form of a terminating decimal. 2. Non-terminating Decimals While expressing a fraction in the form of decimal, when we perform division we get some remainder. If the division process does not end i.e. remainder is not equal to zero, then such dec

Properties of  Integers Addition Properties of Integers: 1. Closure Property:    The sum of two Integers is always an Integers. Example: Let two Integers are -4 and -6 then                            (-4) + (-6) = (-10) is an Integer. 2. Commutative Property:  Let a and b are two Integers, then                                 a + b = b + a   Example: Let two integers are -5 and 6 then                    (-5) + 6 = 1                   6 + (-5) = 1   then        (-5) + 6 = 6 + (-5)  Hence commutative property is verified 3. Associative Property:   Let a, b and c are three integers, then                                             a + ( b + c) = (a + b ) + c Example: Let three integers are 3, 5 and (-6) then                   3 + { 5 + (-6) } = 3 + {5 - 6 } = 3 + (-1) = 2                   {3 + 5 } + (-6)  =  8 - 6  = 2  then          3 + { 5 + (-6) } =   { 3  + 5 } + (-6)  Hence Associative property is verified 4. Existence of Additive Identity: 

Integers An  integer  is a number that can be written without a fractional component that. It can be positive, negative or zero.It is denoted by " ℤ ". ℤ  = { .......  –3,  –2,  –1, 0, 1, 2, 3, 4, 5, ........ } Type of Integers  1. Positive     Integers 2. Negative       Integers 3. Non-negative Integers 1. Positive Integers Whole numbers greater than zero are called Positive Integers . It is also called Natural Numbers   and denoted by " ℤ + ".  ℤ +  = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ....... } 2. Negative Integers A  negative integer  is an  integer  to the left of zero on the number line. It is less than zero.It is denoted by " ℤ − ". ℤ −   = { ........... , −7,  −6,  −5,  −4,  −3,  −2,  −1} 3. Non-Negative Integers An  integer  that is either 0 or positive. It is also called Whole Numbers and denoted by " ℤ ≠  ". Some Authors use symbol "  ℤ * ".   ℤ ≠  = { 0,  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, .......

Properties of Whole Numbers Addition Properties of Whole Numbers: 1. Closure Property:   The sum of two whole numbers is always a whole numbers. Example: Let two whole numbers 5 and 7 then                            5 + 7 = 12 is a whole number. 2. Commutative Property:  Let a and b are two whole numbers,                                                a + b = b + a  Example: Let two whole numbers are 5 and 6 then                   5 + 6 = 11                   6 + 5 = 11   then        5 + 6 = 6 + 5  Hence commutative property is verified 3. Associative Property:   Let a, b and c are three whole numbers,                                             a + ( b + c) = (a + b ) + c Example: Let three whole numbers are 3, 5 and 6 then                   3 + ( 5 + 6 ) = 3 + 11 = 14                   (3 + 5 ) + 6  =  8 + 6  = 14  then          3 + ( 5 + 6 ) =   (3  + 5 )  + 6  Hence Associative property is verified 4. Existence of Additive Identity:   For any whole n

Whole Numbers All natural numbers together with zero form the collection of all whole numbers. It is denoted by "W" W = { 0,1,2,3,4,5,6,7,8,9,10, ............... } Type of Whole Numbers 1. Even Numbers 2. Odd  Numbers 1. Even Numbers A number which is divisible by 2, it is called even number. It is denoted by "E".  E = { 0,2,4,6,8,10,12,14,16,18,20,  ........ } 2. Odd Numbers A number which is not divisible by 2, it is called odd number. It is denoted by "O". O = { 1,3,5,7,9,11,13,,15,17,19,21, ....... }  We can understand whole numbers from a picture

Properties of Natural Numbers Addition Properties of Natural Numbers: 1. Closure Property:   The sum of two natural numbers is always a natural numbers. Example: Let two natural numbers 2 and 5 then                            2 + 5 = 7 is a natural number. 2. Commutative Property: Let a and b are two natural numbers,                                                a + b = b + a  Example: Let two natural numbers are 5 and 7 then                   5 + 7 = 12                   7 + 5 = 12   then        5 + 7 = 7 + 5  Hence commutative property is verified 3. Associative Property:   Let a, b and c are three natural numbers,                                             a + ( b + c) = (a + b ) + c Example: Let three natural numbers are 3, 5 and 6 then                   3 + ( 5 + 6 ) = 3 + 11 = 14                   (3 + 5 ) + 6  =  8 + 6  = 14  then          3 + ( 5 + 6 ) =   (3  + 5 )  + 6  Hence Associative property is verified 4. Existence of Additive Identity:

Natural Numbers Counting numbers are known as natural numbers.  It is denoted by "N"  N = { 1,2,3,4,5,6,7,8,9,10,11,12,13, ....... } Type of Natural numbers  1. Prime numbers  2. Composite numbers   1. Prime numbers A number which have only two prime factor, it is called a prime numbers. It is denoted by "P".  P = { 2,3,5,7,11,13,17,19, ....... } e.g. "2" have two factors 1 and itself .  Similarly some other examples written below :                     3 -> 1 and 3                     5 -> 1 and 5                     7 -> 1 and 7                                                      11 -> 1 and 11 etc.  2. Composite numbers A numbers which have more than two factors, it is called a composite numbers.      { 4,6,8,9,10,12,14,15,16,18, ...... } e.g. "4" have more than two factors 1, 2 and 4 Similarly some other examples written below :                     6 -> 1, 2, 3 and 6