JMATHSLEARNING

History Thales (640 BC – 546 BC) The word ‘Geometry’ is derived from the Greek words ‘Geo’ means ‘Earth’ and ‘Metrein’ means to ‘ To Measure’. A Greek mathematician, Thales is credited with giving the first known proof. This proof was of the statement that a circle is bisected (i.e., cut into two equal parts) by its diameter. One of Thales’ most famous pupils was Pythagoras (572 BC), whom you have heard about. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued till 300 BC. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise, Euclid  (325 BC – 265 BC) called ‘Elements’.          He divided the ‘Elements’ into thirteen chapters, each called a book. These books influenced the whole world’s understanding of geometry for generations to come. Euclid listed 23 definition in book 1 of the ‘Elements’. A

Formulae for binomial     1.        (a + b) 2 = a 2  + 2ab + b 2 Proof :    Taking L.H.S              (a + b) 2 = (a + b) . (a + b)                           = a (a + b ) + b (a + b)                           = a 2 + ab + ab + b 2            (a + b) 2   = a 2  + 2ab + b 2       …….. (i)     2.        (a – b) 2 = a 2 – 2ab + b 2 Proof :  Putting b = – b in equ n  (i)         (a – b) 2   = a 2  + 2a(–b) + (– b) 2         (a – b) 2 = a 2   –  2ab + b 2     3.        a 2 – b 2 = (a + b) . (a – b) Proof :   Taking R.H.S        (a + b) . (a – b) = a (a – b) + b (a – b)                                  = a 2 –  ab + ab – b 2                                  = a 2 – b 2    4.        (a + b) 3 = a 3 + b 3 + 3ab (a + b) = a 3 + b 3 + 3a 2 b + 3ab 2 Proof : Taking L.H.S    (a + b) 3 = (a + b) . (a + b) 2                 = (a + b) . (a 2  + 2ab + b 2 )                 = a (a 2  + 2ab + b 2 ) + b (a 2  + 2ab + b 2 )                 = a

Remainder Theorem Let p(x) be a polynomial of degree   ≥  1  and let a be any real number. When  p(x) is divided by ( x – a), then the remainder is p(a). Proof : Suppose that when p(x) is divided by (x – a), the quotient is q(x) and the remainder is r.           ∴          Dividend = Divisor x Quotient + Remainder                                 p(x) = (x – a) . q(x) + r        ……….. (i)                                 Putting x = a in (i),                                  We get r = p(a) Thus, when p(x) is divided by (x – a), then the remainder is p(a). Example: Find the remainder when the polynomial                  p(x) = x 3 + 3x 2 – 2x +1 is divided by (x – 4). Solution:        p(x) = x 3 + 3x 2 – 2x +1                            x – 4 = 0  ⇒ x = 4               By the remainder theorem, we that when p(x) is divided by (x – 4), the remainder is p(4). Now, remainder = p(4)                               = (4 3 + 3 x 4 2 – 2 x 4 + 1)       

Constant A symbol having a fixed numerical value is called a constant Examples:  3, -6, 5/6, 0.23, π, etc. are all constants. Variable A symbol which may be assigned different numerical values us known as a variable . Example: We know that the area of a circle is given by the formula A = πr 2 , where r is the radius of the circle. Here, π is constant, while A and r are variables. Algebraic Expression A combination of constants and variables, connected by some or all of the operations +, − , × , and ÷ is known an algebraic expression . Examples: x +2, x 2 +y 2 , x – 1/x, etc. Terms of an Algebraic Expression The several parts of an algebraic expression separated by + or – operations are called the terms of the expression . Examples:   (i). 3 + 4x – 7xy 2 + 9xy is an algebraic expression containing four terms , namely, 3, 4x, 7xy 2 and 9xy     (ii). X 3 + 3x 2 y – 5xy 2 + y 3 – 6 is an algebraic expression containing five terms , namely, X 3 , 3x

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,