MATHS MADE EASY

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Mathematics is a real demon for some students who are preparing for competitive exams. I am starting this blog for making students comfortable with mathematics and feel how interesting and enjoying the maths is. I am trying to develop an interesting way of doing this. Firstly, i will start with the concepts for IIT aspirants and Intermediate students. Later i will focus on Higher Mathematics and Big Competitive exams like IAS, IFS, BSc and MSc. I will try to solve your queries relating to all maths problems and at least guide you towards the solution. Hope, we enjoy this journey.




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Properties of Modulus & Argument: Complex Number

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Properties of Modulus: Modulus of z is the length of vector representing z form origin to the point z. 1.  If z = x + iy then   2.   &  Since,  ( equality follows when y = 0) and   (equality follows when x = 0) IMP 3.  Since, and,  4.  ,  5.  6.  7.   holds when  i.e. both the vectors are in same direction.  holds when   i.e. both the vectors are in opposite direction In general, 8.  IMP 9.  Explanation: Since,  Therefore,  or,                                                  [Since   (? click here ) ] and  10.     ( Property of Parallelogram ) Properties of Argument: Argument is the angle between the vector representing z from the positive direction of x-axis The Principal Argument belongs to (-pi, pi] 1.  ,   2.   , where k = 0, 1 or -1 3.   , where k = 0, 1 or -1 4.   , z cannot be negative Real Number because for negative Real number Arg(z) is equal to pi and so   will become -pi but i
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Fractional Part function

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10. Fractional Part f(x) = {x}  Every number 'x' can be written as the sum of its integer and fractional parts. x = [x] + {x} Therefore fractional part, {x} = x - [x] For example, 4.7 = 4 + 0.7 or, 0.7 = 4.7 - 4 i.e. fractional part = Number - Integral Part x = 4.7 , [x] = [4.7] = 4 and {x} = {4.7} = 0.7 Consider, {-4.7} {-4.7} = -4.7 - [-4.7] = -4.7 - (-5)  or, {-4.7} = 0.3 if the number is integer, its fractional part is obviously zero. {6} = 6 - [6] = 6 - 6 = 0 Note- fractional part is always positive and it never becomes 1. in other words, fractional part of a number is the difference between the number and its integral part. Remember,  i.e. fractional part is always less than 1 and greater than or equal to zero. When x will be an integer fractional part will become zero i.e. integer has no fractional part. How to draw the graph, note this, , if  because the fractional part is equal to the number if number is between
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Geometrical Meaning of Argument and Modulus - I: Complex Number

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LIKE TO JOIN ON FACEBOOK ( click here ) Students are requested to understand this carefully   1. Arg(z) = pi/3 z = 0 is excluded since z = 0 is the only complex number whose argument is not defined 2. Arg(z) > pi/3 Since principal argument belongs to (-pi, pi],  therefore Argument cannot be greater than pi. therefore Arg(z) > pi/3 =>)  z = 0 is excluded 3. |Arg(z)| < pi/4 => -pi/4 < Arg(z) < pi/4 z = 0 is excluded 4. => Arg(z) = pi since, principal argument belongs to  , therefore Argument cannot be greater than pi. z = 0 is excluded 5.  -pi/3 is included z = 0 is excluded 6. Complex number   , equal only when z is purely real number or '0' IMP 7. Arg(|z|) = 0 Since |z| is a Positive Real number , therefore it lies on right side of x - axis i.e. on positive Real axis and x-axis has '0' argument because it makes 0 radian angle from x-axis For z = 0 , argument does not exist T
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