COMPLEX NUMBERS , DEMOIVRE'S THEOREM & EULER'S FORM

       COMPLEX    NUMBERS

Any  number of the form  a+bi is known as  complex  number where a & b are real numbers & i=  √-1 , a is called real part & b is called imaginary part.

           A complex number is generally denoted by z. If a is 0 then 

      z is purely imaginary & if b is 0 then z is purely real.

NOTE:

  1) i  = √-1  

  2) i² = -1

  3) i³ = i² i =-1.i =-i

  4) i⁴ =i³.i =-i.i =-i² =-(-1) =1

  5) i⁵ = i⁴ .i =1.i =i

  6) i⁶ =i⁵.i = i.i = i² = -1

 7) i⁷ =i⁶ .i =-1. i =-i

 8 ) i⁸ =i⁷.i =-i.i =- i² = -(-1)=1

 9) i⁹ =  i⁸.i = 1.i =i

10)i¹º =i⁹.i =i.i =i² = -1

  &  so on .

Read blog on Line & it's Equation

                ALJEBRA OF COMPLEX NUMBERS

  1)  ADDITION : When two complex numbers 

   are added their sum is also a complex number.

   eg.

  if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁+ z₂ is given by

       z₁+ z₂ = (a₁+b₁i ) + (a₂+b₂i ) 

                  = (a₁+a₂) + (b₁+b₂)i

2)   SUBTRACTION : When two complex 

   numbers are subtracted their  difference is 

    also a complex number .

   eg.

    if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁-z₂ is given by

       z₁- z₂ = (a₁+b₁i ) - (a₂+b₂i ) 

                 = (a₁-a₂) + (b₁-b₂)i

2) MULTIPLICATION : When two complex numbers 

    are multiplied their multilication is also a complex number .

   eg.

 if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁. z₂ is given by

       z₁. z₂ = (a₁+b₁i ) . (a₂+b₂i ) 

                 = (a₁a₂-b₁b₂) + ( a₁b₂+b₂a₁)i

                 EQUALITY OF COMPLEX NUMBERS 

Two complex numbers are said to be equal if

1)Their real parts are equal

2)Their imaginary parts are also equal.

 eg. If z₁=a+bi  & z₂=2+3i 

             z₁=z₂ if a+bi  =2+3i i.e a=2 & b=3

         CONJUGATE OF A COMPLEX NUMBER

Two complex numbers which are differ only in the sign of their imaginary parts are said to be conjugates of each other.

If z is complex number then it's conjugate is denoted by z .ie. z bar.

if 2+3i is given complex number the it's conjugate is given by 2-3i.

          

                 z                              conjugate  z

          2+3i                                             2-3i.

               2-3i                               2+3i

             -2+3i                              -2-3i

             -2-3i                                -2+3i

              -3i                                     +3i

             +3i                                      -3i

NOTE : 1)SUM &product of two complex conjugates are real.

               2)difference of two complex conjugates are imaginary.

Read the blog on Pascal Triangle & FIBONACCI NUMBERS

                       MODULUS & ARGUMENTS

If z=a+bi is a complex number then modulus of z is given by r=√a²+b² .

Argument or amplitude of z is given by  𝛩＝tan⊣(y/x) 

i.e tan inverse of y/x

           POLAR FORM OF A COMPLEX NUMBER

IF z=a+bi is a complex number then it's 

   polar form is given by z=r(cos𝛩+ i sin 𝛩)

EXPONENTIAL FORM OF A COMPLEX NUMBER

1)   eⁱӨ = e raise to iӨ =cos Ө+i sinӨ

2) e⁻ⁱө  =e raise to -iө  =cos Ө-i sinӨ

3) eⁱӨ +e⁻ⁱө =2cosө

4) eⁱӨ - e⁻ⁱө =2isinө

  DE MOIVRE'S THEOREM

For any rational number n the value or one of the value of 

(cos Ө+i sinӨ)ⁿ= (cos nӨ+i sin nӨ)

e.g. 1) if (cos Ө+i sinӨ)⁴ = (cos 4Ө+i sin4Ө)

       2) if (cos Ө-i sinӨ)⁴ = (cos 4Ө-i sin4Ө)

      3)  if (cos Ө+i sinӨ)⁻⁴ = (cos 4Ө-i sin4Ө)

      4)  if (cos Ө-i sinӨ)⁻⁴ = (cos 4Ө+i sin4Ө)

EULER'S FORM 

1) if z₁=a₁+b₁i  & z₂=a₂+b₂i then r=√a²+b² .

&     z₁

__________=  ( r₁/r₂) eⁱ(ө₁-ө₂)      

     z₂                

2) z₁ . z₂  = (r₁.r₂) eⁱ(ө₁+ө₂)  

i.e    (r₁.r₂). e raise to i.(ө₁+ө₂)  

 This is known as Euler's form

BASIC ALJEBRA FORMUAE

 

BASIC  ALJEBRA FORMUAE

1) (a+b)¹  =a +b 

2) (a+b)² =a² +2ab +b²

3) (a-b)² =a² -2ab +b²

4) (a+b)³ =a³ +3a²b+3ab²+b³

5)(a-b)³ =a³-3a²b+3ab²-b³

6)(a+b)⁴=a⁴+4a³b¹ +6a²b² +4a¹b³ +b⁴ 

7)(a+b)⁵=a⁵+5a⁴b¹+10a³b²+10a²b³+5a¹b⁴+b⁵   

8) (a²-b²) =(a+b) (a-b)

9) (a+b)(a-b)  =a²-b²

10) (a³+b³)  =(a+b) (a²-ab+b² )

11) (a³-b³)   = (a-b) (a²+ab+b² )

12)(a+b+c)² =a²+b²+c²+2ab+2bc+2ca

13) (a+1/a)² = a²+2+1/a²

  .i.e a²+1/a² = (a+1/a)²-2

14) (a-1/a)² =a²-2+1/a²

i.e. a²+1/a²  = (a-1/a)² +2

15) (a+1/a)³ = a³+3a+3.1/a+1/a³

16) (a+b)²-2ab =a²+b²

17) (a+b)² - 4ab = (a-b)² 

18) (a-b)² +4ab= (a+b)²

19) (x +a) (x+b) =x (x+b) +a (x+b)

                            = x² +x.b +a.x+a.b

20) (x +a) (x-b)  = x (x-b) +a (x-b)

                            =x² - x.b + a.x - a.b

21) (x-a) (x+b)   = x (x+b) - a(x+b)

                           =x² +x.b - a.x - a.b

22) (x-a) (x-b)   = x (x-b) -a (x-b)

                          = x² - x.b - a.x + a.b

READ THE BLOG ON ANGLE & IT'S TYPE

      Laws of Indices  :

1) (a.b)ⁿ=aⁿ.bⁿ  (Power is common base is different)

2) aⁿ. aⁱ= aⁿ⁺ⁱ   (Base is same power is different)

3) 1/aⁱ  = a⁻ⁱ

3) aⁿ/aⁱ  =aⁿ➗ aⁱ = aª⁻ⁱ

4) (aⁿ)ⁱ = aⁿⁱ

5)( (aⁿ)ⁱ )ˡ = (a)ⁿⁱˡ

6) aº  = 1

7) a¹ = a 

8) a⁻ⁿ = 1/ aⁿ

9) (a)¹/ⁿ = n√a  e.g. ∜a = (a)¹/4

10) (a)ⁿ/ˡ  =1 / (a)ˡ/n

11) (a/b)ⁿ  = aⁿ/bⁿ

READ THE BLOG ON " INDIAN RAILWAY "

Note :

1)  (-) × (-)  = +

2) (-) × (+)  = -

3) (+) × (-)  = -

4) (+) × (+) = +

ANGLE & IT'S TYPES

                

                        ANGLE   &  IT'S  TYPES 

ANGLE : An angle is the figure formed by two rays joined together at a common point called vertex .

         Two rays acts as sides of angle.

If OA is initial arm &OB is final arm as in fig1.The rotation of initial arm is in anticlockwise direction .

Then the angle formed is known as positive angle .

If OB is initial arm & OA is final arm as in fig2.The rotation of initial arm is in clockwise direction. Then the angle formed is known as Negative angle.

          Fig.1                                 Fig.2

   Angle formed by two ray generally lie in the same plane.

  Intersection of two planes also formed the angle,which is known as Dihedral angles. 

The angles are measured in 

1)Degree (Sexagesimal system)

2) Radian (Circular system).

There are different types of angles in geometry.

1)Right angle  :  An angle who measure is 90 degree is known as right angle.

                  

2)Acute angle  : An angle who measure is less than 90 degree is known as acute angle.

3)Obtuse angle  : An angle who measure is more than 90 degree is known as obtuse angle.

  Read the Blog on limits , Derivatives & Integration

4)Straight angle  :When two rays join together in opposite direction at a common point they form a straight angle.

  A  straight angle is also called as flat angle .

       It's measure is 180 degree .

5)Zero angle  : An angle whose measure is zero degree is called a zero angle.

  It is formed when both the rays which formed the side of angle lie in the same line or they overlap each other or they coincide with each other .

Ray OA & RayOB Coincide with each other .

6)Reflex angle : A reflex angle is an angle which is more than 180 degree but less than 360 degree.

   If one angle is more than 180 degree than other angle formed in a plane can be either acute or obtuse 

i.e opposite of reflex angle is either acute or             obtuse .

7)Directed angle : The ordered pair of rays (OA,OB) together with the ROTATION OF THE RAY OA to the position of the ray OB is called the directed angle AOB.

  If the rotation of the initial ray is anticlockwise then the measure of directed angle is cosidered positive and if the rotation is clockwise then the measure of the angle is considered as negative.

READ THE BLOG ON DIFFERENCE BETWEEN BACTERIA & VIRUSES

8)One rotation angle :After one complete rotation  if the initial ray OA coincide with the final ray i.e terminal ray OB then the angle so formed is known as one rotation angle. For example 360 Degree.

9)Standard angle : If the vertex of the angle is at the origin and the initial ray is along positive direction of x -axis of the coordinate system , such angle is known as  standard angle.

Angle POX , AngleXOQ , Angle XOR are          standard angles .

10)Quadrantal Angle : A directed angle in standard position whose terminal ray lie along X-axix or Y-axis is called a quadrantal angle.

Angle POQ , Angle POR , Angle POS , Angle POQ (Clockwise) are quadrantal angles.

11)Co- terminal angles : Directed angles of different amount of rotation having the same position of initial and final rays are known as Co-terminal angles.

Angle AOB (30 degree) , Angle AOB (-330 degree) , Angle AOB (390 degree) are Co-terminal angles. 

12)Supplementary angle : When sum of measure of two angle is 180 Degree then those angles are known as supplementary angle. Eg. If one angle is 60 Degree then it's supplementary angle is 120 Degree.

13) Complementary Angle : When sum of measure of two angle is 90 Degree then those angles are known as Complementary angles.

14) Adjacent angles : Two angles are said to be adjacent if 1)They have common vertex 2)they have one side common & 3) both the angles are on opposite side of common side.

Angle  CBD & angle DBE are adjacent angle .

Angle ABC & Angle CBD are adjacent angle . 

But Angle ABC & angle DBE are not adjacent angle .

15)Vertically opposite angles : Two angles are said to be vertically opposite if they are formed when two lines intersect at a common point & they are formed in opposite direction.      

                  Angle AOC & Angle DOB

                 Angle  AOD & Angle COB are                                        vertically opposite angles.

Their measure is same.

READ THE BLOG ON "MASOOM DIL "

16 ) FOR TWO PARALLEL LINE :

 If the two line are parallel (i.e. two line never intersect each other) & a transeverse intersect the two parallel lines at two different points  then following angles are formed .

1)Alternate angles 

2)Corresponding angles

3)interior angle 

4)Exterior angle

5)Vertcally opposite angle

 

          from fig.

1)Alternate angles are of equal measures 

            & they are : 

            Angle c & angle e

            Angle b & angle h

2)Corrosponding angles are ...

            Angle d & angle h

            angle c & angle g

            angle a & angle e

            angle b & angle f

            

3)Interior angles are :

     Angle c , angle h , angle b , angle e

     Aso   angle c+ angle h =180degree

    &  angle b+ angle e = 180 degree 

4) Exterior angles are :

      Angle d & angle g

       Angle a & angle f

5) Vertically opposite angles are :

           angle d & angle b

           angle a & angle c

           angle h & angle f

           angle e & angle g

   

Read the Blog On "INDIAN RAILWAY "

17) Plane Angle  :     An Angle between two Intersecting lines is                            known as plane angle in the same plane .  

                  It is measured in Radian. 

                   It is defined in two dimension .

In case of circle it is define as angle subtended by arc at the centre of the circle .

18) Solid Angle  : It is 3 Dimensional angular volume .

              It is measured in steradians. If the surface covers the                         entire sphere then the solid angle subtended at the center is 4 pi radian . Solid angle concept is generally used in physics.

19) Space Angle :  

It is a angle between a line and a Plane in the space . 

It is also angle between two plane in the space. 

It is also an angle between two line in the space.

Read the Blog on Physical Quantities in PHYSICS

UNITS AND DIMENSIONS OF PHYSICAL QUANTITIES IN PHYSICS ...

                                                    UNITS  AND DIMENSIONS 

      SR.NO.                   PHYSICAL  QUANTITIES                 UNIT                                DIMENSION

       1                               DISTANCE                                       m , cm                                 M   L   T

       2                               DISPLACEMENT                           m  , cm

       3                               SPEED                                            m/sec , cm/sec

       4                               VELOCITY                                  m/sec , cm/se

       5                                ACCELERATION                        m/sec² , cm/sec²

       6                                MASS                                             kg  .gm

       7                                TIME                                               sec

       8                                FORCE                                       Newton ,dyne

       9                                WORK                                         joule , erg

      10                               POWER                                             Watt , j/sec

      11                              AREA                                                  m² , cm²

      12                              VOLUME                                            m³ ,  cm³ 

      13                              LENGHT                                                m , cm

      14                             DENSITY                                          kg/m³ ,  gm/cm³

      15                             MOMENTUM                              kg m/sec  , gm cm/sec

      16                             PRESSURE                                  Newton/m² . dyne/cm²

      17                             TEMPRATURE                          kelvin

      18                              TORQUE                                   Newton m , dyne cm

      19                              IMPULSE                                 Newton sec

      20                              CURRENT                               Ampere

      21                              CHARGE                                 coulomb

     22                             KINETIC ENERGY                    joule , erg

     23                             POTENTIAL ENERGY               joule ,erg

     24                             SURFACE TENSION                   N/m . dyne/cm

     25                             SURFACE  ENERGY                     joule

     26         RADIUS OF GYRATION                                   m , cm

     27         FREQUENCY                                                      Hertz  , cps

    28          ENERGY                                                                         joule

    29          ELECTRIC POTENTIAL                                             volt

    30          ELECTRIC  RESISTANCE                                      ohm

    31         ELECTRIC  CONDUCTANCE                             /ohm , Siemen

    32         ELECTRIC      CAPACITANCE                              farad

    33         MAGNETIC     FLUX                                                   Weber

   34          INDUCTANCE                                                                Henry

   35          MAGNETIC  FLUX   DENSITY                             Wb/m²

   36          ILLUMINATION                                                  lux

   37         LUMINOUS  FLUX                                                  cd 

   38         SPECIFIC  HEAT                                           J/Kkg

   39         SPECIFIC    VOLUME                              meter cube/kg

  40          STEFAN'S  CONSTANCE

  41          STRESS

 42           STRAIN

 43           TEMPERATURE   GRADIANT

 44           VELOCITY    GRADIANT

 45           UNIVERSAL GRAVITATIONAL   CONSTANT

 46           UNIVERSAL GAS   CONSTANT 

 47           ANGLE

 48           SOLID ANGLE

 49           WEIGHT

 50           RADIOACTIVITY

 51           INTENSITY OF LIGHT

 52          QUANTITY  OF  SUBSTANCE

 53          CURRENT  DENSITY

 54          MAGNETIC  INDUCTION

 55          ANGULAR    DISPLACEMENT

 56          ANGULAR  VELOCITY

 57          ANGULAR  ACCELERATION

 58          ANGULAR MOMENTUM

 59          CENTRIPETAL  FORCE

 60          FARADAY  CONSTANT

 61          HEAT

 62         HEAT    CAPACITY

 63         MAGNETIC MOMENT

 64         MAGNETIC DIPOLE MOMENT

  65        MAGNETIC SUSCEPTIBILITY

  66        COEFFICIENT  OF  VISCOSITY

 67         WAVE LENGHT

 68         WAVE NUMBER

 69         THERMAL   CONDUCTIVITY

 70        ELECTRICAL   CONDUCTIVITY

 71       ELECTRICAL   DIPOLE MOMENT

 72        ELECTRIC  FIELD

 73        ELECTRIC  FLUX                                 Nm²/C

 74       PERMITIVITY  OF FREE  SPACE            H/m

 75      PERMEABILITY OF SPACE              

 76         SPRING  CONSTANT                           N/m

77        MODULUS   OF  ELASTICITY             N/m²

78            YOUNG,S   MODULUS                      N/m²

79             BULK  MODULUS                           N/m²

80               RIGIDITY  MODULUS                     Newton/square meter

81              CONDUCTANCE                                /ohm

82               CONDUCTIVITY                                /ohm meter

83            RESITIVITY                                               Ohm meter

84         COEFFICIENT OF   FRICTION               NO UNIT

85          EQUIVALENT  CONDUCTANCE            /ohm m²

86        MOLAR    CONDUCTANCE                      /ohm .m² . /mole

DIFFERENCE BETWEEN BACTERIA AND VIRUSES

   DIFFERENCE  BETWEEN  BACTERIA  AND VIRUSES

There are two common causes of disease in human being .

1. Bacteria

2.Viruses

Bacteria are single celled living organisms . Most bacteria are not harmful but small percentage of bacteria are harmfull and causes disease. Humans have trillions of bacteria in and on our bodies. Most of them are usefull for our bodily functions . Normally bacterial infections are limited to one part of our body like ear infection, throat infection  etc. Antibiotics  stops or slow downs the spread of bacteria.

Viruses are not considered living organisms in nature that means they need a live host in order to survive.They infect the cells of living being , including  humans. Most viruses are very harmful but some can be beneficial.Once we come in contact with viruses , they attacks healthy cells in our body , taking them over to reproduce and they spread throughout our body. Coronavirus or COVID-19 is 

one of them , it is caused by the SARS-Co-2 virus.

 Symptoms of this virus is fever , cough and shortness of breath. 

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Viruses can't be treated with antibiotics .Vaccines can be useful for both bacteria as well as viral infections .But Corona virus do not have any vaccine till date. They never die. Once immune system of 

person decreases it will overcome and it ends with death .

Bacteria and viruses both are susceptible to hand soap and sanitizers. It is found that proper hand washing ( around 20 sec and more ) kills and removes the bacteria and viruses from our skin. 

 So we should wash our hands frequently after touching  communal surfaces such as elevator buttons , ATMs etc.

In addition to this try to avoid touching our face  frequently , covering coughs and sneezes is also very important.We can Prevent

the spread of both bacteria and viruse by proper practising good hygiene.

If we are sick stay at home, try to be away from the others to avoid the spread of viruses or bacteria . Get quarantine by him self.

However  Healthy people can over come the  infections fast but  elderly people over 60 years of age and childrens are at high risk for developing complications from these infections.

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                                                Fig. Latest Coronavirus from SARS isolated in FRhK-4 cells. (WHO )