Archive for February, 2014

Locus of a point (mathematics)

Locus :-

If a point travels according to the given condition then path created by point is known as the locus of that point.

Or locus is a geometric shape where every point satisfies the given condition.

Equation of the locus:

A algebraic relations got from the geometric condition for any variable point (x,y) is known as equation of the locus.

 

Example:-

Find the locus of a point, which is always at equal distance from two points (-1,2) and (4,0).

 

Let the coordinate that point is P(h,k).

Now according to the given condition

Distance from p(h,k) to (-1,2) = distance from P(h,k) to (4,0)

\sqrt{(h+1)^2 +(k-2)^2} = \sqrt{√(h-4)^2+(k-0)^2}

(h+1)^2 +(k-2)^2 = (h-4)^2+(k-0)^2

h^2+2h+1 + k^2 -4k +4 = h^2 -8h + 16 +k^2

2h+1-4k +4 +8h - 16 = 0

10h -4k - 11 =0

So locus of point p(h,k) is 10x - 4y -11 =0

Some important formulas of coordinate geometry

Some important formulas of coordinate geometry :

Distance between two points (x_1,y_1) and (x_2,y_2)

= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

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Distance of a point (x_1,y_1) from the center (0,0)

= \sqrt{x_1^2+y_1^2}

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Coordinate of a point , who divides a line joining the two points (x_1,y_1) and (x_2,y_2) in ratio of m_1 : m_2

Internal division (\frac{m_1x_2+m_2x_1}{m_1+m_2},\frac{m_1y_2+m_2y_1}{m_1+m_2})

External division (\frac{m_1x_2-m_2x_1}{m_1-m_2},\frac{m_1y_2-m_2y_1}{m_1-m_2})

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Mid point formula : Mid point coordinate of a  line segment by joining two points (x_1,y_1) and (x_2,y_2) is :

=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})

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Distance between Two points (Two Dimensions)

Distance between Two points (Two Dimensions) :

Let P(x_1,y_1) and Q(x_2,y_2) are two points in a plane,distance between them is d.

Now we have to find the distance d, draw two perpendiculars on x axes from P and Q points, which are PM and QN, now draw a perpendicular PR from point P on QN.

distance formula

According to the above figure:

OM =x_1 ; PM = y_1

ON = x_2 ; QN = y_2

So PR = MN = ON – OM = x_2 -x_1

And QR = QN – RN  = QN – PM = y_2 -y_1

Now In right triangle PRQ :

PQ = \sqrt{PR^2 + QR^2}

Or PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Or d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}  ….( distance formula)

Introduction of co-ordinate Geometry

First it was studied by French Mathematician Rene Descartes in 1637. In this geometry , position of the points is denoted by some special numbers we call them coordinates and we denote different shapes ( lines, circles etc ) with the help of algebraic equations. We use coordinates in this branch of mathematics, so we call it Co-ordinate geometry.

Cartesian Coordinates:

We draw two perpendicular lines in a plane, both are intersecting to each other at a point O, horizontal line is XOX’ and vertical line is YOY’, we call them Coordinate axes or Rectangular coordinate axes or rectangular axes.

coordinate axes

Line XOX’ is known as x-axes and line YOY’ is known as y-axes and intersecting point is O, we call it origin.

Let there is a point P in a plane, draw two perpendiculars from point P to x and y axes. Which are PM and PN ( shown in the below figure ), we denote the distance of P point from x-axes is x and from y-axes is y. we call x as x-coordinate or abscissa and y as y-coordinate or ordinate of point P. and these x and y are the coordinates of the Point P. we write them like (x,y).

coordinates

Quadrant and sign of coordinates:

Coordinate axes divides the plane into four parts, these four parts are XOY, YOX’, X’OY’ and Y’OX. We call them first, second, third and fourth quadrant.

From the right and upper side of the origin all distances are positive on OX and on OY and at left and down side from the origin all distances are negative on OX’ and on OY’.

quadrant

How to find Domain and Range of a given function

How to find Domain and Range of a given function:

If we have given a function f in the form of ordered pair then we can directly find the domain and range, set of fist element of all pair will be a set of domain and second element of all pairs will be a set of Range.

Example:

Given function :  f = (1,2),(1,3),(1,4),(5,2),(5,3),(5,4),(7,2),(7,3),(7,4)

Domain is {1, 5, 7} and Range is {2, 3, 4}

 

If we have given a function f in the form of equation ( x and y form)then we can find the value of y by putting some values of x, here set of all values of x for which y is defined is domain and set of all values y is Range.

Example:

Given function f :  x + y = 7 , where x,y є N

If x = 1, then 1 + y = 7 => y = 6

If x = 2 then 2 + y = 7 => y = 5

If x = 3 then 3 + y = 7 => y = 4

If x = 4 then 4 + y = 7 => y = 3

If x = 5 then 5 + y = 7 => y = 2

If x = 6 then 6 + y = 7 => y = 1

If x = 7 then 7 + y = 7 => y = 0 (0 is not a natural number)

So Domain = {1,2,3,4,5,6}  and  Range = {6,5,4,3,2,1}

What is function and type of Functions

What is function? Function (mathematics) is defined as if each element of set A is connected with the elements of set B, it is not compulsory that all elements of set B are connected; we call this relation as function.

f: A → B ( f is a function from A to B )

function

Not Function

Types of function:

    1. One-one Function or Injective Function : If each elements of set A is connected with different elements of set B, then we call this function as One-one function.one to one function
    2. Many-one Function : If any two or more elements of set A are connected with a single element of set B, then we call this function as Many one function. many to one function
    3. Onto function or Surjective function : Function f from set A to set B is onto function if each element of set B is connected with set of A elements. surjective function
    4. Into Function : Function f from set A to set B is Into function if at least set B has a element which is not connected with any of the element of set A.   into function
    5. One-one Onto Function or Bijective function : Function f from set A to set  B is One one Onto function if(a)   f is  One one function(b)   f is Onto function.   bijective function

 

What is relation and types of relations

What is relation (Mathematics) and types of relations:-

Relation (mathematics) is same as in real life, it is define as two things connected to each other or it is the property of things to connect in a certain manner.

Example:

  • Akbar was the son of Humayun
  • 25 is the multiple of 5.
  • In a triangle ABC, AB is the base.

Domain and Range of a Relation:  If R is a relation defined between set A and set B then set of first elements of ordered pair is known as Domain and set of second elements of the ordered pair is known as Range means Domain of R is {a|(a,b) є R} and Range of R is {b|(a,b) є R}.

Example :

If R = { (1,2),(1,4),(1,6),(2,2),(2,4),(2,6)} ( A relation between A and B )

So Domain of R  = {1, 2} = A  and Range of R = {2,4,6} = B

 

Types of Relation:-

1. Reflexive Relation:  If every element of a set A is related to itself In a Relation R it means that Relations R is reflexive relation.

(a,a) є R for all a є A.

 

2. Symmetric Relation: A relation R defined in a set A, if element ‘a’ is related to ‘b’ and ‘b’ is also related to ‘a’ in the same way then this relation is known as Symmetric Relation.

Relation R is symmetric if (a,b) є R => (b,a) є R for all a,b є A.

 

3. Anti symmetric Relation: A relation R defined in a set A. if element ‘a’ is related to ‘b’ and ‘b’ is related to ‘a’, both will be true only when a=b, then this relation is known as anti symmetric Relation.

(a,b) є R and (b,a) є R => a = b, for all a,b, є A

 

4. Transitive Relation: A relation R defined in a set A, if element ‘a’ is related to ‘b’ and ‘b’ is related to element ‘c’ from this if ‘a’ has relation to c then this relation is known as Transitive relation.

If (a,b)є R and (b,c)є R => (a,c)єR , for all a,b,c, є A

 

5. Equivalence Relation: we call equivalence relation  to a Relation R if

R is reflexive

R is Symmetric

R is Transitive

 

6. Partial order Relation:  we call Partial order relation  to a Relation R if

R is reflexive

R is Anti Symmetric

R is Transitive

 

7. Total order Relation: we call Partial order relation  to a Relation R if

R is Partial Order relation

And for every a,b є A either (a,b) є R or (b,a) є R or a = b are true.

Pythagorean Theorem : The Oldest Mathematics Theorem

Pythagorean Theorem: The Oldest Mathematics Theorem

Pythagorean Theorem is the oldest mathematics theorem or development by the human, this theorem is combination thought of different branches of the mathematics like algebra, geometry. It is developed by Pythagoras in 500 BC but we get some evidence that Babylonians know about it in1800BC as we get a clay tablet known as Plimpton 322 , Which systematically lists a large number of integer pairs (a, c) for which there is an integer b satisfying.

a2 + b2 = c2

Pythagorean Theorem or Pythagoras’ Theorem is a theory that states, that sum of two square’s areas whose sides are equal to the base (length b) and perpendicular (length a) of the right triangle is equal to the area of the square constructed on the hypotenuse (length c) of the right triangle. Where a,b and c are the side length of three squares.

Pythagorean Theorem

So c2 = a2 + b2

Theorem can be written as an equation:

(hypotenuse)2 = (base)2 + (perpendicular)2

History Of Mathematics

History of the mathematics: mathematics is very important part of the human culture from birth of the human, as they were use it for count the hunting animals to live the life, and then they use it to trade goods. Knowledge of the mathematics spread from one culture to others. Proof tells us that mathematics was started from Babylon and Egypt in 1800 BC, after it main contribution to the development goes to Greek, china, India and some part of Europe.

Study of the subject is started in the 6th century BC form the Pythagoreans, they first named this subject as mathematics, it is taken from the ancient Greek ‘mathema’, its meaning is “subject of instruction”. First development or written proof is Pythagorean Theorem in the history of mathematics.

Current form of mathematics is not possible without the contribution of some great humans. Here are few mathematicians who contributed to mathematics.

  • Euclid (Egypt)
  • Archimedes (Greek)
  • Brahmagupta (India)
  • Muhammad ibn Musa al Khwarizmi (Republic of Iraq)
  • Omar Khayyam (Iran)
  • Rene Descartes (France)
  • Pierre de Fermat (France)
  • Isaac Newton (England)
  • Gottfried Whilhelm von Leibniz (Germany)
  • Leonhard Euler (Switzerland)
  • Joseph Louis Lagrange (Italy)
  • Carl Friedrich Gauss (Germany)
  • Niels Henrik Abel (Norway)
  • Evariste Galois (France)
  • Bernhard Riemann (Germany)
  • Felix Klein (Germany)
  • Henri Poincare (France)
  • Davis Hilbert (Germany)
  • Emmy Noether (Germany)
  • Hermann Weyl (Germany)
  • Srinivasa Ramanujan (India)