1. INTRODUCTION
A graph is a collection of edges (arcs or lines) and vertices (or points). Each edge has two vertices. Number of edges coming out of a vertex is called the degree of that vertex.
Figure-1
For example: In the graph given in Figure-1 has four vertices v_1,v_2,v_3 ,v_4 and five edges v_1 v_2,v_1 v_3,
v_1 v_4,v_2 v_3,v_3 v_4 . Here deg〖v_1 〗=3 as three edges are coming out of the vertex v_1, similarly deg〖v_2 〗=2,
deg〖v_3 〗=3,deg〖v_4 〗=2. We get complete details about graphs in Deo [1], Johnsonbaugh [2] and Ram [3].
Here in this article, the graphs will be used in a different way. An edge will represent a line segment and that line segment will represent a commutative and associative binary operation which will be applied on the numbers written along it. In place of vertices, numbers will be written along the line segment.
For example: If the line segment given below in Figure-2 represents the multiplication of the numbers written along it, then we get abc
Figure-2
Similarly, if the line segment given below in Figure-2 represents the addition of the numbers written along it, then we get a+b+c
Figure-2
The numbers written along any line segment will be multiplied and if two line segments are connected by one end of each, then product obtained from the numbers written along them will be added.
For example:
From the graph given in Figure-3, we get the products ab and ac from each line segment and so the resultant is ab+ac which is obtained on adding the products ab and ac obtained from each edge
2. CONVENTION OF REPRESENTAION OF MULTIPLICATION OF NUMBERS BY A GRAPH:
The algebraic identities like 〖(a+b)〗^2=a^2+2ab+b^2 , 〖(a+b)〗^3=a^3+3a^2 b+3ab^2+b^3 can be represented by graphs by adopting a definite convention of representing any algebraic expression with the help of geometrical line segments. In this way, the algebraic expressions can be converted into interesting figures which are the best options for aid to memory.
It will be assumed, that binary operation of multiplication of numbers defined by graph obeys commutative and associative laws.
The algebraic expression ab of product of two numbers a,b can be represented by a line segment so that the numbers a,b lie along this line segment:
The algebraic expression abc of product of three numbers a,b,c can be represented by a line segment so that the numbers a,b,c lie along this line segment as shown below:
Similarly, the algebraic expression a_(1 ) a_(2 ) a_(3 )…a_(n ) of product of n numbers a_(1 ),a_(2 ),a_(3 ),…,a_(n )can be represented by a line segment so that the numbers a_(1 ),a_(2 ),a_(3 ),…,a_(n ) lie along this line segment as shown below:
The algebraic expression ab+cd sum of product of two numbers a,b and product of two numbers c,d can be represented by two line segments connected at one end of each so that the numbers a,b lie along one line segment and c,d lie along other line segment as shown below:
The algebraic expression abc+de sum of product of three numbers a,b,c and product of two numbers d,e can be represented by two line segments connected at one end of each so that the numbers a,b,c lie along one line segment and d,e lie along other line segment as shown below:
The algebraic expression a_(1 ) a_(2 ) a_(3 )…a_(m )+b_(1 ) b_(2 ) b_(3 )…b_(n ) sum of product of m numbers a_(1 ),a_(2 ),a_(3 ),…,a_(m ) and product of n numbers b_(1 ),b_(2 ),b_(3 ),…,b_(n ) can be represented by two line segments connected at one end of each so that the numbers a_(1 ),a_(2 ),a_(3 ),…,a_(m ) lie along one line segment and b_(1 ),b_(2 ),b_(3 ),…,b_(n ) lie along other line segment as shown below:
4. REPRESENTATION OF SOME FORMULAS BY GRAPHS:
The expressions like ab+bc+ca can be represented as below:
Algebraic identities like 〖(a+b)〗^2=a^2+2ab+b^2 and 〖(a-b)〗^2=a^2-2ab+b^2 can be converted into the following graphical expressions:
Algebraic identity like 〖(a+b+c)〗^2=a^2+b^2+c^2+2ab+2bc+2ca can be converted into the following graphical expressions:
Algebraic identity like a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca) can be converted into the following graphical expressions:
Algebraic identity like〖 (a+b)〗^3=a^3+3a^2 b+3ab^2+b^3=a^3+b^3+3ab(a+b) can be converted into the following graphical expressions:
Algebraic identity like〖 (a-b)〗^3=a^3-3a^2 b+3ab^2-b^3=a^3-b^3-3ab(a-b) can be converted into the following graphical expressions:
Algebraic identity (a+b)(a-b)=a^2-b^2 can be converted into the following geometrical expressions:
Algebraic identity 〖 (a+b)〗^4=a^4+4a^3 b+6a^2 b^2+4ab^3+b^4 can be converted into the following graphical expressions:
Algebraic identity
〖 (a+b+c)〗^3=a^3+b^3+c^3+3a^2 b+3ab^2+3b^2 c+3bc^2+3c^2 a+3ca^2+6abc
can be converted into the following graphical expressions:
5. REPRESENTATION OF SOME TRIGONOMETRIC FORMULAS BY GRAPHS:
Trigonometric formula sin〖(x+y)=sin〖x cos〖y+cos〖x siny 〗 〗 〗 〗 and
sin〖(x-y)=sin〖x cos〖y-cos〖x siny 〗 〗 〗 〗 can be converted into the following graphical expressions:
Trigonometric formula cos〖(x+y)=cos〖x cos〖y-sin〖x siny 〗 〗 〗 〗 and
cos〖(x-y)=cos〖x cos〖y+sin〖x siny 〗 〗 〗 〗 can be converted into the following graphical expressions:
Trigonometric formula cos〖(x+y)=cos〖x cos〖y-sin〖x siny 〗 〗 〗 〗 and
cos〖(x-y)=cos〖x cos〖y+sin〖x siny 〗 〗 〗 〗 can be converted into the following graphical expressions:
expressions:
Trigonometric formula sin〖(x+y)-sin〖(x-y)〗=2cos〖x siny 〗 〗 can be converted into the following graphical expressions:
Trigonometric formula cos〖(x+y)+cos〖(x-y)〗=2cos〖x cosy 〗 〗 can be converted into the following graphical expressions:
expressions:
Trigonometric formula sinA+sinB=2 sin〖(A+B)/2〗 cos〖(A-B)/2〗 can be converted into the following graphical expressions:
Trigonometric formula sinA-sinB=2 cos〖(A+B)/2〗 sin〖(A-B)/2〗 can be converted into the following graphical expressions:
Trigonometric formula cosA+cosB=2 cos〖(A+B)/2〗 cos〖(A-B)/2〗 can be converted into the following graphical expressions:
Trigonometric formula cosA-cosB=-2 sin〖(A+B)/2〗 sin〖(A-B)/2〗 can be converted into the following graphical expressions:
Trigonometric formula sin2A=2 sinA cosA and cos2A=cos^2A-sin^2A can be converted into the following graphical expressions:
Trigonometric formula sin3A=3 sinA-4 sin^3A and cos3A=4 cos^3A-3 cosA can be converted into the following graphical expressions:
6. REPRESENTATION OF SUM BY GRAPHS:
The sum of first n natural number is 1+2+⋯+n=∑▒n=n(n+1)/2, it can be represented by graph as follows:
The sum of squares of first n natural number is 1^2+2^2+⋯+n^2=∑▒n=n(n+1)(2n+1)/6, it can be represented by graph as follows:
The sum of cubes of first n natural number is 1^3+2^3+⋯+n^3=∑▒n=[n(n+1)/2]^2, it can be represented by graph as follows:
Therefore, in this way, we can convert almost all algebraic and trigonometric formulae into easily learnable graphical expressions which can be committed to memory with help of suitable graphic expression.
REFERENCES
[1] Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall India (1979).
[2] Richard Johnsonbaugh, Discrete Mathematics, Pearson Education India (2003)
[3] Babu Ram, Discrete Mathematics, Pearson Education India (2018)
REFERENCES
[1] Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall India (1979).
[2] Richard Johnsonbaugh, Discrete Mathematics, Pearson Education India (2003)
[3] Babu Ram, Discrete Mathematics, Pearson Education India (2018)