Componendo and Dividendo

Componendo and Dividendo is an algebraic fundamental, mostly helps us to simplify the complicated functions and also math problems. It plays significant role in all mathematical concepts. In this article you are going to know the mathematical proof of Componendo and Dividendo rule.

Consider a, b, c and d are four numbers which belong to all numbers group including complex numbers. Assume that the ratio of a and b are equal to the ratio of c and d. it can be expressed in mathematical form. However the value of a shoulder not equal to value of b. Similarly, the value of c shoulder not equal to value of d.

a b

=

c d

Add and subtract the left hand side with a unit value. In fact we are not adding anything because by adding and subtracting a unit value, you are adding a zero to function.

a b

+ 1 - 1

=

c d

The negative unit value can be moved to right hand side and it become positive unit value at right hand side area.

a b

+ 1

=

c d

+ 1

It can be further simplify it by transferring right hand side part into function and also left hand side into one part. Consider it as equation 1.

a + b b

=

c + d d

(1)

As we already considered a technique to add a unit value to both sides but now move unit value to right hand side part and it become negative unit value. However, keep negative unit value in left hand side part.

a b

+ 1 - 1

=

c d

 

»

a b

- 1

=

c d

- 1

Now, it can be simplified further and considered it as equation 2.

»

a - b b

=

c - d d

(2)

If you observe the equation 1 and 2, you understand that the denominator of left and right hand sides of both equations are same. Therefore we can divide one equation with another equation to get the required solution.

a + b b	=	c + d d
a - b b		c - d d

You finally get a ratio which is known as Componendo and Dividendo rule.

a + b a - b

=

c + d c - d