Algebraic fractions

Arithmetic fraction
 A fraction describes how many parts of a certain size there are, for example, one- half, three-eighths, one- quarter, three quarters etc.
The shortcomings of natural numbers become wholly evident when an integer is to be divided by another integer, by one that is not its divisor, and then the idea of the quotient becomes impossible to form without introducing the notion of a fraction.
We have only to imagine a line of 7 feet in length and nobody can doubt the possibility of dividing this line into 3 equal parts and of forming an idea of the length of one of these parts.
We shall quote the great mathematician Euler who says this of the ‘notion of a fraction’:
Since therefore we may form a precise idea of the quotient obtained, though that quotient may not be an integer number, we consider a particular species of numbers, called fractions, or broken numbers; of which the instance adduced furnishes an illustration.
For if we have to divide 7 by 3, we easily conceive the quotient which should result, and express it by ; placing the divisor under the dividend, and separating the two numbers by a stroke, or line .
 
The concept of using letters to generalize numbers
All of this might very well be evident; now we have only to mention, in algebra, in order to generalize numbers, we represent them by letters, as a, b, c, x, y, z etc. Thus, the expression a + b, signifies the sum of the numbers represented by the letters a, b; and these numbers can be either very great, or very small. In the same manner, the expression a/b, signifies the quotient of a number ‘a’ which is divided by another number ‘b’.
On the concept of algebraic expressions, Professor Wentworth writes: ‘An algebraic expression is a number written with algebraic symbols. An algebraic expression may consist of one symbol, or of several symbols connected by sign.
\Thus, a, 3abc, 5a + 2b – 3c,  are algebraic expressions. ‘ .
 
The algebraic fraction
So, in general, when the number ‘a’ is to be divided by the number ‘b’, we represent the quotient by , and call this form of  algebraic expression an algebraic fraction
The fractions a/b, ab/cd,  a/5, b/7a,  5/ab  are all examples of algebraic fractions
 
 The fraction a/b and division of the literal ‘a’ by ‘b’
On the topic of division of simple expressions, Professor De Morgan attempts to delineate the algebraic fraction. He says:
‘In dividing ‘a’ by ‘b’ we find the answer to this question: If ‘a’ be divided into ‘b’ equal parts, what is the magnitude of each of those parts? The quotient is, from the definition of a fraction, the same as the fraction , and all that remains is to see whether that fraction can be represented by a simple algebraic expression without fractions or not; just as in arithmetic the division of 200 by 26 is the reduction of the fraction to a whole number, if possible.’

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