We read about fractions in the articles titled ’Arithmetic fractions’ and ‘Algebraic fractions’. There are a few properties of fractions that are essential to a clear understanding of the topic. In this article, we will read about the equivalent property of fractions
In his expository book, De Morgan says the following of fractions –
‘A fraction (arithmetic) is not altered by multiplying or dividing both its numerator and denominator by the same quantity.
A fractional expression (algebraic) is not altered by multiplying or dividing both its numerator and denominator by the same expression.’
It is therefore possible to have multiple representations of the same fraction, each of which is equal to the others. These fractions are termed equal fractions.
It may be evident that the following fractions etc. make an integer, i.e., ‘1’ and that consequently they are all equal to one another.
The same equality prevails in the following fractions,etc, since 2 is their common value; for the numerator of each, divided by its denominator, gives 2.
So all the fractions,etc, are equal to one another, since 3 is their common value.
On the topic of Equal fractions, the great mathematician Leonhard Euler says-
“We may likewise represent the value of any fraction in an infinite variety of ways. For if we multiply both the numerator and the denominator of a fraction by the same number, which may be assumed at pleasure, this fraction will still preserve the same value. For this reason, all the fractions,
&c. are equal, the value of each being . Also,
&c. are equal fractions, the value of each being .
The fractions,&c. have likewise all the same value.’’
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.
Generalizing equivalent fractions to include algebraic fractions
In the article entitled ‘algebraic fractions’ we discussed how algebraic fractions are essentially algebraic expressions.
To re –iterate, we quote Professor Wentworth who says this of algebraic expressions-
‘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. ‘ .
Hence we may conclude, in general, that the fraction may be represented by any of the following expressions, each of which is equal to; namely &c.