Distinction between algebraic and arithmetical fraction

We all agree that math educators should strive to produce good mathematics. But how does one define good mathematics? It is fairly evident that there are many different types of mathematics which can be designated ‘good’. Good mathematics may refer to good mathematical problem solving, it may refer to good mathematical technique, and it may refer to good mathematical application or good mathematical theory.
But these are not the types of ‘good mathematics’ we seek of the math educators. Math educators need to have good mathematical insight i.e., the ability to provide major conceptual simplifications; and have the ability to draw analogies, provide unifying principles and heuristics. They also need to be proficient in good mathematical pedagogy i.e., they need to have a good lecture or writing style that enables others to learn and do mathematics more effectively or make contributions to mathematical education.
The distinction between algebraic and arithmetical fraction is one that you might not find in many popular math text books. It may often be omitted owing to the obviousness of the distinction; i.e., algebraic fractions are fractions where the numerator and denominator are both algebraic terms while arithmetic fractions have specific numbers as numerator and denominator.
Yet, numbers and algebraic expressions do not belong to exclusive dichotomous domains as one might imagine. Elucidating the differences between algebraic and arithmetical fractions is one of the hallmarks of a good math educator.
To recap, an algebraic fraction is one in which the numerator and denominator are comprised of algebraic expressions
What are algebraic expressions?
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.’
An example of an algebraic fraction could be 2a/b or
Arithmetic fraction is one in which the numerator and denominator are comprised of numbers. For example the quotient of two numbers that are divided, for ex- 7 and 3, is given by the arithmetic fraction 7/3. In the words of the great mathematician Euler the arithmetic fraction 7/3 is expressed –
‘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’.
One can find the definitions of arithmetic fractions and algebraic fractions in many popular math books, but to look for an articulate distinction between the two ideas one has to read Professor De Morgan’s book.
In his expository mathematics book, math educator De Morgan writes –
We must here observe that a distinction must be drawn between algebraic and arithmetical fractions. For example, a + b / a – b is an algebraic fraction, that is, there is no expression without fractions which is always equal to a + b / a – b. But it does not follow from this that the number which a + b / a – b represents is always an arithmetical fraction; the contrary may be shown.
Let a stand for 12, and b for 6, then a + b / a – b is 3.
Again, a2+ab is a quantity which does not contain algebraic fractions, but it by no means follows that it may not represent an arithmetical fraction. To show that it may, let a = and b = 2, then a2 + ab = .
Other examples will clear up this point if any doubt yet exists in the mind of the student. Nevertheless, the following propositions of arithmetic and algebra, which only differ in this, that “whole number” in the arithmetical proposition is replaced by “simple expression” (By a simple expression is meant one which does not contain the principal letter in the denominator of any fraction) in the algebraical one, connect the two subjects and render those demonstrations which are in arithmetic confined to whole numbers, equally true in algebra as far as regards simple expressions.

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