Arithmetic fractions

We all have read the concept of fractions in school. The idea of fraction has been used ubiquitously since the time of the ancient Egyptians and continues to be used in our contemporary world.  The word ‘fraction’ has become part of our everyday vocabulary. We use the word as synonymous with the word part. Are all fractions parts? Are fractions a type of number? Read on to know the answers to more such questions.
The word fraction is derived from the Latin word ‘fractus’ which has the same meaning as broken. When used in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, three-eighths, eight-fifths, three-quarters etc.
Need for fractions
How do you measure the amount of milk consumed by one person when a family of 5 consumes 3 L of milk in a day? What do you call an apple that has been cut evenly in two pieces?
The natural/counting numbers are not sufficient to describe all the quantities found in nature.
The need for fractions as numbers is most evident, when we are endeavoring to ascertain the quotient of two numbers, of which, one is not the divisor of the other.
Mr. Gary S Goldman illustrates the need for fractions by means of a pizza-example to better connect to children, as follows –
‘You have two of your best friends over for a pre-algebra study group (at least that is what you have told your parents). Instead of working on pre-algebra problems and learning some math techniques from each other, you order a large pizza and the study turns into more of a party! The pizza arrives in a large box—pre-cut into eight slices. How can the pizza be divided equally among three people? Well, you divide the 8 pieces of pizza by 3 people to obtain 8/3. ‘
Nobody can deny the fact that it is possible to divide a pizza of 8 pieces among 3 people.
Clearly, whole numbers are not enough to quantify all the different measurements/ quantities in daily life. There is a need for numbers that measured parts of quantities.
The amount of pizza each person receives cannot be signified by means of natural numbers and thus an entirely new species of numbers i.e., fractions was imagined.
In his book ‘Elements’ the great mathematician ‘Leonhard Euler’ provides an example that illustrates the need for a new species of numbers – fractions.
He writes:
 ‘When a number, as 7, for instance, is said not to be divisible by another number, let us suppose by 3, this only means, that the quotient cannot be expressed by an integer number; but it must not by any means be thought that it is impossible to form an idea of that quotient. Only imagine a line of 7 feet in length; nobody can doubt the possibility of dividing this line into 3 equal parts, and of forming a notion of the length of one of those parts’
Since therefore we may form a precise idea of the quotient obtained in similar cases, though that quotient may not be an integer number, this leads us to 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 7/3; placing the divisor under the dividend, and separating the two numbers by a stroke, or line.’
Representing fractions
Of course, a new species of numbers require a new form of notation. It is very important to have precisely defined mathematical notations to represent the different numbers to avoid instances where they may be misinterpreted.
In his book ‘First steps in algebra’, Professor Wentworth writes ‘ The dividend is called the numerator and the divisor is called the denominator; and the numerator and denominator are called the terms of the fractions’
Thus, in all fractions the lower number is called the denominator, and that above the line the numerator.
The word ‘numerator’ is derived from Latin word numerātor which means counter or numberer (used for a person who counts or numbers). In arithmetic, the numerator denotes the top number in a fraction. The numerator shows us the number of parts that we have.
The word ‘denominator’ has its roots in Medieval Latin and is used in arithmetic to denote the number of equal parts into which a unit is divided. Numerator shows how many parts we have. The denominator denotes the bottom part of a fraction.
Clearly, being able to read a fraction is as important as writing a fraction. Leonhard Euler writes-  ‘In a fraction, which we read seven thirds, 7 is the numerator and 3 the denominator.’ 
We must also read:2/3 , two thirds; 3/4, three fourths; 3/8, three eighths; 12/100, twelve hundredths; and 1/2, one half; &c.’

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