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Unit fractions

If we want to succeed in algebra, it is imperative that we learn all the basic definitions associated with the concept.

The definition of the term ‘unit’ is one of chief importance in algebra, and possibly, in the whole of mathematics.

Defining units

The term ‘unit’ has been put in beautiful language in the book ‘First Steps in Algebra’, and was published in 1904 by ‘G. A. Wentworth’. It is a good plan to read his definition aloud, and make comments upon them, and ask questions about them. Of units, he says-

In counting separate objects or in measuring magnitudes, the standards by which we count or measure are called unit.

Thus, in counting the boys in a school, the unit is a boy; in selling eggs by the dozen, the unit is a dozen; in selling bricks by the thousand, the unit is a thousand bricks; in measuring short distances, the unit is an inch, a foot, or a yard; in measuring long distances, the unit is a rod or a mile.

Of numbers, Wentworth writes ‘Repetitions of the unit are expressed by number’ i.e., the repetition of a specific unit is counted using numbers.

We talked about algebraic and arithmetic fractions in the previous articles. To re-iterate, a fraction is a number that is expressible in the form a/b, where the number ‘a’ is an integer and the number ‘b’ is a positive whole number.

Defining fraction

In his exhaustively expository book, mathematician Leonhard Euler defines a fraction as follows

‘The nature of fractions is frequently considered in another way, which may throw additional light on the subject. If, for example, we consider the fraction, it is evident that it is three times greater than. Now, this fraction means, that if we divide 1 into 4 equal parts? This will be the value of one of those parts; it is obvious then, that by taking 3 of those parts we shall have the value of the fraction.

In the same manner we may consider every other fraction; for example,; if we divide unity into 12 equal parts, 7 of those parts will be equal to the fraction proposed.

From this manner of considering fractions, the expressions numerator and denominator are derived. For, as in the preceding fraction, the number under the line shows that 1 2 is the number of parts into which unity is to be divided; and as it may be said to denote, or name, the parts, it has not improperly been called the denominator.

Farther, as the upper number, viz. 7, shows that, in order to have the value of the fraction, we must take, or collect, 7 of those parts, and therefore may be said to reckon or number them, it has been thought proper to call the number above the line the numerator.

As it is easy to understand what is, when we know the signification of , we may consider the fractions whose numerator is unity, as the foundation of all others. Such are the fractions,

&c.’

We call the fractions whose numerator is unity, as unit fractions. The numerator of a unit fraction is the same as 1, and the denominator is any positive integer. A unit fraction is therefore the reciprocal of an integer and is aptly considered to be the foundation of all other fractions. All other fractions are simply multiples of unit fractions.