lent to 357, but having for divisor a whole number. An answer to be expected is, 3507; at any rate, he can be led to this result, and, as this quotient is seen to be 50, he can conclude that 35÷7 50. An examination of a few more examples will show how always to proceed.

It would be well to have the student, in his earlier practice, write out a full statement of what he does. Suppose he is required to find the quotient 1.3754 ÷ 23; his solution should stand somewhat as follows:

The quotient of 1.3754 by 23 is the same as (multiplying each number by 100) that of 137.54 by 23. 2 3)1 3 7.5 4(5.98

115

22.5

2 0.7

1.84

1.84

.·. 1·3754 ÷ ·23 — 5·98.

The Relation of Decimals to Vulgar Fractions.The simple rules being understood, we may now consider the conversion of decimals to the ordinary fractional form, and the conversion of ordinary fractions into the decimal form.

From the definition, 273 = 10 + 180 + 1000 = 273 and the student sees at once how to write a decimal in the form of a fraction.

10009

We may next ask the pupil to divide 1 by 2, as an exercise in the division of decimals.

2)1.0(5

But the quotients 1÷2, 34, 78 have up to this point been taken as equivalent to 1, 1, 3.

As an exercise these results might be verified thus:

A method has now been found for converting an ordinary fraction into a decimal; at the same time another method has been suggested in the verification above

75. The latter method is very valuable from the point of view of theory, and the pupil should work several examples in this way.

We shall next consider the example .

to give rise to a recurring decimal. Let us now seek to convert to a decimal by the other method. According to it, we multiply the denominator by some number which will change it into 10, 100, 1000, etc.—that is, into

some power of 10. Now, any such power is made by multiplying 10 by itself some number of times; but 10 itself is made by multiplying 2 and 5; therefore every power of 10 is made up wholly of the factors 2 and 5, and in equal number. We can not, then, multiply 3 by any number that will make it into a power of ten— that is, we can not convert into an ordinary decimal with a finite number of digits. We have thus a complete view of the case. The following are examples of recurring decimals:

='16666 the 6's recurring, and this is

written = .16.

Here the first figure of the decimal does not recur, and is said to give rise to a mixed recurring decimal, those formerly met with being called pure recurring decimals. Similarly,

If we try to apply the second method to these examples,

The examples given lead up to the following propositions, for the truth of which it will be easy to state the general argument:

Proposition I.-A fraction whose denominator contains only the factors 2 or 5 leads to a decimal, the number of whose digits is the same as the number of times the factor 2 or the factor 5 is contained in the denominator, according as the former factor or the latter occurs the greater number of times.

Proposition II.-A fraction whose denominator contains neither the factor 2 nor the factor 5 leads to a pure recurring decimal.

Proposition III.-A fraction whose denominator contains in addition to the factors 2 and 5 a factor prime to these factors, leads to a mixed recurring decimal, the number of digits that are before the period being the same as the number of times the factor 2 or the factor 5 is contained in the denominator, according as the former factor or the latter occurs the greater number of times.

Questions similar to the following afford a valuable exercise on this part of the work:

(1) In the case of fractions, such as 4,, etc., leading to pure recurring decimals, what limit is there to the number of figures in the period?

=

(2)+ 142857 explain why any other fraction with denominator 7 will lead to a recurring decimal with a period consisting of the same digits following one another in the same circular order.

It will now be in place to consider the converse process of changing recurring decimals into their equivalent vulgar fractions. A difference of opinion exists as

to the best mode of dealing with these decimals. The method here presented is for many reasons thought to be the best:

=3333... (the 3's repeated without end); ·3 × 10 = 3.3333... (the 3's repeated without end); and 3 3333... (the 3's repeated without end);

=

which may be tested by converting into decimal form.

i7 × 99 = 17; ...

which in its turn may be tested.

Next, to find the fractional equivalent of ·279

The pupil, after working a few examples, will be in a position to formulate a rule for writing down the fraction which is equal to any given recurring decimal.

This treatment has the advantage of furnishing the pupil a direct and definite method of procedure. Against