comprehend the operation and be able to interpret the results in every case. Practised from the first in using the idea of correlation-of number defining the measur ing unit and number defining the measured whole-in both multiplication and division, he can tell on the instant which of these factors is demanded in any problem. (3) There does not seem to be any necessity for beginning formal division by the "long division" process. The pupil knows that 2 x 5 = 10, and that 10 ÷ 5 = 2, whatever may be the unit of measure. He knows that

ten ones divided by 5 is two ones, that ten tens divided by 5 is two tens, ten hundred-units divided by 5 is two hundred-units, etc. He has learned that 12 units of any order in the decimal system when divided by 5 gives 2 units of that order, with 2 units of that order, or 20 units of the next lower order, remaining; which 20 units on division by 5 gives 4 units of that order, making the total quotient 24. In short, if the pupil has been taught to divide a number of any two digits by any of the single digits, he can divide any number by a single digit. Thus, suppose 4976 is to be divided by 8: here eight will not divide 4 giving a quotient of the same order-i. e., in the thousand units; the 4 is changed to 40 units of the next lower order, making, with the 9 of that order, 49. This divided by 8 gives 6, with 1 over. Similarly this 1 is 10 of the next order, which, with the 7 of that order, makes 17; this divided by 8 gives 2, with 1 over; this 1 is 10 of the next order, and with the 6 makes 16 of that order, which, divided by 8, gives 2, the last figure of the quotient. No matter what the series of figures, the process is the same, and the pupil should experience

8)4976

622

no real difficulty if rational method and practice have been followed. A few practical points may be noted:

(1) The division by any power of 10 is as easy as multiplication by any power of 10—is, in fact, derived directly from it.

(2) So with division by factors of the divisor, which is directly connected with multiplication by factors of the multiplier. To the pupil it will prove an interesting exercise to discover the "true remainder." Take, for example, 5795 ÷ 48.

8)5795
6)724.
120

3 rem. in ones, the quotient being 724 eights. 4 rem. in 8-unit groups;

hence remainder in ones is 8 × 4 + 3 = 35. This is the old rule: Multiply the first divisor by the second remainder and add the product to the first remainder. The same method is applicable to the case of three or more factorial divisors; apply the rule to the last two divisions, and use the result with the first divisor and first remainder. Or, reduce each remainder to units as it occurs; for example, divide 2231 by 90 (= 3 × 5 × 6). 3)2231

5)743 unit-groups of 3 with rem. 2 units;

6)148 unit-groups of 15 with rem. 3 groups of 3-9 units; 24 groups of 90 with rem. 4 groups of 15 = 60 units.

The remainder is therefore 60+9+2 = 71. Otherwise, applying the rule with the last two divisions : 5 × 4+3=23; use this as the "second remainder" X from the "first divisor," and remainder 23 × 3+2=71.

(3) In long division the multiplications and subtrac

tions may be combined, as described under multiplication and subtraction—e. g., 635040 ÷ 864.

864)635040(735
3024

4320

(1) Seven 4's, 28 and two = 30-carry 3. (2) Seven 6's, 45 and zero 45-carry 4. (3) Seven 8's, 60 and three 63. This gives 302, which, with 4 brought down, makes the first remainder. Proceed similarly with 3 and 5, the other figures in the quotient. The student may note the application of the method in a longer operation: Divide 217,449,898,579 by 56437. The following is the work:

3852967

56437)217449898579

481388

298929

167448

545745

378127

395059

Three 7's, 21 and eight = 29-carry 2.

-carry 1.

Three 3's, 11

and three 14-carry 1. Three 4's, 13 and one = 14 Three 6's, 19 and eight 27-carry 2. Three 5's, 17 and four = 21. This gives 48138, which, with the 8 (heavy-faced type) brought down, makes the complete first remainder. With this proceed exactly as before, and so on with the other remainders.

(4) Casting out the Nines.—It is seen that 9 (and

of course 3) is a measure of 9, 99, 999, 9999, etc.—that is, of 101, 1001, 1000 - 1, etc. Hence, if from any number there be taken all the ones, and 1 from every 10, 1 from every 100, etc., the remainders from the tens, the hundreds, the thousands, etc., constitute a number which is a multiple of 9. The original number will therefore be a multiple of 9, if the total of the deductions made is a multiple of 9; this total is the number of ones the number of tens the number of hundreds, etc.-that is, this total is the sum of the digits of the given number. For example, is 39273 divisible by 9?

Hence the given number is exactly divisible by 3, but leaves a remainder of 6 when divided by 9, because 249 leaves 6 remainder. The principle is any number divided by 9 leaves the same remainder as the sum of its digits divided by 9.

To cast the nines out of any number, therefore, is to find the remainder in dividing the number by 9. In casting out the nines from the sum of the digits we may conveniently omit the nines from the partial sums as fast as they rise above 8.

Proofs of Division.—(1) By repeating the calculation with the integral part of the quotient for divisor. (2) By multiplying the divisor by the complete quo

tient. (3) By casting out the nines, as in multiplica

9's out of divisor.. 8 X 5.. out of quotient.

4 .. out of dividend.

If there is a remainder the method can still be applied. Test the accuracy of

155

3,893,865,378 ÷ 179 = 21,753,437

179

where the remainder is 155.

9's out of divisor (

6 .. out of 8 X 5 + 2. 8X5, 2.. out of quotient and remainder. 6 .. out of dividend.

The disadvantages of this proof are similar to those in the proof of multiplication by casting out the nines.

Fundamental Principles connecting Multiplication and Division.-From the theory of number as measurement and numerical operations as a development of the measuring idea, there are certain fundamental principles-fundamental also in fractions-connecting the operations of multiplication and division. The principal of these are the following:

(1) If equals be multiplied by equals, the products are equal.

(2) If equals be divided by equals, the quotients are equal.

(3) If an expression contains a series of multipliers and divisors, changing the order of the multipliers and divisors does not change the value of the expression.