hundreds, we take the other 1 and call it 10 tens. Leaving 9 of the tens, we take the other 1, call it 10 units, and add to the 3 units, making 13 units.

In practice, it is unnecessary to write the numbers in this changed form, nor is it necessary to go through the above process mentally. When we have to take one from a higher term, we always have 10 to add to the term from which we are subtracting; the term from which we take the 1 is of course reduced by 1; and, if O's intervene in the minuend, when we come to subtract we shall always have 9's in the place of the O's. What we have to do, then, in the problem just given, is this: As we cannot take 7 from 3, we take 7 from 13; instead of 3 from 0, we take 3 from 9; instead of 8 from 0, 8 from 9; and instead of 4 from 6, 4 from 5. All this we can see as we go along, without actually performing the reduction even mentally.

The sign of multiplication is an oblique cross.

3 x 6 18. Three times six equals eighteen. =

3 is called the multiplier; 6, the multiplicand; 18, the product; 3 and 6, factors of the product.

42

3

3

3 3x6 = 6 x 3.

3

3 18 is a multiple of 6 and of 3. 6x7=

3

Sum 18

Multiplicand 246
Multiplier 10
Product

2460

6 x 318, +4=22 6 x 424, +2=26

For, 10 x 246 = 246 × 10, or 246 tens. Or thus: If we annex a cipher to a number, each digit is removed one place to the left, and consequently its value is ten times as great as it was before. And if the several parts of the number are each made ten times as great, the whole number is made ten times as great.

Multiply 4672 by 31, and annex two ciphers.

Multiply 436 by 23, and annex five ciphers.

Multiplicand 472163

Multiplier 51002

944326

472163

2360815

Product 24081257326

Disregard the intermediate ciphers of the multiplier.

DIVISION.

If we start with 7, we can subtract 2 three times, leaving a

divisor; 3, the quotient; 1, the remainder.

The sign of division is a short horizontal line, like that of subtraction, with a dot above and a dot below. The line may be omitted or the dots may be omitted, the dividend taking the place of the upper dot, the divisor that of the lower.

Multiplication begins with 0, and proceeds by successive additions of the same number. If, reversing the process, we subtract this number as many times as we have added it, we run back to 0.

Multiplication reversed is division.

Whenever a division comes out even, the process is the reverse of multiplication.

The two factors of a product are not similar. 3×6 means three sixes. 6 is the number taken, and 3 the number of times it is taken. Given the product 18, and 6, the number which is taken some number of times, we find how many times it is taken by reversing the process of multiplication; that is, by a process which is equivalent to subtracting 6 from 18, 6 from the remainder, and so on as long as we can do it, noting the number of subtractions.

Now, suppose, instead of 18 and 6, we have given 18 and 3, and are required to find the number which taken 3 times equals 18. How shall we do this? Simple as the problem may appear, there is no direct method by which it can be performed.

We have seen that division reduces to multiplication and subtraction, multiplication to addition, addition and subtraction to counting. Every operation in arithmetic is reducible

to the simple process of counting, either forward or backward, and counting will not solve the problem. But, as before noticed, 3 x 6 produces the same product as 6 x 3; and so for our present purpose we may consider 18 as made up of 3's instead of 6's. By division, we find there are six 3's in 18, so 6 is the factor we are seeking.

So, while division always involves the process of finding how many times one number is contained in another, it may be for the purpose of finding the number which taken a certain number of times amounts to a certain other number; or, employing a convenient and common metaphor, of finding one of the equal parts of a number.

One of the two equal parts of a number is called one-half of the number; one of the three equal parts, one-third; one of the four equal parts, one-fourth; two of the three equal parts, twothirds; three of the four equal parts, three-fourths; etc.

In practice, we consider 3 x 6 as three 6's, or six 3's, or rather we think of neither 6's nor 3's, but of 6 and 3 multiplied together, and this is sufficiently exact for purposes of computation, though in the final analysis there can be no such thing as "multiplied together."

It is by virtue of this assimilation of factors that we are enabled to have several factors of a product, which would otherwise be impossible. Thus, 2 × 3 × 7 = 42; 2, 3, and 7 being all considered as factors of 42, at one and the same time.

Divide 2379 by 8.

8) 2379

8 in 23 hundreds, 2 hundred 297 and 3 remainder. times, with 7 hundreds remainder. 7 hundreds 70 tens, 7 tens 77 tens. 8 in 77 tens, 9 tens times, with 5 tens remainder. 5 tens 50,959. 8 in 59, 7 times, with 3 remainder.