3. Simple Multiplication.

SIMPLE MULTIPLICATION is the method of finding the amount of a given number, by repeating it any proposed number of times; as, 6 repeated 4 times, or 4 times 6 is 24.*

In Multiplication there must be at least two numbers given to find a third.

The two given numbers spoken of together are called factors. Spoken of separately, the number to be repeated, or multiplied, is called the multiplicand, the number by which it is repeated, or multiplied, is called the multiplier, and the number found by the operation is called the product.

Before the scholar can proceed in this rule, the following table must be thoroughly committed to memory.

Multiplication and Division Table.

Use of the preceding Table for Multiplication. Find the multiplier in the left hand column and the multiplicand in the upper line; the product will be found in the line with the multiplier, directly under the multiplicand. Thus 48 the product of 6 and 8, is found in the line with 6 under 8.

For Division.-Find the divisor in the left hand column, run your eye along to the right hand till you find the dividend, and right over it in the upper line is the quotient. Thus 48 divided by 6 the quotient is 8.

* Multiplication is only an abridged method of Addition, and all the questions in Multiplication may be solved by that rule. Thus 6 multiphed by 4, is the same as 4 sixes added together, or 6×4—6+6+6+6=24 But the solution by addition would be extremely tedious, particularly when the multiplier is a large number.

Rule.*

1. Write the multiplier under the multiplicand, with units under units, tens under tens, and so on, and draw a line under them.

2. Begin at the right hand and multiply all the figures of the multiplicand separately by each figure of the multiplier, setting the first figure of the product directly under the figure of the multiplier, which is employed, and carrying for the tens as in Addition.

3. Add these several products together, and their sum is the total product, or answer required.

Proof†

Make the former multiplicand the multiplier, and the former multiplier the multiplicand, and proceed as before; if the product be equal to the former, the product is right.

* When the multiplier is a single digit, multiplying every figure of the multiplicand by that digit, is evidently multiplying the whole by it; and carrying for the tens is only assigning the several parts of the product to their proper places. This must be obvious from the following analysis of the first examples. Multiplicand. 3 7 6 Multiplier.

4

Product. 15 0

or 3 7 6
4

24 280 1200

1504

Here 4 times 6 is 24, 7 being in the place of tens, is 70, and 4 times 70 is 280, and 3 being in the place of hundreds, is 300, and 4 times 300 is 1200. Here the multiplicand is divided into parts, and each of the parts multiplied by 3. Their product added together amounts to 1504, the same as by the rule.

Where the multiplier consists of more than 1 digit, it is considered to be divided into as many parts as there are digits, and the whole multiplicand being multiplied by each of these parts, is evidently multiplied by the whole multiplier. The product arising from multiplying by the second figure in the multiplier, or the figure in the place of tens, is ten times as great as it would be if that figure occupied the unit's place, and that arising from the third figure one hundred times as great, and so on, and these values are truly expressed by writing the first figure of each product directly under the figure by which we multiply, as will be evident by inspecting the operation; hence the sum of the several products is the product of the whole multiplicand into the whole multiplier.

+ This method of proof depends upon the proposition that two numbers being multiplied together, either of them may be made the multiplier, or the multiplicand, and the product will still be the same, which may be thus proved.-Suppose the two factors to be 6 and 3.

1, 1, 1, 1, 1, 1,

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

Now if we write three lines of 1s with six 1s in a line, it is evident that the whole number of 1s will make as many times 6 as there are lines, that is, 3 times 6, and as many times 3 as there are columns, that is, 6 times 3. Hence it

I. When there are ciphers on the right hand of one or both the factors,

RULE.

Neglecting the ciphers, proceed as before, and place on the right hand of the product, as many ciphers as were neglected in both the factors. EXAMPLES.

Prod. 3 3 2 0 6 4 0 0 0 0 0 Prod. 3 5 4 0 3 0 8 0 0 0 0 0

is plain that 3 times 6 are the same number of units, or give the same product, as 6 times 3, and the same may be shown of any other two factors.

There are several other methods of proof. The following by division will be found very convenient after becoming acquainted with that rule.

I. Divide the product by the multiplier, and if it be right the quotient will be equal to the multiplicand.

Another method much practised, is by casting out the nines.

RULE.

II. Cast the nines out of the multiplicand and multiplier; multiply the two excesses together, cast the nines out of their product and write down the excess; then cast the nines out of the product of the sum, and if the excess be equal to the former, the work is supposed to be right.

work is right. This method may generally be depended upon, but it is liable to the same inconvenience as in Addition, that a wrong operation sometimes appears to be right, and for the reason mentioued under that rule. There are other methods of proving Multiplication, but these are deemed sufficient.

II. When the multiplier is a composite number.

A composite number is one which is produced by the multiplication of other numbers, and the component parts are the numbers employed in producing the composite number. Thus 4 times 6 is 24. Here 24 is a composite number, and 4 and 6 are its component parts.

Rule.*

Multiply by one of the component parts, and that product by the other.

III. When the multiplier is 9, 99, or any number `of nines.

Rule.†

Annex as many ciphers to the multiplicand as there are nines in the multiplier, and from the sum thus produced, subtract the multiplicand, the remainder will be the product.

*The reason of this rule is obvious; for in the first example, the product of 6 times 2478 multiplied by 6, is as evidently 36 times 2478, as 6 times 6 is 36. A composite number may have 2, 3, or more component parts. Thus 30 is a composite number whose component parts may be 6 and 5, or 3 and 10, or 5, 3 and 2. &c.

+ The reason of this rule will appear by considering that annexing a cipher to any sum, is the same as multiplying it by 10, annexing two ciphers the same as multiplying by 100, &c. Now when the muitiplier is 9, annexing a cipher to the multiplicand, multiplies it by 10, which repeats it once more than is proposed by the multiplier; therefore if we take the multiplicand from this sum, we have the amount of the multiplicand nine times repeated, or the product arising from multiplying by 9. When there are two nmnes in the multiplier, annexing two ciphers to the multiplicand multiplies it by 100, which repeats it once more than proposed by the multiplier. Hence, taking the multiplicand once from this sum, we have the true product arising from multiplying it by 99, and the same reasoning is applicable to any number of nines.

Application.

QUESTIONS.

4. What are they called, spoken of 13. How do you proceed when the