CHAPTER V.

INVOLUTION AND EVOLUTION.

SIMPLE APPLICATIONS OF ARITHMETICAL RULES.

Contracted Multiplication and Division.-The results of all experiments show at best only an approximation to a true result. The accuracy of an expression is, it is true, increased by extending the number of decimal figures in the result, but it should be carefully noted that the accuracy of any result does not depend on the number of decimal places to which the result is calculated, but on the accuracy with which the observations were made.

In any result obtained the last decimal place may not be accurate, but the figure preceding should be as accurate as possible. It is advisable for the sake of accuracy to carry the result to one place more than is required in the result.

It will at once be evident that to multiply together two numbers (in each of which several decimal figures occur), and afterwards to reject several decimals from the product, loss of time will be experienced. Especially is this the case in practical questions in which the result is only required to be correct to the second or third place of decimals; in such cases what is known as Contracted Multiplication is useful.

Contracted Multiplication. In this method the multiplication by the highest figure of the multiplier is first performed. By this means the first partial product obtained is the most important one.

The method can be shown and best understood by an example.

Ex. 1. Multiply 006914 by 8.652.

66

The product of the two numbers can of course be obtained by the ordinary methods; to compare the two methods, ordinary" and "contracted," the product is obtained as shown:

The ordinary method will be easily made out from the working shown; in the contracted method it will be seen that the figures in the multiplier are reversed, and the process continued as follows: Multiply first by 8, so obtaining 55312; next by 6-this step we will follow in detail -6×4=24, the 4 is not written down (but if written down it is cancelled as shown), and the 2 is carried on. Continuing, 6×1=6, and adding on 2 gives 8. Next, 6×9-54, the 4 is entered; and 6×6 is 36, this with the 5 from the preceding figure gives 41, hence the four figures are 8, 4, 1, 4 as shown.

In the next line, multiplying by 5, we can obtain the two figures 0 and 7 as shown, but as these are not required unless there is some number to be carried, it is only necessary to obtain 69 × 5, and write down the product 345, add 1 for the figure rejected making 346 as shown; finally, as 2 × 9 will give 18, and therefore we have to carry 1, we obtain 2×6=12, together with the 1 carried from the preceding figure gives 13, add 1 gives 14. Adding all these partial products together we obtain the product required.

Thus in the second row one figure is rejected, in the next row two figures, and in the last row three figures are left out.

It must be noticed that when the rejected figure is 5 or greater, the preceding figure is increased by 1, also that the last figure of the product is not reliable. Having noted (or cancelled) the rejected figures, as will be seen from the example, the decimal point is inserted as in the ordinary method. The position of the decimal point can, in the majority of cases, be effected by mere inspection, and the

ability to ascertain without error the position of the decimal point is necessary when the slide-rule is used. The student is advised to get the practice necessary to do this without effort.

Contracted Division.-It is assumed that the student is familiar with the ordinary method of obtaining the quotient in the case of division, but this long process of division can also be advantageously contracted. The method of doing this will be clear from the following worked example.

Ex. 1. Divide 03168 by 4.208.

We shall work this example by the contracted method alone.

4208) 31680 (7529

29456

2224

2104

120

84

36

36

In the above example the number 7 is obtained by the usual process of division. By multiplying the divisor by 7, the product 29456 is arrived at. When this is subtracted from 31680, the remainder 2224 is left. It is seen that if we drop or cancel the 8 from the divisor 4208, thus obtaining 420, it can be divided into the remainder 2224 five times. In multiplying by 5 we take account of the 8 thus, as 5 × 8 is 40, we do not enter the 0 but carry on the 4; but 0x5=0, and adding 4, we see this is the figure to be entered. Now proceed to the next and the following figures, obtaining in the usual way 2104; subtract this from 2224, and the remainder 120 is obtained. Proceeding in like manner with the multiplier 2, we obtain 84, which, subtracted from 120, leaves 36, and our last figure in the quotient is 9. By the method described on p. 52, the answer is 007529.

The above example shows that the method of contracted division consists in leaving out or, as it is called, rejecting a figure at each operation. Any number which would be added on to the next figure by the multiplication of the rejected figure is carried forward in the usual way. To avoid mistakes it may be convenient to either draw a line through each rejected figure of the divisor, or to place a dot under it.

EXERCISES. XIII.

1. Without introducing into the work any unnecessary figures, find, to four significant figures, the values of 01785 × 87.29 and 1.04625 87.

2. 0008679 × 496 038. 4. 3 0964515÷0645. 6. 10.834394÷05309.

8. 8561 02 × 5.6039.

10. 16308362.4 × 1.942. 12. 5.61023 × 597001÷001.

3. 12.607 × 3.08 × 1.0006.

5. 589 0067 x 3.1008. 7. 18596 508÷98760. 9. 4.37642 × 00172. 11. 7·0342 x .003206.

Involution. When a number is multiplied by itself once, twice, or more times, the process is called Involution. The number thus multiplied is called the root, and the products are called the powers of the number. Any number multiplied by itself once is said to be squared, and the product so obtained is called the square or the second power of the original number. The number itself is called the square root of the product.

Thus, 3 x3=9. Here the product 9 is the square or second power of 3, and 3 is the square root of 9.

Instead of writing the expression 3 × 3 a small figure is placed near the top of the number or quantity, and on the right-hand side of it, thus: 32. This indicates how many times the number appears in the product. Thus, we write 3×3 as 32, 3 × 3 × 3 as 33, etc. The smaller figure written near the top of a number in the manner described is called the index or exponent of the number.

Adopting this notation, 31 would be called the first power of 3, 32 the second power, 33 the third power, etc.

The squares, cubes, or even higher powers can be easily obtained if the number is not greater than 10 (higher powers are best obtained by using logarithms). Thus 22=4, 23=8, 24=16, etc.

The powers of 10 itself are easily remembered, and are as follows: 102=100, 103=1000, 106=1,000,000, etc.

10-1=1, 10-2=100, 10-3=1000, etc.

This method of indicating large numbers is very convenient in Physical Science, in which such numbers as 2, 5, or 10 millions, etc., are of frequent occurrence; for in place of writing

5,000,000 for instance, we may write it more shortly as 5 × 106; and, as shown in the chapter on logarithms (p. 148), we can more easily write down the characteristic of the logarithm of the number.

In general terms it may be said that the powers of any quantity a would be written as a2, a3, a1, etc.

The squares of all the numbers from 1 to 10 should be remembered; they are as follows:

In a similar manner the squares of all numbers from 10 to 20 should be written down as well as the cubes of all the numbers from 1 to 10 inclusive.

Evolution. The reverse of Involution is to extract, or, find the roots of any given powers.

The root of a power or number is such a number, that, multiplied by itself a certain number of times, it will produce that power. Thus the square root of a given number is that number, which, when multiplied by itself, is equal to the given number.

The root of a given number may be denoted by the symbol ✔placed before it, with a small figure indicating the nature of the root placed in the angle; thus, the cube root of 27 is denoted by 27, the fourth root of 64 by 64, and so on. The square root in this manner would be denoted by 9, but the 2 is usually omitted, and it is written as

Another, and for many purposes a better method, is to indicate the root by a fraction placed as an index, and referred to as a fractional index. Thus, for example, the square root of 9 is written 9, and is read as nine to the power one-half. Similarly, the cube root of 27 is written as 27, meaning 27 to the power one-third.