SECTION XII.

INVOLUTION.

404. Involution is the process of finding any power of a number

405. The Power of a number is the product arising from using the number several times as a factor. The number itself is called the root of the power, or simply the first power.

406. The Second Power of a number is the product obtained by using the number twice as a factor. Thus, 16 =4×4 is the second power of 4.

407. The Third Power of a number is the product obtained by using the number three times as a factor. Thus, 64 4X4X4 is the third power of 4.

408. The Degree of a power is indicated by a small figure called an exponent, at the right near the top of the number; thus, 42 represents the 2d power of 4, 53 the 3d power of 5.

409. The Exponent shows how many times the number is used as a factor. Thus, 73 denotes that 7 is to be used three times as a factor; that is, 7×7×7, which equals 343.

The second power of a number is also called its square, because the area of a square equals the product of two equal sides. The third power of a number is called the cube, because the product of three equal sides gives the contents of a cube.

410. From the above statements for involution we have the following

RULE-Find the product of the number used as a factor as many times as there are units in the power.

1. Find the square of 16.

SOLUTION. To find the square of 16 we multiply 16 by itself and we have 256. To find the cube of 16 we would multiply 256 by 16.

OPERATION.

16

16

256

411. The pupil will now learn the following principles, which will prepare the way for evolution.

PRIN. I.-The square of a number contains twice as many figures as the number, or twice as many, less one

DEM. The square of 1 is 1, the square of 9 is 81, hence the square of a number of one figure is one or two figures. The square of 10 is 100, the square of 99 is 9801, hence the square of a number of two figures is a number consisting of three or four figures, that is, twice two, or twice two less one, etc.

12 1

92

102

81

100

992 9801

Therefore, etc.

PRIN. II.-The cube of a number contains three times as many figures as the number itself, or three times as many,

one or two.

13 1

93

less

729

103

: 1000

998

: 970299

DEM.—The cube of 1 is one figure and the cube of 9 is three figures, as shown in the margin, hence the cube of any number consisting of one figure is a number consisting of one, two, or three figures. The cube of 10 is four figures, the cube of 99 is six figures, hence the cube of a number consisting of two figures contains four, five, or six figures, that is, three times two, or three times two less one or two. Therefore, etc.

INVOLUTION ANALYTICALLY.

412. The object of this method of involution is to exhibit the law of involution, that we may derive the law and method of evolution.

1. Find the square of 25 by the analytical method.

SOLUTION. Twenty-five equals 20 plus 5, or 2 tens plus 5 units. Writing as in the margin, and commencing at units, we have 5 times 5 equals 5 square, 5 times 20 equals 5 X 20, 20 times 5 equals 5 times 20, 20 times 20 equals 20 square, and adding we

25

25

OPERATION.

20+5

20+ 5

5 X 20 +52

=

50 202+

5 X 20

125=

625=202+2× (5 × 20) + 5a

have 202+2X(5×20)+52; hence the square of 25 equals the square of the tens, plus twice the tens into the units, plus the square of the units.

NOTE.-Let the pupil be careful to understand and state the inference we have put in italics, as this is the law with which he must be familiar to understand evolution.

Square the following numbers analytically :

413. We will now state some of the principles derived from the above solutions, in the form of propositions.

PROP. I. The square of a number of tens and units equals the square of the tens, plus twice the tens into the units, plus the square of the units.

PROP. II.-The square of a number of hundreds, tens, and units equals the square of the hundreds, plus two times the hundreds into the tens, plus the square of the tens, plus two times the sum of the hundreds and tens into the units, plus the the units.

square of

414. These principles may be expressed in an abbreviated form as follows: Let u represent units figure, t tens figure, h hundreds figure, and T thousands figure; then we have (t+u)2=t2 + 2t × u+u2.

(+t+) h+2hXt+t+2(+0Xu+u (T+h+t+u)2 == T2 + 2 T× h + h2 + 2 ( T2 + h) X t + t + 2(T+h+t) Xu + u2.

CUBING NUMBERS.

415. The following method of cubing will exhibit the law of involution, from which we can easily derive the process of evolution.

1. Find the cube of 25 by the analytical method.

equals 25X5 X 20, or 2 × 52 X 20, five times 202 equals 5×202. We next multiply by 20. Twenty times 52 equals 20 × 52, twenty times 2X5X20 equals 2×5×202, twenty times 202 equals 203. Taking the sum of these products and we have first 53; next, once 52 × 20 plus twice 52×20 equals three times 52×20; next, twice 5 × 202 plus once 5202 equals three times 5X 202, and next we have 203; hence 253203+3 X5 X 202 + 3 X 52 × 20+ 53. Therefore the cube of 25 equals the cube of the tens, plus three times the square of the tens into the units, plus three times the units into the square of the tens, plus the cube of the units.

Cube the following numbers analytically :—

416. We will now restate some of the principles derived from the above solutions, in the form of propositions.

PROP. I.-The cube of a number of tens and units equals the cube of the tens, plus three times the square of the tens into the units, plus three times the tens into the square of the units, plus the cube of the units.

PROP. II. The cube of a number of hundreds, tens, and units equals the cube of the hundreds, plus three times the square of the hundreds into the tens, plus three times the hundreds into the square of the tens, plus the cube of the tens, plus three times the square of the sum of the hundreds and tens into the units, plus three times the sum of the hundreds and tens into the square of the units, plus the cube of the units.

417. These principles may be expressed in an abbreviated form as follows: Let u= the units, t the tens, and h=the hundreds; then we have

=

(t+u)3 = t3 +3ť2 × u + 3t × u2 + u3.

(h+t+u)3=h3 + 3h2 × t + 3h × 12 + 1°3 + 3 (h + 1)2 × u +3 / (h+t) Xu2 + u3

EVOLUTION.

418. Evolution is the process of finding one of the several equal factors of a number.

419. One of several equal factors of a number is called a root of the number. Evolution may therefore be defined as the process of finding a root of a number.

420. The Square Root of a number is one of the two equal factors of that number; thus, 7 is the square root of 49, since 49=7×7

421. The Cube Root of a number is one of the three equal factors of that number; thus, 3 is the cube root of 27, since 27 3X 3X3.

422. The Fourth Root is one of the four equal factors, the fifth root is one of five equal factors, the sixth one of six equal factors, etc.

423. The Sign of evolution is ; thus, 64, or 1/64, denotes the square root of 64; 27 denotes the cube root of 27, etc. The little figure at the left of the symbol is called the index of the root.

424. When the number is a perfect power and the factors are easily found, the root of a number can be readily obtained by the following

RULE.-Resolve the number into its prime factors, and for the square root take one of every two equal factors, for the cube root take one of every three equal factors, etc.