SECTION X.

INVOLUTION AND EVOLUTION.

INVOLUTION.

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

609. A Power of a number is the product arising from using the number several times as a factor. The number itself is called the first power.

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

611, The Third Power of a number is the product obtained by using the number three times as a factor. Thus 64 is the third power of 4, since 4×4 × 4 = 64.

612. The Fourth Power of a number is the product obtained by using the number four times as a factor; the Fifth Power, five times as a factor, etc.

613. The Degree of a power is indicated by a small figure, called an exponent, placed at the right and a little above the number. Thus, 52 represents the 2d power of 5, 63, the third power of 6, etc.

614. The Exponent indicates how many times the number is used as a factor. Thus, 83 denotes that 8 is used as a factor three times; that is, 8x8x8, which equals 512.

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

PRINCIPLES.

1. A power of a number is obtained by using the number as a factor as many times as there are units in the degree.

2. The product of any two powers of a number equals a power of the number denoted by the sum of the exponents.

For, if we multiply the cube of a number by the 4th power of the number, we will evidently have the number used seven times as a factor, or the 7th power of the number; thus, 53 X 51 = (5 × 5 × 5) × (5 × 5 × 5 × 5) = 57; and the same may be shown in any other case.

3. A power of a number raised to any power equals a power of the number denoted by the product of the exponents. For, if we square the cube of a number, we will evidently use the number as a factor two times three times, or six times; thus, (53)2= 53 X 53, which, by Prin. 2, equals 56, and the same may be shown in any other case.

NOTE.-By means of this principle we can abbreviate the operation of involution; thus we can raise a number to the sixth power by squaring its cube, or to the 12th power by squaring its sixth power, or cubing its 4th power, etc.

MENTAL EXERCISES.

1. The cube of 4 equals 4 used how often as a factor?

2. How often is 6 used as a factor in finding the 5th power of 6? 3. How often is 5 used as a factor in the cube of the square of 5?

4. What power of 6 is 62 multiplied by 63 ?

5. If we multiply 73 by 74, what power of 7 shall we have?

6. What power of 4 is equal to 43 multiplied by 45?

7. What power of 8 is equal to 82 × 83 × 84?

8. What power of 2 is the square of the square of 23 ?

9. The square of a number equals 8 times that number; what is the number?

10. What number multiplied by 6 gives of the square of the number for a product?

11. What number multiplied by 16 gives of the square of the number for a product?

12. What fraction multiplied by equals of the square of the fraction?

13. What number multiplied by 12 and 9 gives of the cube of the number for a product?

WRITTEN EXERCISES.

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

5. Square 205 Ans. 42025. 9. Cube 99.

Ans. 970299

Ans. 1. 21. (15.5).

Ans. 49.

Ans. 57720.0625.

Ans. ()".

Ans. 85. 22. 42 × 43 × 44.

3

Ans. 79. 23. ()3×(†)*.

Ans. 128. 24. (2.5)a × (2.5) 6. Ans. (2.5)1o Ans. (1)5.25. (3.3)2 × (3.3)3. Ans. (3.3)*

SQUARING NUMBERS.

615. There are Two Methods of squaring numbers, called the Analytic or Algebraic, and the Synthetic or Geometrical.

616. The object of these methods is to find the law of forming the square, and thus prepare for corresponding methods of explaining Evolution.

NOTE.-Teachers who prefer the geometrical method of explaining evolution may allow pupils to omit explaining involution by the analytic method, and vice versa.

1. Find the square of 25 analytically and synthetically.

25=
25 =

125=
50 =

OPERATION.

20+5

20+5

5 × 20+ 52

202+

5 x 20

· 202 + 2 × (5 × 20) †ɔ̃3

ANALYTICAL SOL.-Twenty-five equals 20 plus 5, or 2 tens plus 5 units. Writing this as 20+5, and commencing at units to square, we have 5 times 5 equals 52, 5 times 20 equals 5X 20, 20 times 5 equal 5 × 20, 20 times 20 equals 202, and adding, we have 202† 2 × (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, which we find to be 625.

625 =

GEOMETRICAL SOL.-Let the line AB represent a length of 20 units, and BH, 5 units. Upon AB construct a square, the area will be 202 = = 400 square units. On the two sides DC and BC construct rectangles, each 20 units long and 5 broad, the area of each will be 5 X 20 100 and the area of both will be 2 x 100 200 square units. Now add the little square on CG, whose area is 52 = 25 square units, and the sum of the different areas, 400 200+25=625, is the area of a square whose side is 25.

=

NOTE. When there are three figures, after completing the second square as above, we must make additions to it, as we did to the first square. When there are four figures, there are three additions, etc.

617. The following principles derived from the above solutions are important, and should be committed to memory:

PRINCIPLES.

1. The square of a number of two figures, equals the TENS2+2 times TENS X UNITS+ UNITS2.

2. The square of a number of three figures equals HUNDREDS2+2 times HUNDREDS TENS+TENS2+2(HUNDREDS+

TENS) UNITS + UNITS 2.

618. These principles may also be expressed in symbols. Letu represent units figure, t tens, h hundreds, and T thousands, and a period between two letters denote their multiplication; then we have

(t+u)2 = t2+2t.u+u2.

(h+t+u)? =h2+2h.t+t2+2(h+t).u+? (T+h+t+u)2 = T2+2 T. h+h2+ 2(T+h).t + t2 + 2(T+h+t).u+u2.

CUBING NUMBERS.

619. There are Two Methods of cubing numbers, called the Analytic or Algebraic, and the Synthetic or Geometrical method.

620. The object of these methods is to find the law of forming the cube, and thus to prepare for corresponding methods of explaining Evolution.

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

258

252
25 =

OPERATION.

202+2x (5 x 20) +51

20+ 5

ANALYTICAL SOL.-Squaring 25 by the method already given, we have 202 + 2x (5 X 20)+52. We then multiply this by 20+5. Five times 52 equals 53, 5 times 2 × 5 × 20 equals 2× 5 × 5 × 20, or 2 x 52 x 20, five times 202 equals 5 X 202 We next multiply by 20. Twenty times 52 equals 20 × 52, twenty times 2 x 5 x 20 equals 2 x 5 x 202, twenty times 202 equals 203. Taking the sum of these products and we have first 53; next, once 52 × 20 plus twice 52 x 20 equals three times 52 x 20; next twice 5 X 202 plus once 5 X 202 equals three times 5 X 202, and next we have 208; hence 253= 203 +3 x 5 x 202 +3 x 52 x 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 tens into the square of the units, plus the cube of the

5 × 202 + 2 x 52 x 20 +53

20+2 x 5 x 202 + 52 × 20

203 + 3 x 5 x 202+3x 52 x 20-53

units.

2. Find the cube of 45 by means of the cubical blocks.

OPERATION.

403 = 64000 402 X 5 X 3=24000 40 × 52 X 3 = 3000 53= 125

Hence 453 91125

=

GEOMETRICAL SOL.-Let A, Fig. 1, represent a cube whose sides are 40 units, its contents will be 403: = 64000. To increase its dimensions by 5 units we must add, 1st, the three rectangular slabs, B, C, D, Fig. 2; 2d, the three corner pieces, E, F, G, Fig. 3; 3d, the little cube H, Fig. 4. The three slabs B, C, D, are 40 units long and wide and 5 units thick; hence their contents are 402 x 5 x 3 = 24000; the contents of the corner pieces, E, F, G, Fig. 3, whose length is 40 and breadth and thickness 5, equal 40 x 52 x 33000; and the contents of the little cube H, Fig. 4, equal 53= 125; hence the contents of the cube represented by Fig. 4 are 64000+24000+3000+125 = 91125.