INVOLUTION.

257. If a number be multiplied by itself, the product is called the second power, or square of that number. Thus, 4×4 16: the number 16 is the second power or square of 4.

If a number be multiplied by itself, and the product arising be again multiplied by the number, the second product is called the 3d power, or cube of the number. Thus, 3×3×3=27: the number 27 is the 3d power, or cube of 3.

The term power designates the product arising from multiplying a number by itself a certain number of times, and the number multiplied is called the root.

Thus, in the first example above, 4 is the root, and 16 the square or 2d power of 4.

In the second example, 3 is the root, and 27 the 3d power or cube of 3. The first power of a number is the number itself.

258. Involution is the process of finding the powers of

numbers.

The number which designates the power to which the root is to be raised, is called the index or exponent of the power. It is generally written on the right, and a little above the root. Thus, 4 expresses the 2d power of 4, or that 4 is to be multiplied by itself once: hence,

4o 4x4 16.

For the same reason 33 denotes that 3 is to be raised to the 3d power, or cubed: hence,

33 3×3×3=27: we may therefore write

257. If a number be multiplied by itself once, what is the product called? If it be multiplied by itself twice, what is the product called? What does the term power mean? What is the root? What is the first power of a number?

258. What is Involution? What is the number called which designates the power? Where is it written? What is the exponent of the square of a number? Of the cube? Of the fourth power? How do you raise a number to any power?

4- 4 the 1st power of 4.

42=4×4 16 the 2d power of 4. 4-4X4X4= 64 the 3d power of 4. 4-4x4x4x4= 256 the 4th power of 4. 45=4×4×4×4×4=1024 the 5th power of 4.

Hence, to raise a number to any power,

&c.

Multiply the number continually by itself as many times less 1 as there are units in the exponent, and the last product will be the power sought.

259. We have seen that Involution teaches how to find the power when the root is given. Evolution is the reverse of Involution: it teaches how to find the root when the power is known. The root is that number which being multiplied by itself a certain number of times, will produce the given power.

The square root of a number is that number which being multiplied by itself once, will produce the given number.

The cube root of a number is that number which being multiplied by itself twice, will produce the given number.

259. What is Evolution? What does it teach? What is the root of a number? What is the square root of à number? What is the cube root of a number? Make the sign denoting the square root. How do you denote the cube root?

=

For example, 6 is the square root of 36, because 6×6 36; and 3 is the cube root of 27, because 3 × 3 × 3=27. The sign placed before a number denotes that its square root is to be extracted. Thus, √36-6. The sign √ is called the radical sign, or the sign of the square root.

When we wish to express that the cube root is to be extracted, we place the figure 3 over the sign of the square root: thus, 8=2 and √64-4, and 3 is called

the index of the root.

EXTRACTION OF THE SQUARE ROOT,

260. To extract the square root of a number, is to find a number, which being multiplied by itself once, will produce the given number. Thus,

Before proceeding to explain the rule for extracting the square root, let us first see how the squares of numbers are formed.

The first ten numbers are

1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Roots. 1 4 9 16 25 36 49 64 81 100 Squares. The numbers in the second line are the squares of those in the first; and the numbers in the first line are the square roots of the corresponding numbers of the second.

Now, it is evident that, the square of a number expressed by a single figure will not contain any figure of a

260. What is required when we wish to extract the square root of a number? What is the greatest square of a single figure? What is the highest order of units that can berived from the square of a single figure? How many pertect squares are there among the numbers less than one hundred? What is the square of a number expressed by two figures equal to? In what places of figures will the square of the tens be found? In what places will the product of the tens by the units be found?

higher order than tens; and also, that if a number contains three figures, its root must contain tens and units. The numbers 1, 4, 9, &c., of the second line, are called perfect squares, because they have exact roots. Let us now see how the square of any number may be formed, say the number 36. up of 3 tens or 30, and 6 units. Let the line AB represent the 3 tens or 30, and BC the six units.

F

Let AD be a square H on AC, and AE a square on the ten's line AB.

Then ED will be a square on the unit line 6, and the rectangle EF will be the product of HE which is equal to the ten's line, by IE which is equal to the unit line. Also, the

A

This number is made

30

I

D

30

6

6

6

rectangle BK will be the product of EB which is equal to the ten's line, by the unit line BC. But the whole square on AC is made up of the square AE, the two rectangles FE and EC, and the square ED. Hence,

The square of two figures is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.

Let it now be required to extract the square root of 1296.

Since the number contains more than two places, its root will contain tens and units. But as the square of one ten is one hundred, it follows that the square of the tens of the required root, must be found in the two figures on the left of 96. Hence, we point off 12 96(36 the number into periods of two figures each.

We next find the greatest square contained in 12, which is 3 tens or 30.

9.

66)396
396

We then square 3 tens which gives 9 hundred, and then place 9 under the hundred's place, and subtract.

This takes away the square AE and leaves the two rectangles FE and BK, together with the square ED on the unit line.

Now, since tens multiplied by units will give at least tens in the product, it follows that the area of the two rectangles FE and EC must be expressed by the figures of the given number at the left of the unit's place 6, which figures may also express a part of the square ED.

If, then, we divide the figures 39, at the left of 6, by twice the tens, that is, by twice AB or BE, the quotient will be BC or EK, the unit of the root.

Then, placing BC or 6, in the root, and also in the divisor, and then multiplying the whole divisor 66 by 6, we obtain for a product the two rectangles FE and CE, together with the square ED.

Hence, the square root 1296 is 36; or, in other words, 36 is the side of a square whose area is 1296.

CASE I.

261. To extract the square root of a whole number,

I. Point off the given number into periods of two figures each, counted from the right, by setting a dot over the place of units, another over the place of hundreds, and so on.

II. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of the root from the first period, and to the remainder bring down the second period for a dividend.

III. Double the root already found and place it on the left for a divisor. Seek how many times the divisor is contained in the dividend, exclusive of the right-hand figure,

261. What is the first step in extracting the square root of numbers? What the second? What the third? What the fourth? What the fifth? Give the entire rule.