255. Approximating Square Root. The first step in approximating a square root is to estimate a first approximation.

Example. Approximate the square root of 79.

Solution. 82 = 64 is less than 79, and 92

=

81 is greater than 79. Hence, √79 lies between 8 and 9. Moreover, it is considerably nearer 9 than 8. Hence we take 8.8 as a first approximation.

8.98

Dividing 79 by 8.8, the quotient is 8.98 (nearly). Hence the root lies between 8.80 and 8.98. We take half the sum of these, or 8.89 as a second approximation.

8.8)790 704

860

792

680
8.886

8.89)79000

7112

7880

7112

7680

7112

5680

Dividing 79 by 8.89, the quotient is 8.886 (nearly). Hence the root lies between 8.890 and 8.886. Half the sum of these is 8.888, which is correct to three decimal places.

5334

256. Rule for Finding Square Root. The process of approximating a square root is described in the following rule:

(1) Estimate a first approximation, and divide the number by it, obtaining a first quotient.

(2) Use half the sum of the first approximation and first quotient as a second approximation, and divide the number by it.

(3) Use half the sum of the second approximation and the second quotient as a third approximation, and divide the number by it.

(4) Continue this process until the required closeness of approximation has been obtained.

The root always lies between the last divisor and the last quotient.

WRITTEN EXERCISES

Find each of the following accurate to two places of decimals:

257. Finding First Approximation of Root. The most difficult step in approximating a square root is to make a close first estimate of the root.

Example 1. Estimate the square root of 385.

Solution. The root lies between 19 and 20, because 385 is greater than 361, and less than 202 400. Further, 385 is somewhat nearer 400

192 =

=

than 361. Hence we use 19.6 as a first approximation.

Example 2. Estimate the square root of 741.

Solution. The root lies between 27 and 28 because 741 is greater than 729, and less than 282 784. Further, 741 is considerably nearer 729

=

272 =
than it is to 784. Hence we use 27.2 as a first approximation.

It makes little difference whether we use 27.2 or 27.3 as the first approximation.

Divide 741 by 27.2. The quotient is 27.24. gives 27.22 as a second approximation.

This

If we use 27.3

as a first approximation, we get a quotient 27.14 and this likewise gives 27.22 as a second approximation.

1160

This method for approximating square roots has the following advantages over the usual method.

1. It can be understood in a very few minutes.

2. It forces on the mind the essential nature of an approximated root.

3. It gives constant drill in estimating roots.

4. The operation itself is equally as short as the other.

WRITTEN EXERCISES

Find the square root of each of the following correct to two places

258. Rule for Finding First Approximation. To estimate a first approximation to the square root of a larger number, we make use of the following:

102

100, and 1002

=

10,000. Hence the square root of any

number between 100 and 10,000 lies between 10 and 100.

Again, 10002 = 1,000,000. Hence the square root of any number between 10,000 and 1,000,000 lies between 100 and 1000.

Following is a rule for estimating a first approximation to the square root of a number.

(1) Divide the number into groups of two digits in each, beginning at the right side of the number or at the decimal point if there is one.

(2) Estimate the root of the last group to the left, and annex enough zeros to make one digit for each group.

Example. Find the square root of 578,946, correct to two decimals.

Solution. We estimate the root of 57, the left group in 57,89,46 to be 7.6. Annex one zero and remove the decimal point, obtaining 760. 578946 760

=

761.77. Hence 760.88 is the second approximation. 578946 ÷ 760.88 760.890. Hence 760.8850 is the third approximation. This is certainly accurate to three places of decimals.

=

To approximate the square root of a decimal, divide it into groups of two digits, beginning at the decimal point, and going toward the right. Estimate the root of the first group containing figures other than zero, and then prefix one zero for each group preceding this one, placing a decimal point to the left of the last one.

Thus, to estimate the square root of .000746, we write .00,07,46, and estimate the root of 7 as 2.7. Then we prefix one zero, obtaining .027 as a first approximation.

9. Find the length in rods of a side of a square piece of land

containing one acre (160 square rods).

CHAPTER XIX

MENSURATION

259. Uses of Mensuration. Mensuration (indirect measurement, see § 262) is of constant use in many trades, professions, and in many kinds of business. The manifold and frequent uses of mensuration will appear throughout this chapter. It is one of the most important parts of business arithmetic.

260. There are two kinds of measurement, namely direct measurement and indirect measurement.

261. Direct Measurement. Direct measurement consists in applying a unit to a magnitude of the same kind to see how many times the unit is contained in it.

Thus, we may measure the length of an object by applying a foot measure or a yard measure directly to the object to see how many times the unit is contained in its length. Similarly, we may measure the contents of a milk can by finding how many times a quart measure may be filled from it.

262. Indirect Measurement. Indirect measurement consists in measuring a quantity by any means other than the direct application of the unit of measure to the quantity to be measured.

Thus, we measure the area of a rectangle by taking the product of its length and width. The unit of measure (the square foot or the square rod) is not applied at all to the quantity to be measured. Similarly, the volumes of bins, cisterns, and tanks are obtained indirectly by measuring the dimensions (length, width, depth) by means of a linear unit, and then computing the volume.

263. Definition of Mensuration. - Mensuration is the art of indirect measurement.

264. Measurement of Lengths.

Indirect measurement of length

is, in general, more difficult than indirect measurement of area or volume. In this book, the only method given for measuring length indirectly is the one given in §§ 270, 271.

265. Angles. Two lines having a common end point are said to

form an angle (Figure 1).

If two lines cut each other so that all four angles formed are equal, then each of these angles is called a right angle (Figure 2).

Figure 1.

Figure 2.

Figure 3.

266. Rectangles, Squares. A four-sided figure which has four right angles is called a rectangle (Figure 3).

If a rectangle has all its sides equal, it is a square.

AREAS

267. Area of a Rectangle. It is seen by inspection of the figure that the number of unit squares contained in a rectangle is equal to the product of its length and width.

Thus, if the rectangle is five inches long and four inches wide, it contains 4 x 5 square inches, since there are 4 rows with 5 square inches in each.

The general rule may be stated as follows:

To use this rule, the length and the width must be measured in the same unit.

If the length and width are measured in feet, the area will be given in square feet; if the length and width are measured in rods, the area will be given in square rods; etc.

If the length and width are given in different units, they must be reduced to the same unit before finding the area.