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(Gottfried Wilhelm von Leibnitz, 1646–1716)

Celebrated for his interest and ability in all branches of learning, especially in the fields of mathematics and philosophy. A contemporary of Newton and regarded as sharing with him the honor of inventing the Calculus.

The word radical is used in connection with other roots than square roots. Thus, V10 means the cube root of 10; that is, the number which when used as a factor three times gives 10. Similarly, 6 means the fourth root of 6, etc. All such numbers represent perfectly definite magnitudes, as did √2 in Fig. 74, yet we cannot express them exactly by means of decimals.

In general, the nth root of any number a is written Va, and this is known as a radical of the nth order. The number n is here called the index of the root, and the number a itself is called the radicand.

When no index is expressed, the index 2 is understood. Thus, √3 means

3.

These definitions apply also to algebraic expressions. Thus, 3 xy2 and V9x2y2 are both radicals, although 9 x2y2 is a perfect square, thus making it possible to write √9x2y2=3xy.

EXERCISES

In the following list, pick out those values that can be expressed exactly without radicals, and those that cannot. For each state the index and the radicand.

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of a radical correct to two or more places of decimals usually calls for a rather long process, as we have seen in § 135,

where we found the value of 550. For this reason, the radicals which are needed most in ordinary life (that is, those coming from square roots and cube roots) have been carefully worked out and placed together in a Table at the end of this book, on p. 314. For the sake of completeness, the squares and cubes of the numbers are also shown, thus making the table very convenient for all kinds of mathematical work. Just how to use the table is described on page 311, which the pupil should now read carefully. Below are a few illustrative examples:

EXAMPLE 1. Find √7 from the table.

SOLUTION. By the top number in the third column of p. 326 (Table) we see that √7=2.64575, this value being correct to 5 decimal places.

EXAMPLE 2. Find 7 from the table.

SOLUTION. By the top number in the sixth column of p. 326 we have 7=1.91293, this value being correct to 5 decimal places. EXAMPLE 3. Find

70.

SOLUTION. By the top number in the fourth column of p. 326 we have √70=8.36660+. The sign which we have placed over the last digit indicates that the number is correct only up to the last decimal place.

EXAMPLE 4. Find 70.

SOLUTION. 70-4.12129+ from 7th column top of p. 326.
EXAMPLE 5. Find 700.

SOLUTION. 700-8.87904+ by top of 8th column, p. 326.

141. Artisan's Method for Finding Square Root. The square root of a number may be easily calculated correct to several places of decimals in case one knows its value correct merely to the first or second decimal place. For example, knowing that √2=1.41+, suppose we wish to calculate √2 to a greater degree of accuracy. We first

divide the 2 by the 1.41, carrying out the work to several places of decimals, as indicated below.

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Now, since 1.41 was known to be slightly less than the true root, it follows that our quotient, 1.41851, is slightly greater than the true root. In other words, the true root itself lies somewhere between these two numbers. So we now calculate the number which lies halfway between them (their average) by taking half their sum. Thus,

1.41 1.41851 22.82851 1.41426

This average, or 1.41426, gives us a very close approximation to the true root. In fact, our answer in the present instance is the correct value of √2 to four places of decimals.†

† In general, if the number whose square root is to be found is greater than one, the new result will be correct at least to twice as many decimal places as the first estimate. The first estimate need not be at all exact, however. If any arbitrary number is taken as the first estimate, the process will give accurate answers after a sufficient number of repetitions.

Similarly, this method may be used to secure a good approximation for the square root of any number in case the value of the root is already known correct to only one or two decimal places. If still greater accuracy is desired, simply repeat the same process.

Because of the convenience and simplicity of this process, especially in rapid calculations such as are made by engineers, mechanics, or other workmen, it is commonly known as the "artisan's method."

EXERCISES

Using the tables, find the approximate values of each of the following quantities.

1. √5. 2. √50. 3. V5. 4. 50. 5. 500.

6. √51.

[HINT. 51=5.1X10. So use the information given for 5.10 on page 322 of Table.]

7. 51. 8. √82. 9. 820. 10. √7.7 11. √7.71 12. √77.1 13. √771.

SOLUTION. 771=7.71×100. Therefore 771 is the same as √7.71 except that the decimal point in the root must be moved one place farther to the right. (See p. 311.) Now, √7.71=2.77669+, so it follows that √771=27.7669+. Ans.

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