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$6 $6 $6 $6 $6 $6 $6 $6 $6

$6 $6 $6 $6 $6 $6

But the cash value of the first annuity is $100, and the cash value of the second is (188) of $100.

... The cash value of an annuity of $6 beginning now, running for three years, is $100 – (188)3 of $100 or {1-(188)'} of $100.

It will be easy to obtain a general formula, and also to find the value of a deferred annuity running for a definite number of years.

CHAPTER XVI.

EVOLUTION.

Square Root.-The product of 3 and 3 is 9; of 5 and 5 is 25. The measures of squares whose sides measure 3 and 5 are 9 and 25. We say that 9 is the square of 3, and that 25 is the square of 5; 3 is the square root of 9, and 5 the square root of 25. The square of 3 is written 3', the square root of 3 is expressed thus: 3. The pupil can write at once the table of squares:

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He will note that the square of any number expressed by one digit is a number expressed by one digit or by two digits, while the lowest number expressed by two digits-viz., 10-has for its square 100, a number expressed by three digits.

It is plain that the square of any number expressed by two digits has for its square a number expressed by three digits or by four digits. Also the square root of a number* expressed by three digits is a number expressed by two digits, and the tens digit is known from the first digit on the left; for example, 625 (if it has an exact square root), lying as it does between 400 and 900, will have for square root a number lying between 20 and 30-that is, the tens figure of the root will be 2. Similarly, if a number is expressed by four digits its square root is expressed by two digits, and the tens digit of the root can be determined from the first two digits (to the left) of the number; thus the square root of 2709-a number lying between 2500 and 3600—will have 5 for a tens digit, and this is determined by the 27 of the number 2709.

Write next the table of squares:

10' = 100

20' = 400

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Now take 13 and square it: the result is 169. We wish to arrive at a method of recovering 13 from 169.

* In general, when we speak of the square root of a number, we suppose that it has an exact square root.

To do this we shall examine how the 169 is formed from the 13:

13

13

9

30

30

100

169

Thus 13, which is made up of two parts, 10 and 3, has for its square a number 169, which is seen to be made up of 100, the square of 10; 9, the square of 3; and twice the product of 10 and 3. This is familiar to the pupil who has worked algebra, and may be illustra

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of the root, so that we know one of the parts of the root, viz., 10. The square of this part is 100, and the rest of the given number, 69, must be 2 times 10,

multiplied by the other part, together with the square of the other part.

169/10

100

69

Accordingly, if we multiply 10 by 2 and divide 69 by this product we get a clue to the other part. Dividing 69 by 2 × 10, or 20, we see that the quotient is a little greater than 3; if, then, after taking 3 times 20 from 69 there is left the square of 3, we have the root. Plainly this is the case:

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Now, this work might be written somewhat more neatly, thus:

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It may be further simplified by leaving out unnecessary zeros, thus:

169 13

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The pupil is now in a position to find the square root of all numbers expressed by three or four digits. It would be well, before considering the square root of

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