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In practice, the answers to exercises like the preceding would be taken from an annuity table, which, at the present day, is always employed in the calculation of Contingent or Life annuities.

The principles involved in the construction of Life-annuity tables cannot be fully entered into here. It may be observed, however, that the probable duration of the life of any particular person is determined, by what is called the “Doctrine of Probabilities,” from such materials as are supplied by the records of births and deaths in different localities; those materials being considered in connexion with the person's age, health, habits, occupation, &c. For instance, it has been ascertained that, of 10,000 infants born alive in Great Britain and Ireland, onlyon an average6,460 1

[ 10 years
6,090
5,642

30
5,075

40
4,397
live to the age of

50
3,643

60 2,401

70 953

So 142

90

20

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Taking, therefore, the case of an ordinarily healthy person 30 years old, we find that his chances of living to the age of

40

5,0757 50

4,397 to 5,642 60

3,643 &c.

&c. So that,ABSOLUTE CERTAINTY being represented by unitythe probability of his reaching the age of

5075 40

5642 50

4397 is measured by the fraction

5642 60

3643

L5642

&c.

&c. In order to understand the use which is made of such fractions as these, let us suppose that the person whose case we are considering purchases a life annuity—to be paid in annual instalments of £50 each. Now, if there were an absolute

50

certainty of his being alive at the end of 10 years, the present value of the roth instalment—the rate of interest being, say, 3 per cent.-would be £

l'ozio; but as the payment of this instalment is contingent upon his living to claim it, and as the probability of his completing his 40th year is represented by 5075, it is evident that the present value of the 10th instalment 5642' not £

50
£
In like manner,

the present value of the 20th instalment is £

4397

; of the

is (n

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'I "

5642

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frozm* 3642

50 I'0330

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30th instalment, £

; &c. The sum of the present values, found in this way, of all the instalments which the annuitant might possibly live to claim, constitutes the present value of the life annuity.

The present value of an annuity falling due in annual instalments of £ I each, and continuing during the annuitan life, is technically spoken of as the value of that life.

DIFFERENT SYSTEMS OF NOTATION.

BINARY
TERNARY

five

It is by no means improbable that, instead of the DECIMAL system of notation, we should have a

r two) one if human beings were three QUATERNARY furnished with only four

>fingers each. QUINARY &c.

&c. In order to realize any of these systems, we have merely to return to the illustration employed at page 3, and suppose that each of the boys there referred to has exactly the number of fingers indicated by the “base” of the system.

Thus, if the boys had only two fingers each, a finger held up

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by the

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So that, in a BINARY system, the values of the digit I would be

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If the boys had only three fingers each, a finger held up by the Ist boy

one nut
2nd

three nuts
would represent
3rd

nine
4th

twenty-severly &c.

&c. So that, in a TERNARY system, the values of I would be

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- group of three

- twenty-seventh

UNIT

- third

me ninth

I

I

I I I If the boys had only four fingers each, a finger beld up by the Ist boy

one nut 2nd

four nuts would represent

sixteen 4th

sixty-four &c.

&c. So that, in a QUATERNARY system, the values of I would be

3rd ,

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It is unnecessary to proceed further with these illustrations. We

e see that, in any system similar in principle to the decimal, I would have the following values—if the base were represented by B:

-B4

B3

=UNITY

B4

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B

I

B

I

I

I

I

11 I 1 I 1 I 1 And by doubling, or trebling, or quadrupling, &c. these values, we should, of course, obtain the corresponding values of 2, of 3, or of 4, &c.—as the case may be.

The number of different digits in any system would be less by i than the base of the system; the base indicating the number of different figures, and one of those figures being, in every instance, a cipher.* Thus, as the figures employed in the decimal system are o, 1, 2, 3, 4, 5, 6, 7, 8, 9-so, the figures employed in a binary

o, ternary

O, I, 2 quaternary

O, 1, 2, 3 quinary

system would be O, I, 2, 3, 4 senary

0, 1, 2, 3, 4, 5 septenary

0, 1, 2, 3, 4, 5, 6 octary

0, 1, 2, 3, 4, 5, 6, 7 ronary

0, 1, 2, 3, 4, 5, 6, 7, 8 In a system having a higher base than 10, one or more new digits would be required. For instance: in an undenary, system [base=eleven] there should be a character for ten; in a duodenary system [base=twelve] two new characters would be necessary-one for ten, and one for eleven ; &c.

It is obvious that, in any system, the removal of the units' point (we can no longer say “ decimal” point) n places, to the right or left, would multiply or divide-as the case may bea combination of figures by B”; the base of the system being represented by B.

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* In the illustration at page 3, each boy is supposed, on finding all his fingers up, to “ begin again "_being relieved (so to speak) by his neighbour, who puts up one finger. If, therefore, the boys were furnished with B fingers each, no boy would, at the close of the reckoning, have more than B-I fingers up; so that, in a system of notation having B for base, the value of the highest digit would, in the units' place, be B-1.

The transposition of a number from the decimal to a different system, or vice versa, will be understood from the following examples :

a

EXAMPLE I.— Transpose 1398 from the decimal to the quinary system.

Dividing 1398 as often as possible by 5 (the base of the quinary system), and annexing the several remainders—begin- 5)1398 ning with the last—to the last

5) 279+3(units) quotient, we find that, in the quinary system, the given num

5)55+4 (groups each :5) berwould be 21043; i.e., 2 groups

5)11to (group = 5) each 54, I group =

5', [no 2+1 (group=58) group=52], 4 groups each = 5, and 3 units :

1398=279x5+3=(55X5+4) X5+3=55x52 +4X5+3= (11X5+0) x 5+45+3=11753+0x5? + 4x5+3=(2x5 +1)* 53+0x5'+4X 5+3=2X51+1 X 59+0x 5+4X5+3.

EXAMPLE II.-Transpose •864 from the decimal to the ternary system.

Here we have to convert a number-consisting of 8 tenths, 6 hundredths, and 4 thousandths-into thirds, ninths, twentysevenths, &c. This conversion is effected in pretty much .864 the same way as the conversion of a decimal of a pound

3 into shillings and pence. Multiplying .864 by 3 (the

2'592 base of the ternary system), we obtain 2 thirds, and a

3 decimal (-592) of a third. Multiplying this decimal by

1.776 3, we obtain i ninth, and a decimal (776) of a ninth.

3 Multiplying the decimal of a ninth by 3, we obtain 2 twenty-sevenths, and a decimal (328) of a twenty

2'328 seventh. Multiplying this last decimal by 3, we obtain

3 I eighty-first (nearly). So that, in the ternary system,

984 the given number would be -2121.

a

ExamPLE III.— Transpose 23456.54321 from the decimal to the septenary system.

The integral portion of this number must be converted into groups

of

seven, groups of forty-nine, groups of three hundred and forty-three, &c. each; and the decimal portion into sevenths, forty-ninths, three hundred and forty-thirds, &c. We therefore proceed as follows, and find that, in the septenary system, the given number would be 125246-354215:

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