connexion with the person's age, health, habits, occupation, &c. For instance, it has been ascertained that, of 10,000 infants born alive in the United Kingdom, only— on an average

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

So that, ABSOLUTE CERTAINTY being represented by unity, the probability of his reaching the age of

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

3643

5642

&c.

paid in annual instalments of £50 each. Now, if there were an absolute certainty of his being alive at the end of 10 years, the present value of the 10th instalment-the rate of interest being, say, 3 per cent. per annum-would be

; but as the payment of this instalment is con1.0310 tingent upon his living to claim it, and as the probability of his completing his 40th year is represented by 5075,

5642'

it is evident that the present value of the 10th instal

ment

is (

(not £1-5310, but)

5075 of £50

5642 10310°

In like manner, the present value of the 20th instalment is

£ 50 4397; of the 30th instalment, £50 3643

&c.

X ;of

103205642

X

103305642

;

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 1 each, and continuing during the annuitant's life, is technically spoken of as the VALUE of that life.

DIFFERENT SYSTEMS OF NOTATION.

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

In order to realise any of these systems, we have merely to return to the illustration employed at pp. 3-4, 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 by the—

So that, in a BINARY system, the values of the digit 1

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

So that, in a TERNARY system, the values of 1 would be—

If the boys had only four fingers each, a finger held up

by the

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

It is unnecessary to proceed further with these illustrations. We see that, in any system similar in principle to the DECIMAL, I would have the following values-if the base were represented by B:

And by doubling, trebling, quadrupling, &c., these values, we should, of course, obtain the corresponding values of 2, 3, 4, &c., respectively.

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 0, 1, 2, 3, 4, 5, 6, 7, 8, 9: so, the figures employed in a—

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=12] 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 divideas the case may be-a combination of figures by B"; the base of the system being represented by B.

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

EXAMPLE I.-Transpose 1398 from the Decimal to the Quinary system.

* In the illustration at pp. 3-4, each boy is supposed, on finding all his fingers up, to "begin again"-being relieved (so to say) 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-1 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.

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