In this table, unlike Table IV., amplitudes are all expressed as decimals of the mean value of the potential gradient for the corresponding season.
On p. 8, 10.502 is multiplied by 3.216, and the result found to be 33.77443 2; and on pp. 23 and 24 occur decimals not attached to integers, viz.
(1) he saw that a point or separatrix was quite enough to separate integers from decimals, and that no signs to indicate primes, seconds, &c., were required; (2) he used ciphers after the decimal point and preceding the first significant figure; and (3) he had no objection to a decimal standing by itself without any integer.
Briggs also used decimals, but in a form not quite so convenient as Napier.
He prints a bar under the decimals; this notation first appears without any explanation in his "Lucubrationes" appended to the Constructio.
We may take it to (say) 4 places of decimals; or we may suppose it to be taken to 1000 places.
This is the foundation of the use of recurring decimals; thus we can replace = s s = 1 o o /(' - 1 + 0 -)1 by .363636(=36/102 +36/ 104 +3 6 / 106), with an error (in defect) of only 36/(10 6.99).
The logarithms to base io of the first twelve numbers to 7 places of decimals are log 1 =0.0000000 log 5 log 2 = 0.3010300 log 6 log 3 =0.477 121 3 log 7 log 4 =0.6020600 log 8 The meaning of these results is that The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa.
John Thomson of Greenock (1782-1855) made an independent calculation of logarithms of numbers up to 120,000 to 12 places of decimals, and his table has been used to verify the errata already found in Vlacq and Briggs by Lefort (see Monthly Not.
To construct this table Briggs, using about thirty places of decimals, extracted the square root of io fifty-four times, and thus found that the logarithm of 1.00000 00000 00000 12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of this form (i.e.
To 276 places of decimals, and deduced the value of log e lo and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is Mo = 0-43429 44 81 9 03251 82765 11289 18916 60508 22 943 97 00 5 80366 65661 14453 78316 58646 4920-8870 77 47292 2 4949 33 8 43 17483 18706 106 74 47 6630-3733 64167 92871 58963 90656 92210 64662 81226 58521 27086 56867 03295 9337 0 86965 88266 88331 16360 773849 0514 28443 48666 76864 65860 85135 56148 212 34 87653 43543 43573 25 which is true certainly to 272, and probably to 273, places (Proc. Roy.
Taking, for example, the number 1.087678, the object is to destroy the significant figure 8 in the second place of decimals; this is effected by multiplying the number by 1- 08, that is, by subtracting from the number eight times itself advanced two places, and we thus obtain 100066376.
Multiply 970224 By The Year Of The Hegira, Cut Off Six Decimals From The Product, And Add 621.5774.
The following are approximate values of the arm in decimals of a foot :
That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.
As an arithmetical calculator he was not only wonderfully expert, but he seems to have occasionally found a positive delight in working out to an enormous number of places of decimals the result of some irksome calculation.
Sums and Differences of Decimals 6.7 77.
Product of Decimals 6.8 78.
On the other hand, in writing decimals, the sequence (of negative powers) is from left to right.
The points of the compass might similarly be expressed by numbers in a binary scale; but the numbers would be ordinal, and the expressions would be analogous to those of decimals rather than to those of whole numbers.
Again, in the use of decimals, it is unusual to give less than two figures.
If there is this tendency to adopt too as a basis instead of to, the teaching of decimals might sometimes be simplified by proceeding from percentages to percentages of percentages, i.e.
To add or subtract decimals, we must reduce them to the same denomination, i.e.
To multiply two decimals exactly, we multiply them as if the point were absent, and then insert it so that the number of figures after the point in the product shall be equal to the sum of the numbers of figures after the points in the original decimals.
In actual practice, however, decimals only represent approximations, and the process has to be modified (§ 111).
There was, however, no development in the direction of decimals in the modern sense, and the Arabs, by whom the Hindu notation of integers was brought to Europe, mainly used the sexagesimal division in the ' " "' notation.
Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations.
For instance, .143 represents correct to 3 places of decimals, since it differs from it by less than 0005.
Thus 143 and 14 286 represent respectively and i ° ° to the same number of places of decimals, but the latter is obviously more exact than the former.
If, for instance, the values of a and b, correct to two places of decimals, are 3 58 and 1 34, then 2 24, as the value of a - b, is not necessarily correct to two places.
2, as base, and take as indices the successive decimal numbers to any particular number of places of decimals, we get a series of antilogarithms of the indices to this base.
Thus, if we go to two places of decimals, we have as the integral series the numbers 1, 2, 4, 8,.
For elementary work the multiplicand may come immediately after the multiplier, as in D; the last figure of each partial product then comes immediately under the corre up to the multiplication of decimals and of approximate values of numbers, is to place the first figure of the multiplier under the first figure of the multiplicand, as in E; the first figure of each partial product will then come under the corresponding figure of the multiplier.
Expression in L and decimals of LI is usually recommended, but it depends on circumstances whether some other method may not be simpler.
A better method is to ignore the positions of the decimal points, and multiply the numbers as if they were decimals between I and i o.
Thus, to divide 8 5.9 by 3.14159 2 7 to two places of decimals, we in effect divide.
3.1416 is a little less than 3 + 7 - s o Recurring Decimals are a particular kind of series, which arise from the expression of a fraction as a decimal.