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first and second significant figures, the number 1.29684 is found between 1.29664 = 2.18 and 1.29862 2.19. The first difference is 198; the second difference is 20; and 20 ÷ 198 = .101+, or .10. Hence, the first five significant figures are 21810; the number of figures in the integral part of the cube is 3 X 32 = 7; and 129.6843 2,181,000, correct to five sig

nificant figures.

=

=

(b) 7.64420 occurs between 7.64032 = 446 and 7.64603 447. The first difference is 571; the second difference is 388; and 388 ÷ 571 .679+, or .68. Hence, the first five significant figures are 44668; the number of ciphers following the decimal point is 3 X 0 = 0; and .764423.44668, correct to five significant figures.

=

=

(c) 3.2425 falls between 3.24278 34.1 and 3.23961 34.0. The first difference is 317; the second difference is 289; 289 317.911+, or .91. Hence, the first five significant figures are 34091; the number of ciphers following the decimal point is 3 X 1 + 1 = 4; and .0324253 = .000034091, correct to five significant figures.

α

RECIPROCALS.

1 b

The reciprocal of a number is 1 divided by the number. By using reciprocals, division is changed into multiplication, since a b = = ax The table gives the reciprocals of all numbers expressed with three significant figures to six significant figures. By proceeding in a manner similar to that just described for powers and roots, the reciprocal of any number correct to five significant figures may be obtained. The decimal point in the result may be located as follows: If the given number has an integral part, the number of ciphers following the decimal point in the reciprocal will be one less than the number of figures in the integral part of the given number; and if the given number is entirely decimal, the number of figures in the integral part of the reciprocal will be one greater than the number of ciphers following the decimal point in the given number. For example, the reciprocal of 3370 = .000296736 and of .00348 287.356.

When the number whose reciprocal is desired contains more than three significant figures, express the number to six significant figures (adding ciphers, if necessary, to make six figures) and find between what two numbers in the

1 n

column headed the significant figures of the given number

falls; then proceed exactly as previously described to determine the fourth and fifth figures.

EXAMPLE.

(a) The reciprocal of 379.426 =? (b)

SOLUTION. (a) 379426 falls between .378788

1

2.63

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1440;

379426 802; 802 1440

.380228 The first difference is 380228 - -378788 the second difference is 380228 .557, or .56. Hence, the first five significant figures are 26356, and the reciprocal of 379.426 is .0026356, to five significant figures.

1

1

2.13

and .467290 = 2.14

(b) .469200 falls between .469484 The first difference is 2194; the second difference is 284; 284

1

2194 = .129+, or .13. Hence, five significant figures.

2131.3, correct to

.0004692

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1.06

1.1236

1.07

1 1449

1.08

1.1664

1.09

1.1881

1.10

1.2100

1.01 1.0201 1.03030 1.00499 3.17805 1.00332 2.16159 4.65701 .990099 1.02 1.0404 1.06121 1.00995 3.19374 1.00662 2.16870 4.67233 .980392 1.03 1.0609 1.09273 1.01489 3.20936 1.00990 2.17577 4.68755 .970874 1.04 1.0816 1.12486 1.01980 3.22490 1.01316 2.18278 4.70267 .961539 1.05 1.1025 1.15763 1.02470 3.24037 1.01640 2.18976❘ 4.71769 .952381 1.19102 1.02956 3.25576 1.01961 2.19669 4.73262 .943396 1.22504 1.03441 3.27109 1,02281 2.20358 4.74746 .934579 1.25971 1.03923 3.28634 1.02599 2.21042 4.76220 .925926 1.29503 1.04403 3.30151 1.02914 2.21722 4.77686 .917431 1.33100 1.04881 3.31662 1.03228 2.22398 4.79142 .909091

1.11

1.2321

1.12

1.2544

1.13

1.2769

1.14

1.15

1 36763 1.05357 3.33167 1.03540 2.23070 4.80590
1.40493 1.05830 3.34664 1.03850 2.23738
1.44290 1.06301 3.36155 1.04158 2.24402
1.2996 1.48154 1.06771 3.37639 1.04464 2.25062
1.3225 1.52088 1.07238 3.39116

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1.81585 1.10454 3.49285 1.23 1.5129 1.86087 1.10905 3.50714 1.24 1.5376 1.90662 1.11355 3.52136 1.07434 2.31459 4.98663 .806452 1.25 1.5625 1.95313 1.11803 3.53553 1.07722 2.32080 5.00000 .800000 1.26 1.5876 2.00038 1.12250 3.54965 1.08008 2.32697 5.01330 .793651 1.27 1.6129 2.04838 1.12694 3.56371 1.08293 2.33310 5.02653 .787402 1.28 1.6384 2.09715 1.13137 3.57771 1.08577 2.33921 5.03968 .781250 1.6641 2.14669 1.13578 3.59166 1.08859 2.34529 5.05277 .775194 1.6900 2.19700 1.14018 3.60555 1.09139 2.35134 5.06580 .769231

2.29577 4.94609 .826446 1.06853 2.30208 4.95968 1.07144 2.30835 4.97319

.819672 .813008

1.35

1.36

1.37

1.38

1.39

1.40

1.31 1.7161 2.24809 1.14455 3.61939 1.09418 1.32 1.7424 2.29997 1.14891 3.63318 1.09696 1.33 1.7689 2.35264 1.15326 3.64692 1.09972 1.34 1.7956 2.40610 1.15758 3.66060 1.10247 1.8225 2.46038 1.16190 3.67423 1.10521 1.8496 2.51546 1.16619 3.68782 1.10793 2.38696 5.14256 .735294 1.8769 2.57135 1.17047 3.70135 1.11064 2.39280 5.15514 .729927 1.9044 2.62807 1.17473 3.71484 1.11334 2.39861 5.16765 1.9321 2.68562 1.17898 3.72827 1.11602 2.40439 5.18010 1.9600 2.74400 1.18322 3.74166 1.11869 2.41014 5.19249

2.35735 5.07875 .763359 2.36333 5.09164 .757576 2.36928 5.10447 .751880 2.37521 5.11723 .746269 2.38110 5.12993 .740741

.724638

.719425

.714286

1.44

1.45

1.41 1.9881 2.80322 1.18743 3.75500 1.12135 2.41587 5.20483 1.42 2.0164 2.86329 1.19164 3.76829 1.12399 2.42156 5.21710 .704225 1.43 2.0449 2.92421 1.19583 3.78153 1.12662 2.42724 5.22932 .699301 2.0736 2.98598 1.20000 3.79473 1.12924 2.43288 5.24148 .694444 2.1025 3.04863 1.20416 3.80789 1.13185 2.43850 5.25359 .689655

.709220

1.46 2.1316 3.11214 1.20830 3.82099 1.13445 2.44409 5.26564 1.47 2.1609 3.17652 1.21244 3.83406 1.13703 2.44966 5.27763 .680272 1.48 2.1904 3.24179 1.21655 3.84708 1.13960 2.45520 5.28957 .675676 1.49 2.2201 3.30795 1.22066 3.86005 1.14216 2.46072 5.30146 .671141 1.50 2.2500 3.37500 1.22474 3.87298 1.14471 2.46621 5.31329 .666667

.684932

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1.51

2.2801 3.44295 1.22882 3.88587 1.14725 2.47168 1.52 2.3104 3.51181 1.23288 3.89872 1.14978 2.47713 5.33680 1.53 2.3409 3.58158 1.23693 3.91152 1.15230 2.48255 5.34848 .653595 2.3716 3.65226 1.24097 3.92428 1.15480 2.48794 2.4025 3.72388 1.24499 3.93700 1.15729 2.49332

5.32507

.662252

.657895

5.36011 .649351

5.37169.645161

1.54 1.55 1.56 2.4336 1.57 2.4649 1.58 2.4964 1.59 2.5281 1.60 2.5600 1.61 2.5921 4.17328 1.26886 4.01248 1.17204 2.52508 5.44012 .621118 1.62 2.6244 4.25153 1.27:79 4.02492 1.17446 2.53030 5.45136 .617284 1.63 2.6569 4.33075 1.27671 4.03733 1.17687 2.53549 5.46256 .613497 1.64 2.6896 4.41094 1.28062 4.04969 1.17927 2.54067 5.47370 .609756 1.65 2.7225 4.49213 1.28452 4.06202 1.18167 2.54582 5.48481 .606061 1.66 2.7556 4.57430 1.28841 4.07431 1.18405 2.55095 5.49586 .602410 1.67 2.7889 4.65746 1.29228 4.08656 1.18642 2.55607 5.50688 .598802 1.68 2.8224 4.74163 1.29615 4.09878 1.18878 2.56116 5.51785 1.69 2.8561 4.82681 1.30600 4.11096 1.19114 2.56623 5.52877 1.70 2.8900 4.91300 1.30384 4.12311 1.19348 2.57128 5.53966 1.71 2.9241

3.79642 1.24900 3.94968 1.15978 2.49866 5.38321 .641026 3.86989 1.25300 3.96232 1.16225 2.50399 5.39469 .636943 3.94431 1.25698 3.97492 1.16471 2.50930 5.40612 .632911 4.01968 1.26095 3.98748 1.16717 2.51458 5.41750 .628931 4.09600 1.26491 4.00000 1.16961 2.51984 5.42884 .625000

.595238

.591716

.588235

5.00021 1.30767 4.13521 1.19582 2.57631 5.55050 1.72 2.9584 5.08845 1.31149 4.14729 1.19815 2.58133 5.56130 1.73 2.9929 5.17772 1.31529 4.15933 1.20046 2.58632 5.57205 1.74 3.0276 5.26-02 1.31909 4.17133 1.20277 2.59129 1.75 3.0625 5.35938 1.32288 4.18330 1.20507 2.59625 1.76 3.0976 5.45178 1.32665 4.19524 1.20736 2.60118 1.77 3.1329 5.54523 1.33041 4.20714 1.20964 2.60610 5.61467 1.78 3.1684 5.63975 1.33417 4.21900 1.21192 2.61100 5.62523 1.79 3.2041 5.73534 1.33791 4.23084 1.21418 2.61588 5.63574 1.80 3.2400 5.83200 1.34164 4.24264 1.21644 2.62074 5.64622

.584795

.581395

.578035

5.58277 .574713

5.59344.571429

5.60408 .568182

.564972

.561798

.558659 .555556

.552486

5.66705 .549451

1.81 3.2761 5.92974 1.34536 4.25441 1.21869 2.62558 5.65665 1.82 3.3124 6.02857 1.34907 4.26615 1.22093 2.63041 1.83 3.3489 6.12849 1.35277 4.27785 1.22316 2.63522 5.67741 .546448 1.84 3.3856 6.22950 1.35647 4.28952 1.22539 2.64001 5.68773 .543478 1.85 3.4225 6.33163 1.36015 4.30116 1.22760 2.64479 5.69802 .540541

5.72865 .531915

5.73879.529101

5.74890 .526316

1.86 3.4596 6.43486 1.36382 4.31277 1.22981 2.64954 5.70827 .537634 1.87 3.4969 6.53920 1.36748 4.32435 1.23201 2.65428 5.71848 .534759 1.88 3.5344 6.64467 1.37113 4.33590 1.23420 2.65900 1.89 3.5721 6.75127 1.37477 4.34741 1.23639 2.66371 1.90 3.6100 6.85900 1.37840 4.35890 1.23856 2.66840 1.91 3.6481 6.96787 1.38203 4.37035 1.24073 2.67307 5.75897 .523560 1.92 3.6864 7.07789 1.3-564 4.38178 1.24289 2.67773 5.76900 .520833 1.93 3.7249 7.18906 1.38924 4.39318 1.24505 2.68237 5.77900 .518135 1.94 3.7636 7.30138 1.39284 4.40454 1.24719 2.68700 5.78896 1.95 3.8025 7.41488 1.39642 4.41588 1.24933 2.69161 5.79889 1.96 3.8416 7.52954 1.40000 4.42719 1.25146 1.97 3.8809 7.64537 1.40357 4.43847 1.25359 3.9204 7.76239 1.40712 4.44972 1.25571 3.9601 7.88060 1.41067 4.46094 1.25782 2.70989 4.0000 8.00000 1.41421 4.47214 1.25992 2.71442

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2.01

2.02

4.0401 8.12060 1.41774
8.24241 1.42127
8.36543 1.42478

8.48966 1.42829
8.61513 1.43178

4.48330 1.26202 2.71893 5.85777 .497512 4.0804 4.49444 1.26411 2.72343 5.86746 .495050 2.03 4.1209 4.50555 1.26619 2.72792 5.87713 .492611 2.04 4.1616 4.51664 1.26827 2.73239 5.88677 .490196 2.05 4.2025 4.52769 1.27033 2.73685 5.89637 .487805 2.06 4.2436 8.74182 1.43527 4.53872 1.27240 2.74129 5.90594 .485437 2.07 4.2849 8.86974 1.43875 4.54973 1.27445 2.74572 5.91548 .483092 2.08 4.3264 8.99891 1.44222 4.56070 1.27650 2.75014 5.92499 .480769 2.09 4.3681 9.12933 1.44568 4.57165 1.27854 2.75454 5.93447 .478469 2.10 4.4100 9.26100 1.44914 4.58258 1.28058 2.75893 5.94392 .476191 9.39393 1.45258 4.59347 1.28261 2.76330 5.95334 .473934 9.52813 1.45602 4.60435 1.28463 2.76766 5.96273 .471698 9.66360 1.45945 4.61519 1.28665 2.77200 5.97209 .469484 9.80034 1.46287 4.62601 1.28866 2.77633 5.98142 .467290 9.93838 1.46629 4.63681 1.29066 2.78065 5.99073 .465116 10.0777 1.46969 4.64758 1.29266 6.00000 .462963 10.2183 1.47309 4.65833 1.29465 2.78924 6.00925 .460830 2.18 4.7524 10.3602 1.47648 4.66905 1.29664 2.79352 6.01846 .458716 2.19 4.7961 10.5035 1.47986 4.67974 1.29862 2.79779 6.02765 .456621 2.20 4.8400 10.6480 1.48324 4.69042 1.30059 2.80204 6.03681 .454546 2.21 2.22 2.23

2.11 2.12 2.13

4.4521

2.14

2.15

4.4944
4.5369
4.5796
4.6225

2.16 4.6656 2.17 4.7089

2.78495

2.24

4.9284
4.9729
5.0176

2.25 5.0625

2.26 5.1076

4.8841 10.7939 1.48661 4.70106 1.30256 2.80628 6.04594 .452489 10.9410 1.48997 4.71169 1.30452 2.81051 6.05505 .450451 11.0896 1.49332 4.72229 1.30648 2.81472 6.06413 .448431 11.2394 1.49666 4.73286 1.30843 2.81892 6.07318 .446429 11.3906 1.50000 4.74342 1.31037 2.82311 6.08220 .444444 11.5432 1.50333 4.75395 1.31231 2.82728

6.09120.442478

2.39

2.27 5.1529 11.6971 1.50665 4.76445 1.31424 2.83145 6.10017 .440529 2.28 5.1984 11.8524 1.50997 4.77493 1.31617 2.83560 6.10911 .438597 2.29 5.2441 12.0090 1.51327 4.78539 1.31809 2.83974 6.11803.436681 2.30 5.2900 12.1670 1.51658 4.79583 1.32001 2.84387 6.12693.434783 2.31 5.3361 12.3264 1.51987 4.80625 1.32192 2.84798 2.32 5.3824 12.4872 1.52315 4.81664 1.32382 2.85209 2.33 5.4289 12.6493 1.52643 4.82701 1.32572 2.85618 2.34 5.4756 12.8129 1.52971 4.83735 1.32761 2.86026 6.16224 2.35 5.5225 12.9779 1.53297 4.84768 1.32950 2.86433 6.17101 2.36 5.5696 13.1443 1.53623 4.85798 1.33139 2.86838 6.17975 2.37 5.6169 13.3121 1.53948 4.86826 1.33326 2.87243 6.18846 .421941 2.38 5.6644 13.4813 1.54272 4.87852 1.33514 2.87646 5.7121 13.6519 1.54596 4.88876 1.33700 2.88049 6.20582 .418410 2.40 5.7600 13.8240 1.54919 4.89898 1.33887 2.88450 6.21447 .416667 2.41 5.8081 13.9975 1.55242 4.90918 1.34072 2.88850 6.22308 .414938 2.42 5.8564 14.1725 1.55563 4.91935 1.34257 2.89249 6.23168 .413223 2.43 5.9049 14.3489 1.55885 4.92950 1.34442 2.89647 6.24025 .411523 2.44 5.9536 14.5268 1.56205 4.93964 1.34626 2.90044 6.24880 .409836 2.45 6.0025 14.7061 1.56525 4.94975 1.34810 2.90439 6.25732 .408163

6.13579 .432900 6.14463.431035

6.15345 .429185

.427356

.425532

.423729

6.19715.420168

2.46 6.0516 14.8869 1.56844 4.95984 1.34993 2.47 6.1009 15.0692 1.57162 4.96991 1.35176 2.48 6.1504 15.2530 1.57480 4.97996 1.35358 2.49 6.2001 15.4382 1.57797 4.98999 1.35540 2.50 6.2500 15.6250 1.58114 5.00000 1.35721

2.90834 6.26583 .406504 2.91227 6.27431 .404858 2.91620 6.28276 .403226 2.92011 6.29119 .401606 2.92402 6.29961 .400000

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