Complete school algebra
Hawkes, Herbert E. (Herbert Edwin), 1872-1943 | Luby, William A. (William Arthur), 1872- | Touton, Frank C. (Frank Charles), 1880-1936
9 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 9 49 6902 6911 6920 6928 6937 6946 6965 6964 6972 6981 9 50 6990 6998 7007 7016 7024 7033 7042 7060 7059 7067 9 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 8 62 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 8 58 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 8 54 7324 7332 7340 7848 7366 7364 7372 7380 7388 7396 8 LOGARITHMS 496 N 0 1 2 8 4 5 6 7 8 9 D 65 7404 7412 7419 7427 7436 7443 7451 7459 7466 7474 8 66 7482 7490 7497 7505 7613 7520 7628 7636 7643 7551 8 67 7659 7566 7674 7682 7689 7597 7604 7612 7619 7627 8 58 7634 7642 7649 7657 7664 7072 7679 7686 7694 7701 69 7709 7716 7723 7731 7738 7746 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7826 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7946 7962 7959 7966 7973 7980 7987 63 7993 8000 8007 8014 8021 8028 8036 8041 8048 8065 64 8062 8069 8076 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8196 8202 8209 8216 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 6 68 8326 8331 8338 8344 8351 8357 8363 8370 8376 8382 6 69 8388 8396 8401 8407 8414 8420 8426 8432 8439 8445 6 70 8461 8467 8463 8470 8476 8482 8488 8494 8600 8506 6 71 8513 8519 8526 8531 8537 8543 8549 8555 8561 8667 6 72 8573 8579 8586 8591 8697 8603 8609 8615 8621 8627 6 73 8633 8639 8646 8651 8657 8663 8669 8675 8681 8686 6 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 6 76 8761 8756 8762 8768 8774 8779 8786 8791 8797 8802 6 76 8808 8814 8820 8826 8831 8837 8842 8848 8854 8859 6 77 8866 8871 8876 8882 8887 8893 8899 8904 8910 8915 6 78 8921 8927 8932 8938 8943 8949 8954 8960 8966 8971 6 79 8976 8982 8987 8993 8998 9004 9009 9016 9020 9025 6 80 9031 9036 9042 9047 9063 9058 9063 9069 9074 9079 6 81 9086 9090 9096 9101 9106 9112 9117 9122 9128 9133 6 82 9138 9143 9149 9154 9159 9165 9170 9176 9180 9186 6 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 6 84 9243 9248 9263 9258 9263 9269 9274 9279 9284 9289 5 85 9294 9299 9304 9309 9315 9320 9325 9330 9336 9340 5 86 9345 9350 9356 9360 9365 9370 9375 9380 9385 9390 5 87 9396 9400 9406 9410 9416 9420 9425 9430 9435 9440 5 88 9446 9450 9456 9460 9465 9469 9474 9479 9484 9489 6 89 9494 9499 9504 9609 9613 9518 9523 9528 9533 9638 6 90 9642 9547 9552 9567 9562 9566 9571 9576 9681 9586 6 91 9690 9595 9600 9606 9609 9614 9619 9624 9628 9633 5 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 6 93 9686 9689 9694 9699 9703 9708 9713 9717 9722 9727 6 94 9731 9736 9741 9746 9760 9754 9759 9763 9768 9773 6 95 9777 9782 9786 9791 9796 9800 9806 9809 9814 9818 6 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 6 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 4 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 4 99 9966 9961 9966 9969 9974 9978 9983 9987 9991 9996 4 496 COMPLETE SCHOOL ALGEBRA <:H-iw^cr-: Compute, using logaritluus : 1. Solution. log 376 = 2.5752. Log 376 =.8584. Divide by 8. Then antilog 8584 = 7.218. 2. -v^TSS. 3. -v^nsF. 4. -4^3421. 5. -^.000639. Solution. log.000639 = 1.8055. If one divided 4.8055 as it stands by 3, he would be likely to confuse the negative characteristic and the positive mantissa. This and other difficulties may easily be avoided by adding to the characteristic and subtracting from the resulting logarithm any integral multiple of the index of the root which will make the characteristic positive. Thus log.000639 = 2.8055 - 6. Log 376 =.9352 - 2 (dividing by 8)- Then antilog 2.9352 =.08614. 6. V.0756. 11. (-6.387) 14. *v^269. 7..07624. 12837 =,,, V^f^^. 8. 4^006679. a mi^f. 16. ^VWWI. 9. (d».4;. 1(43.56)'. 7.984 10. (4926)* y (T.623)' 17. -^^ •v'.07241. 18. Determine the logarithms of 5732, 573.2, 57.32, and 5.732 to the base 10 and to the base 100. Compare the results. What act about logarithms do these results emphasize? LOGAEITHMS 49T Note. The preceding four-place table will usually give results correct to one half of one per cent. Five-place tables give the mantissa to five decimal places of the numbers from 1 to 9999 and, by interpolation, the mantissa of numbers from 1 to 99,999. Such tables give results correct to one twentieth of one per cent, a degree of accuracy which is sufficient for most engineering work. Six-place tables give the mantissa to six decimals for the same range of numbers as a five-place table, but the labor of using a six-place table is not sufficient for most engineering work. Place table is much greater than that of using a five-place one. Seven-place tables contain the mantels of the numbers from 1 to 99,999. Such tables are needed in certain kinds of engineering work and are of constant use in astronomy. In place of a table of logarithms, engineers often use an instrument called a slide rule. This is really a mechanical table of logarithms arranged ingeniously for rapid practical use. Results can be obtained with such an instrument far more quickly than with an ordinary table of logarithms, and that without recording or even thinking of a single logarithm. A slide rule ten inches long usually gives results correct to three figures. In work requiring greater accuracy, a larger and more elaborate instrument which gives a five-figure accuracy is used. Exponential equations. An exponential equation is an equation in which the unknown occurs in an exponent. Many exponential equations are readily solved by means of logarithms, since log x x log a. Thus, let a = c. Then x log a = log c. Whence x = log c, log a. EXAMPLE Solve for x 8 = 324. Solution, log 8 = log 324, or ar log 8 = log 324. Whence x = 0 = 0 = 2.78+. Log 8 = 9031 The student must overcome his hesitation actually to divide one logarithm by another if, as here, it is necessary. COMPLETE SCHOOL ALGEBRA MISCELLANEOUS EXERCISES 1. Can you find the logaxithm of a negative number to a positive base? Explain. Find, without reference to the table, the  log) + 3 log (7 log) + 4 log (7 log) + 4 log (7 log) + 4 log (7 log) + 5 log (7 log) + 6 log (7 log) + 6 log (7 log) + 6 log (7  a circle whose circumference is 3281 centimeters. LOGARITHMS 499 21. The area of a circle is 1.5. (a) Find the area of a circle whose radius is 6.672 feet. (b) Find the radius of a circle whose radius is 67.37 square feet. 22. The area of the surface of a sphere is 4.5. (a) The radius of the earth is 3958.79 miles. Find its surface. (b) Find the length of the equator. (a) Find the radius of a sphere whose volume is 86 cubic feet. (b) Find the diameter of a sphere whose volume is 47 cubic inches. 24. If the hypotenuse and one leg of a right triangle are given, the other leg can always be computed by logarithms. In the adjacent figure, let a and c be given and x required. Then X = Vc^ — a^ = V(c + a) + i log(c — a). Whence log(o + a) + i log(c — a). The hypotenuse of a right triangle is 377 and one leg is 288. Find the other leg. The hypotenuse of a right triangle is 1286 and one leg is 924. Find the other leg. The area of an equilateral triangle whose side is 8 is J V3. Find in square feet the area of an equilateral triangle whose side is 34.23 inches. Solve for x : 26. 3^ = 25. 30. 3 = (1.04)*. 34. 1(F^ = 3. 27.64^ = 4. 31^ = 64. 35.8^+2 = 6. 28. 16^ = 1024. 32. 4^^ + ^ = 84. 36. (.3)"^ = 5. 29. (-2y = 64. 33. 3^ + » = 6561. 37. (07)* = 9. 600 COMPLETE SCHOOL ALGEBRA Find the number of digits in: 88. (a) 3"; (b) 2*«; (c) 2«. 3« • P. 89. In how many years will double itself at 3% interest compounded annually? Solution. At the end of one year the amount of $1 at 3% is $1.03; at the end of two years it is $(1.03) (1.03), or $(1.03); at the end of three years it is $(1.03), and at the end of two years it is $(1.03). If X is the number of years required, (1.03) = 2. Taking the logarithms of both members of the equation, X log 1.03 = log 2. ^^^^' '-bii:o3==:oi28-^^-^+- 40. In how many years will $1 double itself at 6^ interest compounded annually ? 41. In how many years will any sum of money treble itself at 4^ interest compounded annually ? 42. In how many years will $450 double itself at 3j^ interest compounded annually? 43. In how many years will $4000 amount to $7360.80 at 5^0 interest compounded annually ? 44. About 300 years ago the Dutch paid $24 for the island of Manhattan. At 4^ compound interest, what would this payment amount to at the present time ? 45. In how many years will $12 double itself at 3^ interest compounded semiannually? 46. Show that the amount of P dollars in t years at r^ interest compounded annually is P(l+ r)*; compounded semi- annually is p(l + -) ; compounded quarterly is p(H--) ; and compounded monthly is P f 1 -f r^ ) ' * In making computations of this nature by the aid of logarithms, care must be exercised not to retain more significant figures in the result than are given with accuracy by the process. LOGARITHMS 601 47. Find the amount of $5000 at the end of four years, interest at 4% compounded (a) annually; (b) semiannually; (c) quarterly. 48. Find the amount of $4.12 at the end of five years, interest at 4%, compounded quarterly. 49. Set up and solve the equation used to determine the amount which should be paid for a $5 certificate to be paid in five years, interest at 4^, compounded quarterly. Note. It is not a little remarkable that just at the time when Galileo and Kepler were turning their attention to the laborious computation of the orbits of planets, Napier should be devising a method which simplifies these processes. It was said a hundred years ago, before astronomical computations became so complex as they now are, that the invention of logarithms, by shortening the labors, doubled the effective life of the astronomer. Today the remark is well inside the truth. In the presentation of the subject in modern textbooks, a logarithms is defined as an exponent. But it was not from this point of view that they were first considered by Napier. In fact, it was not till long after his time that the theory of exponents was understood clearly enough to admit of such application. This relation was noticed by the mathematician Euler, about one hundred and fifty years after logarithms were invented. It was by a comparison of the terms of certain arithmetical and geometrical progressions that Napier derived his logarithms. They were not exactly like those used commonly today, for the base which Napier used was not 10. Soon after the publication of Napier's work, Henry Briggs, an English professor, was so much impressed with its importance that he journeyed to Scotland to confer with Napier about the discovery. It is probable that they both saw the necessity of constructing a table for the base 10, and to this enormous task Briggs applied himself. With the exception of one gap, which was filled in by another computer, Briggs's tables form the basis for all the common logarithms which have appeared from that day to this. The square roots and the cube roots on the following page are corrected to the nearest digit in the third decimal place. COMPLETE SCHOOL ALGEBRA No. Squares 1 1 2 4 3 9 4 16- 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225. 16 256 17 289 18 324 19 361 20 400 21 441 22 484 23 529 24 576 25 625 26. 676 27 729 28 784 29 841 30 900 31 961 32 1,024 33 1,089 34 1,156 35 1,225 36 1,296 37 1,369 38 1,444 39 1,521 40 1,600 41 1,849 44 1,936 45 2,025 46 2,116 47 2,209 48 2,304 49 2,401 60 2,500 Cubes 1 8 27 64 125 216 343 512 729 1,000 1,331 1,728 2,197 2,744 3,375 4,096 4,913 5,832 6,859 8,000 9,261 10,664 12,167 13,824 15,725 17,876 19,683 21,952 24,389 27,000 29,791 32,768 35,937 39,304 42,875 46,656 50,653 54,872 59,319 64,000 68,921 74,088 79,507 85,184 91,125 97,336 103,823 110,592 117,649 125,000 Square Roots 1,000 1,414 1,732 2,000 2,236 2,449 2,646 2,828 3,000 3,162 3,317 3,464 3,606 3,742 3,873 4,000 4,123 4,243 4,359 4,472 4,583 4,690 4,796 4,899 5,000 5,099 5,196 5,292 5,385 5.477 5.657 5.657 5.745 5.916 6.000 6.083 6.164 6.245 6.325 6.403 6.481 6.548 6.531 6.708 6.782 6.866 6.928 7.000 7.071 Cube Roots 1.000 1.260 1.442 1.587 1.710 1.817 1.913 2.000 2.080 2.080 2.154 2.224 2.289 2.351 2.410 2.466 2.520 2.571 2.621 2.668 2.714 2.759 2.802 2.844 2.884 2.884 2.924 2.962 3.000 3.037 3.072 3.072 3.107 8.141 3.175 3.208 3.240 3.271 3.302 3.370 3.370 3.430 3.430 3.460 3.560 3.657 3.605 3.634 3.669 3.684 No. 51 52 63 54 55 56 56 57 58 58 59 60 61 62 63 64 65 66 66 67 68 69 71 72 73 74 75 76 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Squares 2,601 2,704 2,809 2,916 3,025 3,136 3,249 3,364 3,481 3,600 3,721 3,844 3,969 4,096 4,225 4,356 4,824 4,761 4,900 5,041 5,184 5,329 5,476 5,776 5,929 6,084 6,241 6,400 6,671 6,724 6,889 7,056 7,225 7,921 8,100 8,281 8,464 8,836 9,025 9,216 9,409 9,604 9,801 10,000 Cubes 132,651 140,608 148,877 157,464 166,375 175,616 185,193 195,112 205,379 216,000 226,981 238,328 260,047 262,144 274,625 287,496 300,763 314,432 328,609 343,000 367,911 373,248 389,017 405,224 421,875 438,976 466,533 474,562 493,039 512,000 631,441 551,368 671,787 692,704 614,125 636,066 658,603 681,472 704,969 729,000 763,571 778,688 804,357 830,684 857,375 884,736 912,673 941,192 970,299 1,000,000 Square Roots 7,141 7.211 7.280 7.348 7.416 7.483  67, 93, 186, 249, 254, 261 Archimedes, 274 Arrangement, 75 Associative Law of Multiplication, 70, 71 Axes, X- and y-, 202, 319 Axiom, 39, 40, 176 Babylonians, 157 Bacon, Roger, 186 Base, 480, 485 Binomial, 35, 82 ; square of, 105 Binomial theorem, 465 ; extraction of roots by, 458 ; rth term of (a + b\ 460-461 Binomials, difference of two squares, 124, 297; powers of, 456; product of, 107, 109, 110; sum or difference of two cubes, 133,300 Braces, 64 Brackets, 64 Briggs, 481, 501 Cancellation, 149, 167 Cardan, 279 Cauchy, 452 Characteristic, 483 Check, 36, 42, 46, 46, 61 Circle, area, 16 ; circumference, 191 RB 603 Coefficient, 10 Coefficient, literal, 95 ; polynomial, 38, 96 Commutative Law of Multiplication, 70 Complex number, 387 Conjugate imaginaries, 390 Consequent, 463 Constant, 472, 474, 476 Constant term, 137 Coordinates, 203 Cube root, 11, 114, 251 Cubes, of numbers, 502 ; sum or difference of, 133 Cubic equation, 141, 275 Cylinder, surface, 191 ; volume, 16 Decimal point, 186, 249 Decimals, 249 ; equations containing, 184 ; repeating, 343 Degree, 75 ; of an equation, 137, 141 ; of a term, 75, 76 Denominator, lowest common, 154 Diagonal of rectangle, 248 Diophantos, 278, 279 Discriminant of a quadratic equation, 400 Distance, x- and y-, 202 Division, by logarithms, 491 ; rule for, 88, 91 Egyptians, 157 Ellipse, 408 Equation, 6, 39, 64, 95 ; graph of, 205 ; indeterminate, 217 ; in one unknown, 55 ; in several unknowns, 217 ; in two variables, 202 ; of condition, 54 ; of fourth degree, 275 ; of second degree, 137 ; of third degree, 141 ; root of, 56, 218, 264 ; solution of, 41 ; 604 COMPLETE SCHOOL ALGEBRA solving, 80 ; translation of prob- lem into, 60, 82. See also Linear, Quadratic, Cubic Equations, containing decimals, 184; containing fractions, 175, 180; incompatible, 221; indeterminate system of, 221; literal, 96, 190; literal, in two unknowns, 281; radical, 264. See also Systems, Simultaneous systems Euclid, 82, 157, 261, 884 Euler, 462, 601 Evolution, law of, 880; by logarithms, 498 Exponent, 9, 210; logarithms as an, 481; meaning of  of a cubic function, 862; of an equation, 205; of linear equation, 205, 206, 208; of a linear equation, 360; of a quadratic equation in two variables, 406 Graphical representation, of linear system in three unknowns, 289; of statistics, 210, 412 Graphical solution of an equation in one unknown, 864; the process of, 866 Graphical solution of a quadratic system in two variables, 409 Greeks, 93, 167 Harriot, 5 Highest common factor, 308 Hindus, 5,21, 28, 29, 157, 186, 261, 279 Hyperbola, 406 Hypotenuse, 247 Identity, 54; sign of, 56 Imaginaries, addition and substitution of, 387; conjugate, 890; definitions, 886; division of, 890; equations with imaginary roots, 892; factors involving, 394; mul- tiplication of, 38B; note on the use of, 394 Imaginary, 343 Imaginary roots, 866 Index, 11, 251 Infinite geometrical series, 448 Integer, 1, 260; consecutive, 44; consecutive even, 44; consecutive odd, 44; even, 44; odd, 44; positive, 27 Integral expressions, 96 Interest, 15, 186; simple, 187 Interpolation, 486 Inversion, 466 Involution, law of, 830; by logic, 462 INDEX 505 Irrational number, 260 Italians, 6 Kepler, 408, 601 Klein, Professor Felix, 364 Kronecker, Leopold, 27 La Place, 462 Leibnitz, Gottfried Wilhelm, 465 Lever, 193 Line, straight, 360 Linear and quadratic equations, systems of, 279 Linear equation, 137 In two variable ables, 208, 217, 224 Linear systems in three unknowns, 236 Literal coefficients, 96; equations with, 95, 190, 231, 277 Logarithms, definition of, 480; antilogarithm, 488 ; interpola- tion, 486 ; systems of, 481 ; table of, 494-495 Lowest common denominator, 154 Lowest common multiple, 152 Lowest terms, 149 Mantissa, 483, 491 Means, 464; arithmetical, 435; geometrical, 446 Member, left, 39 ; right, 39 Monomial, 33 ; cube root of, 114 ; square root of, 113 Monomial denominator in equa- tions, 176 Monomial factor, 116 Monomials, addition of, 33, 34; division of , 87 ; division of poly- nomials by, 89 ; multiplication of, 72 ; multiplication of poly- nomials by, 73; subtraction of, 49 Moors, 186 Morse, S.F.B., 276 Motion, uniform, 97, 192 Multiple, lowest common, 162 Multiplication, rules for, 71-74; rule for, by logarithms, 490 Napier, Sir John, 186, 462, 484, 601 Negative numbers, 19, 20, 210, 279. •^'^/t: . l^ '^''' /// -^ / '""^ / ^ f. ^ "hi X *. »'* '■♦ ^/.L* /^ ''^ ■^ ttrti /i uT7.(,^ '/ { 7/(< » ir» Tc: >- -' \''/ f < •iP^y^ ftw». i^ 4 J /<: / UUstf* ^ / L I H .- ^ L h^ • ^' \ /. 