D Ө B C A NOTE 1. A Circle is a figure bounded by a curved line, all parts of the curve being equally distant from the center of the circle. The Circumference is the curve which bounds the circle. An Arc is any portion of the circumference, as A B or B D. An arc equal to a quarter of the circumference, or 90°, is called a quadrant. A Radius is a line drawn from the center to the circumference, as CA or C B. A Diameter is a line drawn through the center and limited by the curve, as A D. Ex. 1. How many seconds in 5s. 25° 48′ 54′′? Ex. 2. Reduce higher denominations. 632934" to 60) 10548 +54′′ 30) 175°48′ 3. Reduce 9s. 20° 55′ 47′′ to seconds. 5 s. +25° Ans. 5s. 25° 48′ 54′′. Ans. 1047347". 4. In 7484925" how many circumferences, signs, etc.? 5. In 3 quadrants, 10° 8′ 5′′ how many seconds? 6. Reduce 984627" to quadrants, degrees, etc. MISCELLANEOUS TABLE. · 110. This table embraces a few terms in common use, and 109. What is a Circle? Circumference? Arc? Quadrant? Radius? Diameter? Ex. 1. How many dozen bottles, each bottle holding 1qt. 1pt. 3gi. will be sufficient to bottle 61gal. 3qt. 1pt. of wine? 2. How many sheets of paper in 3 reams, 18 quires, and 23 sheets? MISCELLANEOUS EXAMPLES IN REDUCTION. 1. Reduce 27£. 14s. 6d. 3qr. to farthings. 3. Reduce 7t. 14cwt. 2qr. 12 lb. 8oz. 6dr. to drams. 9. Reduce 14 lb. 7oz. 15dwt. 23gr. to grains. 10. Reduce 61b 43 33 19 6gr. to grains. 11. Reduce 2548 square inches to higher denominations. 12. Reduce 411 nails to quarters and yards. 13. Reduce 7432 farthings to pence, etc. 14. Reduce 18469874 drams, Avoirdupois, to ounces, etc. 15. Reduce 54896 grains to pennyweights, etc. 16. Reduce 4sq. m. 25a. 3r. 34sq. rd. to square rods. 17. Reduce 8c. yd. 1727c. in. to cubic inches. 18. Reduce 4sq. yds. to square inches. 19. Reduce 4gal. 1pt. to gills. 20. Reduce 2wk. 6d. 8h. 16sec. to seconds. 21. Reduce 4m. 7fur. 39rd. to rods. 22. Reduce 3795 rods to furlongs, etc. 23. Reduce 17yd. 2qr. 3na. to nails. 24. Reduce 10881 links to miles, furlongs, etc. 29. Reduce 310556 square rods to roods, acres, and miles. NOTE. This subject will receive further attention in the articles on Fractions. DEFINITIONS AND GENERAL PRINCIPLES. 111. All numbers are even or odd. An EVEN NUMBER is a number that is divisible by 2 (Art. 74); as 2, 4, 8, 12. An ODD NUMBER is a number that is not divisible by 2; as 1, 3, 5, 11, 19. 112. All numbers are prime or composite. A PRIME NUMBER is a number that is divisible by no whole number except itself and one; as 1, 2, 3, 5, 7, 11, 19. NOTE 1. Two is the only even prime number, for all even numbers are divisible by 2. NOTE 2. Two numbers are mutually prime (i. e. prime to each other) when no whole number but one will divide each of them; thus, 8 and 9 are mutually prime, although neither 8 nor 9 is absolutely prime. A COMPOSITE NUMBER is a number (Art. 61) that is divisible by other numbers besides itself and one; thus, 6 is composite, because it is divisible by 2 and by 3; 12 is composite, because it is divisible by 2, 3, 4, and 6; 25 is composite, because it is divisible by 5 and 5. NOTE 3. A composite number that is composed of any number of EQUAL factors is called a power, and the equal factors are called the roots of the power; thus, 9, which equals 3X3 is the second power or square of 3, and 3 is the second or square root of 9; 64, which equals 4 X 4 X 4, is the third power or cube of 4, and 4 is the third or cube root of 64. NOTE 4. The power of a number is usually indicated by a figure, called an index or exponent, placed at the right and a little above the number; thus, the second power or square of 4 is written 42, which equals 4 X 4 = 16; the third power or cube of 4 is 43, which equals 4 × 4 × 4 = 64. NOTE 5. A ro t may be indicated by the radical sign, ✅; thus, ✅✔9 indicates the second or square root of 9, which is 3. So 3/8 indicates the third or cube root of 8, which is 2. The square root of a number is one of its two equal factors; the cube root is one of the three equal factors of the number. NOTE 6. Every number is both the first power and the first root of itself. 111. What is an Even Number? An Odd Number? 112. A Prime Number? What is the only even prime number? When are numbers mutually prime? What is a Composite Number? A power? A root? How is a power indicated? A root? A number is what power of itself? What root? FACTORING NUMBERS. 113. The FACTORS of a number are those numbers whose continued product is the number; thus, 3 and 7 are the factors of 21; 3 and 6, or 3, 3, and 2 are the factors of 18; etc. NOTE 1. Every number is a factor of itself, the other factor being 1. The prime factors of a number are those prime numbers whose continued product is the number; thus, the prime factors of 12 are 2, 2, and 3; the prime factors of 36 are 2, 2, 3, and 3; etc. NOTE 2. Since 1, as a factor, is useless, it is not here enumerated. 114. To factor a number is to resolve or separate it into its factors. In resolving a number into its factors, The following facts will be found convenient: (a) Every number whose unit figure is 0, or an even number, is itself even, and .. divisible by 2. (b) Any number is divisible by 3 when the sum of its digits (Art. 7) is divisible by 3; thus, 4257 is divisible by 3 because the sum of its digits, 4+2 +5 +7= 18, is divisible by 3. (c) Any number is divisible by 4 when 4 will divide the number expressed by the two right-hand figures; thus, 4 will divide 32, .. it will divide 7532. (d) Any number whose unit figure is 0 or 5 is divisible by 5; as 90, 1740, 35, 34975, etc. (e) Any even number which is divisible by 3 is also divisible by 6; thus, 3528 is divisible by 3 and .. by 6. NOTE 1. For 7 no general rule is known. (f) Any number is divisible by 8 when 8 will divide the number expressed by the three right-hand figures; thus, 8 will divide 816, .. it will divide 175816. 113. What are the Factors of a number? Is a number a factor of itself? What are the prime factors of a number? 114. What is it to factor a number? What number is divisible by 2? By 3? 4? 5? 6? What is said of 7? What number is divisible by 8? (g) Any number is divisible by 9 when the sum of its digits is divisible by 9; thus, 7146 is divisible by 9 because the sum of its digits, 7+1+4+6=18, is divisible by 9. (h) Any number ending with 0 is divisible by 10. (i) Any number is divisible by 11 when the sum of the digits in the odd places is equal to the sum of the digits in the even places; also when the difference of these sums is divisible by 11; thus, 8129, in which 9+1 28, is divisible by 11; also 6280714, in which the sum of the digits in the odd places, 4+7+8+ 6, differs from the sum of the digits in the even places, 1+0+2, by 22, a number divisible by 11 (j) Any number divisible by 3 and also by 4, is divisible by 12; and, generally, any number that is divisible by each of several numbers that are mutually prime, is divisible by the product of those numbers; thus, 84 is divisible by 2, 3, and 7, separately, and.. 84 is divisible by 2 × 3 × 7 = 42; so also 108 is divisible by 4 and 9, and .. by 4 × 9 = 36. NOTE 2. Every prime number, but 2 and 5, has 1, 3, 7, or 9 for its unit figure. For further aid in determining the factors of numbers, we have the following TABLE OF PRIME NUMBERS FROM 1 TO 997. 1 41 101 167 239 313 397 467 569 643 733 823 911 2 43 103 173 241 317 401 479 571 647 739 827 919 3 47 107 179 251 331 409 487 577 653 743 829 929 5 53 109 181 257 337 419 491 587 659 751 839 937 7 59 113 191 263 347 421 499 593 661 757 853 941 11 61 127 193 269 349 431 503 599 673 761 857 947 13 67 131 197 271 353 433 509 601 677 769 859 953 17 71 137 199 277 359 439 521 607 683 773 863 967 19 73 139 211 281 367 443 523 613 691 787 877 971 23 79 149 223 283 373 449 541 617 701 797 881 977 29 83 151 227 293 379 457 547 619 709 809 883 983 31 89 157 229 307 383 461 557 631 719 811 887 991 37 97 163 233 311 389 463 563 641 727 821 907 997 114. What number is divisible by 9? By 10? 11? 12? General principle? |