4. Reduce 376487 seconds to higher denominations. 5. Reduce 365d. 5h. 48m. 50sec. to seconds. 6. In 342698 minutes how many days, hours, etc.?. 7. In 5C. 56yr. 8m. how many calendar months? 8. Reduce 37846 calendar months to centuries, years, etc. 9. Reduce 2419199 seconds to weeks, days, etc. 10. Reduce 34d. 20h. 40m. 50sec. to seconds. CIRCULAR MEASURE. 109. CIRCULAR MEASURE is used in surveying, navigation, geography, astronomy, etc., for measuring angles, determining latitude, longitude, etc. 109. For what is Circular Measure used? Table? Scale? D Ө B C A NOTE. 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 632934" to higher denominations. 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 may be indefinitely extended. 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. 1216. 8oz. 6dr. to drams. 9. Reduce 14 lb. 7oz. 15dwt. 23gr. to grains. 10. Reduce 6lb 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 Frac tious. 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 3 X 3 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 48, which equals 4 X 4 X 4 = €64. NOTE 5. A rot 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 squal 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 .. it will divide 7532. 32, (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 num, ber 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? |