crete? In dividing dollars by an abstract number, is the quotient abstract or concrete? 3. BILLS. What is a bill? Ans. It is a writing given by the creditor to the debtor, showing the amount of the debt. Who is the creditor? the debtor? What is the receipt of a bill? 4. ANALYSIS. What is analysis? Which number forms the basis of the reasoning. 78. GENERAL REVIEW, No. 2. 1. 2875 million +36 thousand + 59481=? 2. Add 567 to the sum of the following numbers: 121; 232; 343; 454; 565; 676; 787; 898. 3. Take 987 from each of the following numbers, and add the remainders: 9876; 5678; 3644; 7573; 2432; 4001. 4. What number must be added to the difference between 58 and 7003 to equal 938425 ? 5. What number, taken from the quotient of 1833000 ÷ 47 leaves 25? 6. What number equals the product of 1785, 394, and (624 —48)? 7. If 5872 is the multiplicand, and half that number the multiplier, what is the product? 8. If 4832796 is the product, and 1208199 the multiplicand, what is the multiplier? 9. If 894869 is the minuend, and the sum of all the numbers in the third example is the subtrahend, what is the remainder ? 10. If 700150 is the dividend, and 3685 the quotient, what is the divisor? 11. If 28936 is the divisor, and 86 is the quotient, what is the dividend? 12. Divide 87 million by 15 thousand. 13. $3.75 $9.32 + $.75 + $10. + $2.185 +4 cents= 14. $19.$.75— $8.25 + $3.54 =? 15. From 18 X $5.873, take $3.68÷4. 16. If $183.30 is the dividend, and $3.90 the divisor, what is the quotient? 17. If $98 60 is the dividend, and 17 the divisor, what is the quotient? For changes, see Key. PROPERTIES OF NUMBERS. 79. SIGNS. + signifies plus, or more. signifies minus, or less. signifies greater than. CAVI+ signifies less than. RECAPITULATION. = signifies equal to. .. signifies therefore. () parenthesis, and, vinculum, signify that the same operation is to be performed upon all the quantities thus connected. DEFINITIONS. 80. Numbers are either integral or fractional. 81. Integral numbers, or Integers, are whole numbers. 82. Fractional numbers are parts of whole numbers. 83. A Factor or Divisor of a number is any number which is contained in it without a remainder; thus, 2 is a factor of 6. 84. A Prime Number is a number which contains no integral factor but itself and 1; as, 1, 2, 3, 11. 85. A Composite Number is a number which contains other integral factors besides itself and 1; as, 4, 6, 8, 25. 86. A Prime Factor is a factor which is a prime number. 87. A composite number equals the product of all its prime factors; thus, 12 = 2 × 2 × 3. 88. Two numbers are said to be prime to each other when they contain no common factor except 1; thus, 8 and 15 are prime to each other. if 89. The Power of a number is the number itself, or the product obtained by taking that number a number of times as a factor. The number itself is the first power; if it is taken twice as a factor, the product is called the second power, or square; three times, it is called the third power, or cube; if four times, Thus, the second power of 3 is 3 × 3 9; the third power of 3 is 3 × 3 × 3 = 27; the fifth power of 3 is 3X 3X 3X 3X 3243. the fourth power, &c. 90. The Index or Exponent of a power is a figure which shows how many times the number is taken as a factor. It is written at the right of the number, and above the line. Thus, in 53, 72, 24, the exponent 2 shows that 5 is taken three times as a factor, 2 that 7 is taken twice, and 4 that 2 is taken four times as a factor. 91. The Root of a number is one of the equal factors which produce that number. If it is one of the two equal factors, it is the second, or square root; if one of the three, the third, or cube root; if one of the four, the fourth root, &c. Thus the square root of 9 is 3, the cube root of 125 is 5. 92. is the Radical Sign, and, by itself, denotes the square root; with a figure placed above, it denotes the root of that degree indicated by the figure; thus, signifies the third root, the sixth root. DIVISIBILITY OF NUMBERS. 93. (1.) Any number whose unit figure is 0, 2, 4, 6, or 8, is even. (2.) Any number whose unit figure is 1, 3, 5, 7, or 9, is odd. (3.) Any even number is divisible by 2. (4.) Any number is divisible by 3 when the sum of its digits is divisible by 3; thus, 2814 is divisible by 3, for 2+8+1+4 15, is divisible by 3. (5.) Any number is divisible by 4, when its tens and units are divisible by 4; for, as 1 hundred, and consequently any number of hundreds, is divisible by 4, the divisibility of the given. number by 4 must depend upon the tens and units; thus, 86324 is divisible by 4, while 6831 is not. (6.) Any number is divisible by 5 if the units' figure is either 5 or 0; for, as 1 ten, and consequently any number of tens, is divisible by 5, the divisibility of the given number by 5 must depend upon the units. (7.) Any number is divisible by 6, if divisible by 3 and by 2. (8.) Any number is divisible by 8, if its hundreds, tens, and Enits are divisible by 8; for, as 1 thousand, and consequently any number of thousands is divisible by 8, the divisibility of the given number by 8 must depend on the hundreds, tens, and units. (9.) Any number is divisible by 9 if the sum of its digits is divisible by 9*; thus, 368451 is divisible by 9, and 23476 is not. (10.) Any number is divisible by 10, 100, or 1000, if it contain at the right 1, 2, or 3 zeros; and so on. (11.) Any number is divisible by 11, if the difference between the sums of the alternate digits is 0, or a number divisible by 11; thus, in 126896, as (1+6+9) — (2+8+6) = 0, the number is divisible by 11; and in 9053, as (9 +5) −(0+3)=11, the number is divisible by 11. (12.) A number is divisible by any composite number, if it is divisible by all the factors of that number. 93. There are no rules of sufficient practical importance for determining when numbers are divisible by other numbers than those spoken of above. Their divisibility must be ascertained by trial. To do this, Divide the number successively by higher and higher primes, until one is found which divides it, or until the quotient is smaller than the divisor. If no divisor is then found, the number is prime; for, if a number contain any prime factor greater than its square root, its corresponding factor must be less. 94. If the odd numbers are written in order, and every third one from 3, every fifth one from 5, every seventh one from 7, and so on, be marked, and the figures 3, 5, 7, &c., be written under the figures as they are marked, the remaining numbers will be primes, and those marked will have for their factors the numbers written beneath; † thus, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 3 3,5 3,7 5 3,9 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, &c. *See Appendix. 3, 13 3,5 9, 15 7 3, 17 Eratosthenes, in the third century B. C., discovered this method of finding primes and factors of numbers, and as he made his table of parchment, cutting out the composite numbers as he found them, this parchment was called Eratosthenes' Sieve. |