: As 60401.8 61754.6 :: 27: 27.6047 the root nearly. Again, assuming 27.6 and working as before, the root will be found 27.60491. 2.2.439. 4. 5.842.16. /.7759. 8. /7121.10216198. 3. 2.363. 5. 3.6199. 7. 10.00. 9. 61218.00121. Q. What is evolution? Q. What is the rule for extracting the square root? Q. How do you find the square root of a vulgar fraction? Q. How do you point off when there are decimals in the given number? Q. Why do you separate the figures of the dividend into periods? Q. How is the operation proved? Q. What is the rule for extracting the cube root? Q. What other method of finding the cube root is given? ARITHMETICAL SYMBOLS. § 162. Certain signs prefixed to numbers indicate the operations that are to be performed. + This sign denotes addition; and is read, plus or more. This sign denotes subtraction; and is read, minus or less. × The sign of multiplication; and is read, multiplied by. The sign of division; and is read, divided by. 67 or 342 Numbers placed like a fraction also denote division; the upper number being to be divided by the lower. The sign of equality; and is read, equal to. = A number placed above another number, a little to the right, is called an exponent; as 32, 53, and denotes that the quantity is to be used as a factor as many times as there are units in the exponent, as 33=3×3×3=27. : is to; ✔or and so is; to; the signs of proportion. Signs of the square root. Signs of the cube and biquadrate root A vinculum is a line drawn over quantities, and it is employed to signify that the total result of these is to be considered but one quantity; which said resulting quantity is to be operated upon by some affixed number or expression. The parenthesis () or bracket [ ] implies the same. EXEMPLIFICATION. The arithmetical expression 3+7+4 +1 is read, three plus seven plus four plus one. The expression 17-8 is read, seventeen minus eight. The expression 9×7 is read, nine multiplied by seven. The expression 488 is read, forty-eight divided by eight. A quantity that is to be divided is more conveniently expressed by placing it over the quantity by which it is to be divided; thus, instead of 69÷9, it may be written 69; and 141÷8 is implied by 141; and 349÷112 is the same as 112° 349 The expression 7+3—1=7+2 denotes that the aggregate of the numbers (7+3-1) on one side of the sign (=), is of the same value as the sum of those (7+2) on the other side of the sign; and therefore, seven plus three minus one, are read, equal to seven plus two. What is the total amount of the expression of 6×5+2— 4? It is read, six times five plus two minus four; now 6x5+24=30+2-4-32-4=28. What is the value of 14-8+4x3? It is read, fourteen minus eight plus four times three; now 14-8+4×3=14 -8+12=26—8—18. What is the value of 4×6×5+8×3—7×2? It is read, four times six times five plus eight times three minus seven times two; now 4×6×5+8×3—7×2=120+24-14= 144-14=130. What is the value of 63÷7+3x8-24÷6? It is read, sixty-three divided by seven plus three times eight minus twenty-four divided by six: now 63+7+3x8-24÷6=9 +24-4-33-4=29. What is the value of 12x8÷6+14? It is read, twelve times eight divided by six plus fourteen; now, 12x8÷6+14 =96÷6+14=16+14=30. What is the value of 5×32÷4×8? It is read, five times thirty-two divided by four times eight; now 5×32÷4×8= 160÷32=5. What is the value of 6x30x12÷4×90-2? It is read, six times thirty times twelve divided by four times ninety minus two; now 6×30×12÷÷4×90—2=6×360÷360—2 =2160÷360—2—6—2=4. What is the value of 96×32÷8×24×2+14—18? It is read, ninety-six times thirty-two divided by eight times twenty-four times two plus fourteen minus eighteen; now 96×32÷÷8 × 24 × 2+14—18-3072÷384+14-18=8+ 14-18-22-18=4. The expression 3+6×7 becomes 9x7=63. The expression 4-3+12x6 becomes 13×6=78. The expression (11-6)x30 becomes 5×30=150. The expression (12—2+3—4)x(10-2) becomes 9×8=72. The expression (14+2)+(29-25) becomes 16÷4=4. The expres sion (6×5-6)×(3×4+2×3) becomes (30-6)×(12+6) =24x18=432. The expression (2x6x3-9+4×2)÷7 becomes (36-9+8)÷7-35÷7-5. The expression (14 -2x3+8)×6÷32 becomes (14—6+8)×6÷32=16×6 ÷32=96÷÷÷32=3. The expression (26-6)×4÷(7x3-5) becomes 20x4 ÷16=80÷÷16=5. The expression (8-2)x(11×3—4× 6)÷2×9 becomes 6×9÷2×9-54÷18=3. The expression (80-2-3x8+5)x(11+6÷3-2×4) becomes (4024+5)×(11+2—8)=21×5=105. The expression (4×6 -11x8+12x14)x(42÷6+9) x (2×3×7-21 x 13+39) becomes (24-88+168)×(7+9) × (42—7) =104×16×35 = 58240. 0 28 21 20 40 The expression [(3+)÷2-3x- becomes ( ÷218)×3-20=-(1303)×3-26 (40)×3-26-11 3 The expression 22 +73-42+34 becomes 4+343—16+ 81-428-16=412. The expression becomes becomes (3.6-1.2)2÷6 (1.728-1.296)÷4 432÷4 .108 (2.4)2÷÷6 =.1125. 5.76-6 .96 The expression (√64-125)x3 becomes (8-5)x3= 3x3=9. The expression 72+128 becomes √(64×2)=6√2+8✅✔✅/2=14√✓/2. (36×2)+ The expression 40+/135- becomes (8×5) +(27×5)—(x5)=2/5+3/5-5-55-1 3/5=433/5. EXAMPLES. 1. What is the value of 3+8+11+17? 12. What is the value of 3×15×6÷2×45—1? 15. What is the value of 6-3+11×6? 16. What is the value of (14-4+5-6)x(12-2)? 17. What is the value 18. What is the value 19. What is the value 20. What is the value 21. What is the value of (12+3)÷(29—26)? of of of (5x4-6)x(2x7+3x2)? of (28-4)x5÷(9x2-3)? 22. What is the value of (7-4)x(12×3—2×7)÷3x11? 23. What is the value of (60÷2-4×9+7)x(12+8÷4 -3×2)? 24. What is the value of (7×3—12×9+10×13)×(49÷ 7+11)×(3×4×5-18×4÷36) ? 25. What is the value of +8878? 26. What is the value of (+)? --3 27. What is the value of (+) ÷ (3—3) ? 5 +.906)÷.92? 31. What is the value of (7+.997-5.504)÷6.75? 32. What is the value of (113+14.775)+(4.375-)? 33. What is the value of 2.1÷(6.39—1.5)+3.75? 13.11-5.25. 34. What is the value of 14-7 .? 35. What is the value of (+3.942)÷6? 36. What is the value of 12+.64÷4-10, (12+.64)÷4-3 53 2 21/ 4 45 40. What is the value of [(3+)÷2—1]×3—1? 41. What is the value of ? (6+.68)÷2—1.2 ' (6+.68)÷(2—1.2) ARITHMETICAL PROGRESSION, § 163. Is when a series of numbers increases or decreases regularly, by the continual addition or subtraction of some equal number, called the common difference; so 1, 2, 3, 4, 5, 6, &c. are in arithmetical progression, by the continual addition of one; and 11, 9, 7, 5, 3, 1, by the continual decreasing or subtracting of two. The numbers which form the series are called the terms of the progression, of which there are five. The first term and the last term, (called the extremes,) the number of terms, the common difference and the sum of all the terms. Any three of these being given, the other two may be readily found. § 164. CASE 1. The first term, the last term, and the number of terms, being given, to find the sum of all the terms. RULE. Multiply the sum of the extremes by the number of terms, and half the product will be the answer. EXAMPLES. 1. The first term of an arithmetical progression is 2, the last term 53, and the number of terms 18: required the sum of the series? |