Cube Root by HandDate: 01/23/98 at 15:48:37 From: Adrian Van Cauwenberghe Subject: Cube root Dear Sir, Many years ago I learned to calculate the square root of a number (without calculators or computers, just pencil and paper). An example will tell you how (square root of 222784) 22.27.84 I 472 16 I --------------------------- ----- I 62.7 I 87 I 942 60.9 I X 7 I X 2 -------- I ------------------------- 188.4 I 609 I 1884 188.4 I ------ 0 I suppose this is relatively familiar to you. Later we learned a comparable but of course more complicated way to calculate the cube root of a number. I forgot almost completely how. What I remember is that we had to split the number up in groups of three figures instead of two (example: 274625 becomes 274.625) and that we had to take the first group (here 274) and find the approximate cube root of that number (here 6 since 6x6x6=216 and 7x7x7=343). But then I am lost. Can you help me further? Many thanks in anticipation. Adrian ( Bruges - Belgium ) Date: 03/03/98 at 16:18:48 From: Doctor Rob Subject: Re: Cube root To find a cube root by the "longhand" method, we proceed very much as we do to find a square root by hand. I intersperse numbered steps with an example. We will find the cube root of 113 to two decimal places. 1. Draw a cube root symbol, or radical, with the number whose root you are seeking underneath. Start with the decimal point and mark off digits in both directions in groups of three. Put a decimal point above the radical, and directly above the other decimal point. . 3/----------- \/ 113.000 000 2. Start with the first group of 1, 2, or 3 digits. Find the largest cube of a single-digit integer less than it. Write the single digit above the radical, and its cube under the first group. Draw a line under that cube, and subtract it from the first group. 4. 3/----------- \/ 113.000 000 64 ------- 49 3. Bring down the next group below the last line drawn. This forms the current remainder. Draw a vertical line to the left of the resulting number, and to the left of that line put three hundred times the square of the number above the radical, a plus sign, thirty times the number above the radical, a multiplication sign, an underscore character, another plus sign, another underscore character, the exponent 2, an equals sign, and some blank space for the answer. 4. 3/----------- \/ 113.000 000 64 ------- 4800+120*_+_^2=???? | 49 000 4. Pick the biggest digit D that would fit into both underscore places, and give a number such that D times it is less than the current remainder. Put it above the radical above the last group of digits brought down, and put it in each of the blanks where the underscore characters are. Compute the number given by the expression, and put it after the equals sign. Multiply D times that number, and put that below the current remainder, draw a horizontal line below that, and subtract, to give a new current remainder. 4. 8 3/----------- \/ 113.000 000 64 ------- 4800+120*8+8^2=5824 | 49 000 46 592 ---------- 2 408 5. If the current answer, above the radical, has the desired accuracy, stop. Otherwise, go back to step 3. Step 3: 4. 8 3/----------- \/ 113.000 000 64 ------- 4800+120*8+8^2=5824 | 49 000 46 592 ---------- 691200+1440*_+_^2=?????? | 2 408 000 Step 4: 4 . 8 3 3/----------- \/ 113.000 000 64 ------- 4800+120*8+8^2=5824 | 49 000 46 592 ---------- 691200+1440*3+3^2=695529 | 2 408 000 2 086 587 --------- 321 413 Step 5: Stop. Thus the cube root of 113 to two decimal places is 4.83. Checking, 4.83^3 = 112.6786, and 4.84^3 = 113.3799, so the answer is correct. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 03/14/98 at 07:24:04 From: Adrien Van Cauwenberghe Subject: Cube root Dear Sirs, I want to thank you very much for your help. I would like to give you a few comments on the answer. 1. The "determination" of D seems to me a bit arbitrary: try 9, if 9 is incorrect try 8, if 8 is incorrect, try 7; if 7 is not correct, try 6, and so on. In a book that I found recently in a shop for second-hand books (edition 1910!), I found large theoretical considerations on the square root and the cube root of a number. To find the cube root of a number up to step 2 they use a similar method as the one you proposed. To determine D they say (for your given example): divide 490|00 by 3*4^2 = 3*16 = 48. Result 10, which of course is too large. Trying 9: still too large, but as we know 8 is correct. In most cases, however, this way of determination of D gives the correct D immediately. 2. Another way to control whether D is correct or not is to calculate (4.8)^3 and to subtract it from the given number 113.000. If the result is positive then D is correct. Here also (113.000 - 110.592) = 2.408; however, this result will also be positive if D is taken too small: nothing is perfect in this world. Date: 03/18/98 at 12:36:59 From: Doctor Rob Subject: Re: Cube root Usually the next digit D is the quotient gotten by dividing the number to the left of the vertical line (the divisor), *ignoring D* (or setting D = 0), into the current remainder. In this case, you would divide 4920 into 49000, and the quotient digit D is guessed to be 9. As you progress to later and later stages in the algorithm, this method is more and more accurate. This problem is exactly analogous to the one encountered when doing long division: what is the next digit? You have to estimate the right answer, and adjust if the remainder is too big or too small. -Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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