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So You Think You Can Divide? So You Think You Can Divide? A History of Division Stephen Lucas Department of Mathematics and Statistics James Madison University, Harrisonburg VA October 10, 2011 Tobias Dantzig: Number (1930, p26) “There is a story of a German merchant of the fifteenth century, which I have not succeeded in authenticating, but it is so characteristic of the situation then existing that I cannot resist the temptation of telling it. It appears that the merchant had a son whom he desired to give an advanced commercial education. He appealed to a prominent professor of a university for advice as to where he should send his son. The reply was that if the mathematical curriculum of the young man was to be confined to adding and subtracting, he perhaps could obtain the instruction in a German university; but the art of multiplying and dividing, he continued, had been greatly developed in Italy, which in his opinion was the only country where such advanced instruction could be obtained.” Ancient Techniques Positional Notation Division Yielding Decimals Outline Ancient Techniques Division Yielding Decimals Definitions Integer Division Successive Subtraction Modern Division Doubling Multiply by Reciprocal Geometry Iteration – Newton Positional Notation Iteration – Goldschmidt Iteration – EDSAC Positional Definition Galley or Scratch Factor Napier’s Rods and the “Modern” method Short Division and Genaille’s Rods Double Division Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division. Also, a is the dividend and b is the divisor. Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division. Also, a is the dividend and b is the divisor. We often write a ÷ b = q r r. Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division. Also, a is the dividend and b is the divisor. We often write a ÷ b = q r r. For rationals and reals, if a = b × c then c = a/b. Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division. Also, a is the dividend and b is the divisor. We often write a ÷ b = q r r. For rationals and reals, if a = b × c then c = a/b. a/b a d ad Note: = × = , division is multiplication by reciprocal. c/d b c bc Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division. Also, a is the dividend and b is the divisor. We often write a ÷ b = q r r. For rationals and reals, if a = b × c then c = a/b. a/b a d ad Note: = × = , division is multiplication by reciprocal. c/d b c bc a/b takes a whole, divides it into b parts, chooses a. Dividing each part into d smaller parts means a/b = (ad)/(bd). Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division. Also, a is the dividend and b is the divisor. We often write a ÷ b = q r r. For rationals and reals, if a = b × c then c = a/b. a/b a d ad Note: = × = , division is multiplication by reciprocal. c/d b c bc a/b takes a whole, divides it into b parts, chooses a. Dividing each part into d smaller parts means a/b = (ad)/(bd). So a/b (ad)/(bd) ad = = in terms of the smaller pieces. c/d (bc)/(bd) bc Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Subtraction As multiplication (of natural numbers) is successive addition, division is successive subtraction. Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Subtraction As multiplication (of natural numbers) is successive addition, division is successive subtraction. For example, 100 − 12 = 88, 88 − 12 = 76, 76 − 12 = 64, 64 − 12 = 52, 52 − 12 = 40, 40 − 12 = 28, 28 − 12 = 16, 16 − 12 = 4, Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Subtraction As multiplication (of natural numbers) is successive addition, division is successive subtraction. For example, 100 − 12 = 88, 88 − 12 = 76, 76 − 12 = 64, 64 − 12 = 52, 52 − 12 = 40, 40 − 12 = 28, 28 − 12 = 16, 16 − 12 = 4, so 100 = 8 × 12 + 4 or 100 ÷ 12 = 8 r 4. Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Subtraction As multiplication (of natural numbers) is successive addition, division is successive subtraction. For example, 100 − 12 = 88, 88 − 12 = 76, 76 − 12 = 64, 64 − 12 = 52, 52 − 12 = 40, 40 − 12 = 28, 28 − 12 = 16, 16 − 12 = 4, so 100 = 8 × 12 + 4 or 100 ÷ 12 = 8 r 4. Or, 100 = 0 · 12 + 100 = 1 · 12 + 88 = 2 · 12 + 76 = 3 · 12 + 64 = 4 · 12 + 52 = 5 · 12 + 40 = 6 · 12 + 28 = 7 · 12 + 16 = 8 · 12 + 4. Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: 1 12 2 24 4 48 8 96 Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: 1 12 2 24 4 48 8 96 100 − 96 = 4 So 100/12 = 8 r 4, Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: and 1652/23: 1 12 1 23 2 24 2 46 4 48 4 92 8 96 100 − 96 = 4 8 184 So 100/12 = 8 r 4, 16 368 32 736 64 1472 Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: and 1652/23: 1 12 1 23 2 24 2 46 4 48 4 92 8 96 100 − 96 = 4 8 184 So 100/12 = 8 r 4, 16 368 32 736 64 1472 1652 − 1472 = 180 Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: and 1652/23: 1 12 1 23 2 24 2 46 4 48 4 92 180 − 92 = 88 8 96 100 − 96 = 4 8 184 So 100/12 = 8 r 4, 16 368 32 736 64 1472 1652 − 1472 = 180 Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: and 1652/23: 1 12 1 23 2 24 2 46 88 − 46 = 42 4 48 4 92 180 − 92 = 88 8 96 100 − 96 = 4 8 184 So 100/12 = 8 r 4, 16 368 32 736 64 1472 1652 − 1472 = 180 Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: and 1652/23: 1 12 1 23 42 − 23 = 19 2 24 2 46 88 − 46 = 42 4 48 4 92 180 − 92 = 88 8 96 100 − 96 = 4 8 184 So 100/12 = 8 r 4, 16 368 32 736 64 1472 1652 − 1472 = 180 Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Successive Doubling – Egyptian Successively doubling the divisor gives powers of two of the divisor to subtract. For example, 100/12 again: and 1652/23: 1 12 1 23 42 − 23 = 19 2 24 2 46 88 − 46 = 42 4 48 4 92 180 − 92 = 88 8 96 100 − 96 = 4 8 184 So 100/12 = 8 r 4, 16 368 32 736 64 1472 1652 − 1472 = 180 1 + 2 + 4 + 64 = 71, so 1652/23 = 71 r 19.
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