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Section 4.2

Math in Our World. Section 4.2. Tools and Algorithms in Arithmetic. Learning Objectives. Multiply using the Egyptian algorithm. Multiplying using the Russian peasant method. Multiplying using the lattice method. Multiply using Napier’s bones. Egyptian Algorithm.

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Section 4.2

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  1. Math in Our World Section 4.2 Tools and Algorithms in Arithmetic

  2. Learning Objectives • Multiply using the Egyptian algorithm. • Multiplying using the Russian peasant method. • Multiplying using the lattice method. • Multiply using Napier’s bones.

  3. Egyptian Algorithm The Egyptian algorithm is an ancient method of multiplication that can be done by hand because it requires only doubling numbers and addition. To multiply two numbers A and B: 1. Form two columns with the first digit of the first number at the top of the first, and the other number to be multiplied at the top of the second. 2. Double the numbers in each column repeatedly until the first column contains numbers that can be added to get A. 3. Add the numbers in the second column that are next to the numbers in the first column that add to A. This sum is the product of A and B.

  4. EXAMPLE 1 Using the Egyptian Algorithm Use the Egyptian algorithm to multiply 13 x 24. SOLUTION Step 1 Form two columns with 1 at the top of the first column and 24 at the top of the second column: Step 2 Double the numbers in each column, and continue to do so until the first column contains numbers that can be added to get the other number in the product, 13: 1 24 2 48 4 96 8 192 We stop here because we can get 13 from adding 1, 4, and 8. Step 3 Add the numbers in the second column that are next to 1, 4, and 8: 24 + 96 + 192 = 312. This is the product of 13 and 24.

  5. Russian Peasant Method Another method for multiplying by hand is known as the Russian peasant method. It’s similar to the Egyptian algorithm, but maybe a bit simpler in that you don’t have to keep searching for numbers that add to one of the factors. To multiply two numbers A and B: 1. Form two columns with A at the top of one column and B at the top of the other. 2. Divide the numbers in the first column by two repeatedly, ignoring remainders, until you reach one. Double the numbers in the second column, with the last result next to the one in the first column. 3. Add the numbers in the second column that are next to odd numbers. This sum is the product of A and B.

  6. EXAMPLE 2 Using the Russian Peasant Method Use the Russian peasant method to multiply 24 x 15. SOLUTION Step 1 Form two columns with 24 and 15 at the top. Step 2 Divide the numbers in the first column by two (ignoring remainders), and double the numbers in the second column, until you reach one in the first column. 24 15 12 30 6 60 3 120 1 240 Step 3 Add the numbers in the second column that are next to odd numbers in the first column: 120 + 240 = 360. This is the product of 24 and 15.

  7. EXAMPLE 3 Using the Russian Peasant Method Use the Russian peasant method to multiply 103 x 19. SOLUTION Form two columns as described previously. 103 19 51 38 25 76 12 152 6 304 3 608 1 1,216 Now add the numbers in the second column that are next to odd numbers in the first: 19 + 38 + 76 + 608 + 1,216 = 1,957. So 103 x 19 = 1,957.

  8. Lattice Method The lattice method for multiplication was used in both India and Persia as early as the year 1010. It was later introduced in Europe in 1202 by Leonardo of Pisa (more commonly known as Fibonacci) in his work entitled Liber Abacii (Book of the Abacus). The lattice method reduces multiplying large numbers into multiplying single digit numbers.

  9. EXAMPLE 4 Using the Lattice Method Find the product 36 x 568 using the lattice method. SOLUTION Step 1 Form a lattice as illustrated with one of the numbers to be multiplied across the top, and the other written vertically along the right side. Step 2 Within each box, write the product of the numbers from the top and side that are above and next to that box. Write the first digit above the diagonal and the second below it, using zero as first digit if necessary.

  10. EXAMPLE 4 Using the Lattice Method Find the product 36 x 568 using the lattice method. SOLUTION Step 3 Starting at the bottom right of the lattice, add the numbers along successive diagonals, working toward the left. If the sum along a diagonal is more than 9, write the last digit of the sum and carry the first digit to the addition along the next diagonal. Step 4 Read the answer, starting down the left side then across the bottom: 36 x 568 = 20,448

  11. EXAMPLE 5 Using the Lattice Method Find the product 2,356 x 547 using the lattice method. SOLUTION Step 1 Step 2

  12. EXAMPLE 5 Using the Lattice Method Find the product 36 x 568 using the lattice method. SOLUTION Step 3 Step 4 Read the answer down the left and across the bottom. 2,356 x 547 = 1,288,732

  13. Napier’s Bones John Napier (1550–1617), a Scottish mathematician, introduced Napier’s bones as a calculating tool based on the lattice method of multiplication. Napier’s bones consist of a set of 11 rods: the first rod called the index and 1 rod for each digit 0–9, with multiples of each digit written on the rod in a lattice column as illustrated on the next slide.

  14. Napier’s Bones

  15. EXAMPLE 6 Using Napier’s Bones Use Napier’s bones to find the product 2,745 x 8. SOLUTION Step 1 Choose the rods labeled 2, 7, 4, and 5 and place them side by side; also, place the index to the left. Then locate the level for the multiplier 8. Step 2 Add the numbers diagonally as in the lattice method. The product is 21,960.

  16. EXAMPLE 7 Using Napier’s Bones Use Napier’s bones to find the product 234 x 36. SOLUTION Step 1 Choose the 2, 3, and 4 rods and place them side by side, with the index to the left. Locate the multipliers 3 and 6. Step 2 Add the numbers diagonally as in the lattice method. The product is 8,424.

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