Section 7.3 Multiplication in Different Bases Chapter 7 Section 7.3 Multiplication in Different Bases

Multiplying Numbers in Different Bases Multiplying numbers in different bases requires the need to have learned both the basic addition and the basic multiplication facts in another base. The table below give the basic addition facts for base four. The reasoning for how we have gotten some of the entries is shown below. 24  24 = 4 (base 10) = 104 24  34 = 6 (base 10) = 124 34  34 = 9 (base 10) = 214  04 14 24 34 104 124 214 234  314 34 1  3 204 1  20 2104 30  3 12004 30  20 20334 13224  34 124 3  2 1204 3  20 21004 3  300 30004 3  1000 112324 The examples to the right show how to use the standard partial products algorithm in different bases. The first shows how to multiply 234  314. The second shows how to multiply 13224  34.

One shift in place value. 256  346 326 1206 2306 10006 14226 45 = 206 = 3 remainder 2 No shift in place value. 42 = 86 = 1 remainder 2 One shift in place value. 35 = 156 = 2 remainder 3 One shift in place value. 32 = 66 = 1 remainder 0 Two shifts in place value. Add up all the numbers A second way to do this is to convert to base 10, do the multiplication like you ordinarily do the convert back to base 6. 256 346 Convert Back 3746 = 62 remainder 2 626 = 10 remainder 2 106 = 1 remainder 4 16 = 0 remainder 1 The answer is: 14226 (Like above) 51 = 5 26 = 12 41 = 4 36 = 18 17 22 Do the base 10 multiplication with these numbers. 17  22 = 374

The Lattice Method Another method for multiplying numbers which provides more structure for how you multiply is called the lattice method of multiplication. It uses a diagonally represented table and fills in the entries with the basic multiplication facts. For Example: to do the problem 317  46 = 14582 1. Fill in the corresponding squares with the basic multiplication facts putting the tens digit above the diagonal. 2. Add down each diagonal starting at the bottom right carrying into the next diagonal. 3 1 7 4 6 1 1 1 2 1 2 4 8 1 4 4 8 6 2 5 8 2 14582 3. The final answer you get by taking the digits going down the left side and along the bottom.

Multiplication of Numbers in Other Bases Using the Lattice Method The lattice method for multiplication can be used to organize how numbers are multiplied. It relies on using the basic multiplication facts. Below to the right we show how to do the base four multiplication problem 3124  2314. I have given the base 4 basic multiplication facts below to the right.  04 14 24 34 104 124 214 3 1 2 1 1 1 1 1 2 2 2 2 1 1 1 3 2 1 3 1 2 3 3 2 2113324 2 6 3 4 1 1 The lattice to the right demonstrates how to do the base 7 multiplication problem 267  347. 2 1 6 4 3 1 3 1 3 1 3 13137