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

2. 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  34.

3. 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

4. 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.

5. 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  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 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