§4.3, Computation in Positional Systems

Learning Targets I will add, subtract, multiply, and divide in bases other than ten. 2

Example 1: Addition in Base Four Solution: We add the right-hand column: 3four+3four=6ten. In base four, the digit symbols are 0, 1, 2, and 3. Since the sum exceeds 3, then we convert this base ten number, 6, to base four: 3

Example 1 continued Next, add the three digits in the fours’ column: Record the sum in the right-hand column. Next, add the three digits in the fours’ column: 1four + 3four + 1four = 5ten 5 is not a digit symbol in base four, so, we must convert 5 into base four. 4

Example 1 continued Record the 11four You can check by converting 33four, 13four, and 112four to base ten and taking the sum of 33four & 13four and make sure it is equal to 112four. We leave this to the student. 5

Example 2: Subtraction in Base Four Solution: Subtract the right column: 1four – 2four. Since 2four is larger than 1four, we borrow from the preceding column: 6

Example 2 continued Next, subtract the second column from the right. Again, you can check by converting 31four, 12four, and 13four to base ten and taking the difference of 31four & 12four and making sure it is equal to 13four. We leave this to the student. 7

Example 3: Multiplication in Base Six Multiply: Solution: Multiply as we do in base ten. That is, multiply the digit 2 by the digit 4. 2six × 4six = 8ten = (1× 6) + (2 ×1) = 12six Record the 2 and carry the 1: 8

Example 3 continued Next, we must involve both multiplication and addition: (2six × 3six) + 1six = 6 + 1 = 7ten = (1 × 6) + (1 × 1) = 11six. Record the 11six in the multiplication problem. As before, we can check this by converting to base ten. This is left to the student. 9

Example 4: Division in Base Four Use the table, showing products in base four, to perform the following division: × 1 2 3 10 12 21 Solution: Divide 22four by 3four. Use the table to find what times 3four is less than or equal to 22four. So, 10

Example 4 continued Multiply 3four × 3four = 21four and write the product under the first two digits of the dividend. Subtract: 22four – 21four = 1four Bring down the next digit, 2four. Use the table to find what times 3four is less than or equal to 12four. We see 2four × 3four = 12four, so the quotient is 32four. 11

HOMEWORK Pg 213, #1 – 24.