1. §4.3, Computation in Positional Systems

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. Example 3 continued
Next, we must involve both multiplication and addition:
(2six × 3six) + 1six = = 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

10. 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

11. 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

12. HOMEWORK
Pg 213, #1 – 24.