© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 4 Number Representation and Calculation
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© 2010 Pearson Prentice Hall. All rights reserved.
CHAPTER 4
Number Representation and Calculation
© 2010 Pearson Prentice Hall. All rights reserved. 2
4.3
Computation in Positional Systems
© 2010 Pearson Prentice Hall. All rights reserved.
Objectives
1. Add in bases other than ten.
2. Subtract in bases other than ten.
3. Multiply in bases other than ten.
4. Divide in bases other than ten.
3
© 2010 Pearson Prentice Hall. All rights reserved.
Example 1: Addition in Base Four
Add
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:
four
four
13
33
4
© 2010 Pearson Prentice Hall. All rights reserved.
Example 1 continued
fourtenfourfour 121241633
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.
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© 2010 Pearson Prentice Hall. All rights reserved.
Example 1 continued
fourtenfourfourfour 1111415131
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.
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© 2010 Pearson Prentice Hall. All rights reserved.
Example 3: Subtraction in Base Four
Subtract:
Solution: Subtract the right column: 1four – 2four.
Since 2four is larger than 1four, we borrow from the preceding column:
four
four
12
31
7
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Example 3 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.
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© 2010 Pearson Prentice Hall. All rights reserved.
Example 5: 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:
six
six
2
34
six
six
six
1
2
2
43
9
© 2010 Pearson Prentice Hall. All rights reserved.
Example 5 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.
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© 2010 Pearson Prentice Hall. All rights reserved.
Example 6: Division in Base Four
Use the table, showing products in base four, to perform the following division:
fourfour 2223
× 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 10 12
3 0 3 12 21
Solution: Divide 22four by 3four. Use the table to find what times 3four is less than or equal to 22four. So,
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© 2010 Pearson Prentice Hall. All rights reserved.
Example 6 continued
12
3
21
2223 four
four 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.
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