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Section 4.3 Computation in Positional Systems

Learn how to perform addition, subtraction, multiplication, and division in bases other than ten. Explore examples and solutions in different bases to enhance your understanding of positional systems.

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Section 4.3 Computation in Positional Systems

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  1. Section 4.3Computation in Positional Systems Objectives Add in bases other than ten. Subtract in bases other than ten. Multiply in bases other than then. Divide in bases other than ten.

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

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

  4. AdditionExample 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 to112four. We leave this to the student.

  5. Subtraction Example: Subtraction in Base Four Subtract Solution: Subtract the right column, or the ones’ column: 1four – 2four. Since 2four is larger than 1four, we borrow from the fours’ column:

  6. SubtractionExample Continued Next, subtract the fours’ column: Again, you can check by converting 31four, 12four, and 13four to base ten and taking the difference of 31four & 12four and make sure it is equal to13four. We leave this to the student.

  7. Multiplication Example: Multiply in Base Six Multiply Solution: Multiply as we do in base ten. That is, multiply the ones’ column. 2six x 4six = 8ten = (1 x 6) + (2 x 1) = 12six Record the 2 and carry the 1:

  8. MultiplicationExample Continued Next, we must involve both multiplication and addition: (2six x 3six) + 1six = 6 + 1 = 7ten = (1 x 6) + (1 x 1) = 11six. Record the 11six. As before, we can check this by converting to base ten. This is left to the student.

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

  10. Division Example Continued Multiply 3four x 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 x 3four = 12four, so the quotient is 32four.

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