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Section 4.4 Computation in Other Bases

Section 4.4 Computation in Other Bases. What You Will Learn. Performing addition, subtraction, multiplication and division in other bases. +. 0. 1. 2. 3. 4. 0. 0. 1. 2. 3. 4. 1. 1. 2. 3. 4. 10. 2. 2. 3. 4. 10. 11. 3. 3. 4. 10. 11. 12. 4. 4. 10. 11. 12. 13.

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Section 4.4 Computation in Other Bases

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  1. Section 4.4Computation in Other Bases

  2. What You Will Learn • Performing addition, subtraction, multiplication and division in other bases

  3. + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 10 2 2 3 4 10 11 3 3 4 10 11 12 4 4 10 11 12 13 Addition • An addition table can be made for any base and be used to add in that base. • Base 5 Addition Table

  4. Example 2: Using the Base 5 Addition Table Add 345 +235 Solution From the table 45 + 35 = 125. Record the 2 and carry the 1. 13 45 + 2 35 25

  5. Example 2: Using the Base 5 Addition Table Solution 13 45 + 2 35 Add the numbers in the second column 15 + 35 + 25 = 115 13 45 + 2 35 1 1 25 The sum is 1125.

  6. Subtraction • Subtraction can also be performed in other bases. • When you “borrow,” you borrow the amount of the base given in the subtraction problem. • If you are subtracting in base 5, when you borrow, you borrow 5. • If you are subtracting in base 12, when you borrow, you borrow 12.

  7. Example 6: Subtracting in Base 12 Subtract 91B12 –2A212 Solution In the units column: 11 – 2 = 9 Second column: 1 – 10, so we need to borrow: 1 + 12 = 13; 13 – 10 = 3 Third column: 9 becomes 8; 8 – 2 = 6 91B12 –2A212 63912

  8. Multiplication • Multiplication can be performed in bases other than 10. • In base 10, 4 × 3 means four groups of three units. • In base 5, 45 × 35 means four groups of three units. • (1+1+1)+(1+1+1)+(1+1+1)+(1+1+1) • Regroup into groups of five: • (1+1+1+1+1)+(1+1+1+1+1)+(1+1) • 45 × 35 = 225

  9. × 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 11 13 3 0 3 11 14 22 4 0 4 13 22 31 Multiplication • Multiplication table for the given base is extremely helpful. • Base 5 Multiplication Table

  10. Multiplication • However, it may be easier to multiply the values in the base 10 system and then change the product to base 5. • Multiplying 4 × 3 in base 10 gives 12. • Converting 12 from base 10 to base five gives 225.

  11. Example 8: Multiplying in Base 7 Multiply 437 × 257 • Solution 5 × 3 = 1510 = (2 × 7) + (1 × 1)= 217 Record the 1, carry the 2 24 37 × 2 55 1

  12. Example 8: Multiplying in Base 7 • Solution (5 × 4) + 2 = 2210 = (3 × 7) + (1 × 1)= 317 Record 31 24 37 × 2 55 311

  13. Example 8: Multiplying in Base 7 • Solution 2 × 3 = 610 = 67 Record 6 24 37 × 2 55 311 6

  14. Example 8: Multiplying in Base 7 • Solution 2 × 4 = 810 = (1 × 7) + (1 × 1)= 117 Record 11, and add in base 7 24 37 × 2 55 311 116 15017

  15. Division • Division is performed in much the same way as long division in base 10. • A division problem can be checked by multiplication. (quotient × divisor) + remainder = dividend

  16. Example 10: Dividing in Base 8 Divide • Solution The multiples of 6 in base 8: • 68 × 18 = 68 68 × 28 = 148 • 68 × 38 = 228 68 × 48 = 308 • 68 × 58 = 368 68 × 68 = 448 • 68 × 78 = 528 • Quotient is 5368, remainder 58

  17. Example 10: Dividing in Base 8 Be careful when subtracting! We had to borrow when subtracting 6 from 0 and when subtracting 4 from 1. Remember that we borrow 108, which is the same as 8 in base 10. • Check • Does (5368 × 68) + 58= 40718? • 5368 • × 68 • 40648 + 58 = 40718True

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