section 4.4 computation in other bases

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 4.4

Computation in Other Bases


What You Will Learn

Performing addition, subtraction, multiplication and division in other bases

4.4-2


Addition An addition table can be made for any base and be used to add in that base.Base 5 Addition Table

131211104412111043311104322104321143210043210+

4.4-3


Add 345

+ 235

Solution From the table 45 + 35 = 125.Record the 2 and carry the 1. 13 45

+ 2 35

25

Example 2: Using the Base 5 Addition Table

4.4-4


Solution 13 45

+ 2 35

Add the numbers in the second column15 + 35 + 25 = 115

13 45

+ 2 35

1 1 25 The sum is

1125.

Example 2: Using the Base 5 Addition Table

4.4-5


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.

4.4-6


Subtract 91B12

– 2A212

Solution In the units column: 11 – 2 = 9Second column: 1 – 10, so we need to borrow: 1 + 12 = 13; 13 – 10 = 3Third column: 9 becomes 8; 8 – 2 = 6 91B12

– 2A212

63912

Example 6: Subtracting in Base 12

4.4-7


MultiplicationMultiplication 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

4.4-8


MultiplicationMultiplication table for the given base is extremely helpful.Base 5 Multiplication Table

3122134042214113031311420243210100000043210×

4.4-9


MultiplicationHowever, 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.

4.4-10


Multiply 437

× 257

Solution5 × 3 = 1510 = (2 × 7) + (1 × 1)= 217

Record the 1, carry the 2

24 37

× 2 55

1

Example 8: Multiplying in Base 7

4.4-11


Solution

(5 × 4) + 2 = 2210 = (3 × 7) + (1 × 1)= 317

Record 31

24 37

× 2 55

311


4.4-12


Solution

2 × 3 = 610 = 67

Record 6

24 37

× 2 55

311

6


4.4-13


Solution2 × 4 = 810 = (1 × 7) + (1 × 1)= 117

Record 11, and add in base 7

24 37

× 2 55

311 116

15017


4.4-14


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

4.4-15


Divide

SolutionThe 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

Example 10: Dividing in Base 8

6

84071

8

68

40718

36

27

22

51

44

5

5368

4.4-16


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.CheckDoes (5368 × 68) + 58= 40718?

5368

× 68 40648 + 58 = 40718 True

Example 10: Dividing in Base 8

4.4-17

section 4.4 computation in other bases

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