14 Arithmetic Operations in Bases Other Than Ten

Transcription

1 14 Arithmetic Operations in Bases Other Than Ten The base-ten arithmetic algorithms discussed in the previous two sections also work in other bases. In this section we apply the algorithms to base five. Addition in Base Five In base five the digits used are 0,1,2,3, and 4. Using blocks one can easily build the following addition table All numerals in the table are written in base five with subscripts omitted. Example 14.1 Compute 12 five + 31 five using blocks. Figure 14.1 shows how to compute 342 five five using blocks. Figure 14.1 Example 14.2 Use a base five line to illustrate 12 five + 20 five. 1

2 Note that 12 five + 20 five = 32 five. Example 14.3 Compute the sum 342 five five (a) using the lattice algorithm (b) using the expanded algorithm (c) using the standard algorithm. To check that the answer to the above addition is correct, we convert everything to base 10 where we feel comfortable. 342 five = = five = = five = = 141 The result is confirmed since = 141. Practice Problems Problem 14.1 Compute the sum 13 five + 22 five using (a) base five blocks (b) expanded algorithm 2

3 (c) lattice algorithm (d) standard algorithm. Problem 14.2 Perform the following computations. Problem 14.3 Complete the following base eight addition table. Problem 14.4 Compute 132 eight + 66 eight Problem 14.5 Computers use base two since it contains two digits, 0 and 1, that correspond to electronic switches in the computer being off or on. In this base, 101 two = = 5 ten. (a) Construct addition table for base two. (b) Write 1101 two in base ten. (c) Write 123 ten in base two. (d) Compute 1011 two two. Problem 14.6 For what base b would 32 b + 25 b = 57 b? Problem 14.7 (a) Construct an addition table for base four. (b) Compute 231 four four. 3

4 Problem 14.8 Use blocks to illustrate the sum 41 six + 33 six. Problem 14.9 Use an expanded algorithm to compute 78 nine + 65 nine. Problem Create a base seven number line and illustrate the sum 13 seven + 5 seven. Problem Construct an addition table in base seven. Problem Use the lattice method to compute the following sums. (a) 46 seven + 13 seven. (b) 13 four + 23 four. Subtraction in Base Five The development of subtraction in base five from concrete to abstract is shown in Figure Figure 14.2 Example 14.4 Use base five number line to illustrate 32 five 14 five. 4

5 Practice Problems Problem Perform the following subtractions: (a) 1101 two 111 two (b) 43 five 23 five (c) 21 seven 4 seven. Problem Fill in the missing numbers. Problem Use blocks for the appropriate base to illustrate the following problems. (a) 555 seven 66 seven (b) 3030 four 102 four. Problem Use both the intermediate algorithm (discussed in Figure 14.2) and the standard algorithm to solve the following differences. (a) 31 four 12 four (b) 1102 four 333 four. Problem Use base five number line to illustrate the difference 12 five 4 five. Multiplication in Base Five Next, consider the multiplication algorithm. A base-five multiplication table will be helpful

6 Example 14.5 Calculate 4 five 3 five using base five number line. So 4 five 3 five = 22 five Example 14.6 Calculate 43 five 123 five using (a) the lattice method for multiplication (b) the expanded algorithm (c) the standard algorithm. Practice Problems Problem Create a base seven number line to illustrate 6 seven 3 seven. Problem Find the following products using the lattice method, the expanded algorithm, and the standard algorithm. (a) 31 four 2 four (b) 43 five 3 five 6

7 Division in Base Five Long division in base five can be done with a long division analogous to the base ten algorithm. The ideas behind the algorithms for division can be developed by using repeated subtraction. For example, 3241 five 43 five is computed by means of repeated-subtraction technique as shown in Figure 14.3(a) and by means of the conventional algorithm shown in Figure 14.3(b). Thus, 3241 five 43 five = 34 five with remainder 14 five. Practice Problems Figure 14.3 Problem Perform the following divisions: (a) 32 five 4 five (b) 143 five 3 five (c) two 11 two. Problem For what possible bases are each of the following computations correct? 7

8 Problem (a) Compute 121 five 3 five with repeated subtraction algorithm. (b) Compute 121 five 3 five with long division algorithm. Problem (a) Compute 324 five 4 five with repeated subtraction algorithm. (b) Compute 324 five 4 five with long division algorithm. Problem (a) Compute 1324 seven 6 seven with repeated subtraction algorithm. (b) Compute 1324 seven 6 seven with long division algorithm. Problem Solve the following problems using the missing-factor definition of division, that is, a b = c if and only if b c = a.(hint: Use a multiplication table for the appropriate base). (a) 21 four 3 four (b) 23 six 3 six (c) 24 eight 5 eight Problem Sketch how to use base seven blocks to illustrate the operation 534 seven 4 seven. 8