Section 4.4
Computation
in Other
Bases
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Inc.
INB Table of Contents
Date
Topic
August 26, 2013
Section 4.4 Examples
10
August 26, 2013
Section 4.4 Notes
11
2.3-2
Page #
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Inc.
What You Will Learn
Performing addition, subtraction,
multiplication and division in other
bases
4.4-3
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Inc.
Addition
An addition table can be made for any base and be used to add
in that base.
Base 5 Addition Table
+
0
1
2
3
4
4.4-4
0
0
1
2
3
4
1
1
2
3
4
10
2
2
3
4
10
11
3
3
4
10
11
12
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Inc.
4
4
10
11
12
13
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 4
5
+ 2 35
25
4.4-5
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Inc.
Example 2: Using the Base 5
Addition Table
Solution
13 4
5
+ 2 35
Add the numbers in the second
column
15 + 35 + 25 = 115
13 4
5
+ 2 35
The sum is 1125.
1 1 25
4.4-6
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Inc.
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.
4.4-7
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Example 6: Subtracting in Base
Subtract
91B
12
12
– 2A2
12
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
4.4-8
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Inc.
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
4.4-9
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Inc.
Multiplication
Multiplication table for the given
base is extremely helpful.
Base 5 Multiplication Table
×
0
1
2
3
4
4.4-10
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
11
13
3
0
3
11
14
22
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Inc.
4
0
4
13
22
31
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.
4.4-11
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Example 8: Multiplying in Base
7
Multiply 43
×
7
257
Solution
5 × 3 = 1510 = (2 × 7) + (1 × 1)=
217
Record the 1, carry the 2
24 3
7
× 2 55
1
4.4-12
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Inc.
Example 8: Multiplying in Base
7
Solution
(5 × 4) + 2 = 2210 = (3 × 7) + (1 ×
1)= 317
Record 31
24
37
× 2 55
311
4.4-13
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Inc.
Example 8: Multiplying in Base
7
Solution
2 × 3 = 610 = 67
Record 6
24
37
× 2 55
311
6
4.4-14
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Inc.
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
4.4-15
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Inc.
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
4.4-16
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Example 10: Dividing in Base
5368
8
Divide 68 40718
Solution
The multiples
68 × 1 8 = 68
68 × 38 = 228
68 × 58 = 368
68 × 78 = 528
68 40718
of 6 in base 8:
68 × 28 = 148
68 × 48 = 308
68 × 68 = 448
36
Quotient is 5368, remainder 58
4.4-17
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Inc.
27
22
51
44
5
Example 10: Dividing in Base
Be
8 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 = 40718 True
4.4-18
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Inc.