Chapter 7

Section 7.2

Addition & Subtraction in Different Bases

Adding Numbers in Different Bases

Adding numbers in different bases requires the need to have learned the basic addition facts in another base. The table below give the basic addition facts for base four.

+ 04 14 24 34

04 04 14 24 34

14 14 24 34 104

24 24 34 104 114

34 34 104 114 124

The reasoning for how we have gotten some of the entries is shown below.

24 + 24 = 4 (base 10) = 104

24 + 34 = 5 (base 10) = 114

34 + 34 = 6 (base 10) = 124

Below is shown how the standard addition algorithm is applied to solve addition problems in base four.

1 2 0 34

+ 1 3 3 24

14023

111

2 3 1 24

+ 2 0 24

04213

11

9 9

+ 1 2 6

2 2 5

Converting to base 10

1 8 2

+ 3 4

2 1 6

Converting to base 10

Addition of Numbers Using the Lattice Method

Another way to organize the addition of numbers is to use the lattice method. It works similar to how you use it with multiplication but you fill in the addition facts in the correct columns. The first problem shows how to use this in base 10 to add 849+5767 and the second shows how it is used in base 4 to add.

8 4 9

+ 5 7 6 71

61

01

550

2 3 1 24

+ 2 0 24

10

01

112

0

6166 3 1 2 04

Try the following addition problems in the given bases. You have to figure out the basic addition facts as you are doing the problems.

2 4 1 35

+ 1 3 4 25

10

10

123

0

4 3 1 05

4 0 5 26

+ 5 3 2 36

05

11

033

1

3 4 1 561

101001112

+ 1101102

21

Adding & Subtracting Binary (Base 2) Numbers

Addition and subtraction of base 2 numbers can be accomplished by know how to do the problems below and following the rules for carrying and borrowing just like in base 10.

0+ 0

0

1+ 0

1

0+ 1

1

1+ 110

1

0+ 0

1

1

1+ 010

1

0+ 110

1

1+ 111

No Carrying digit. Carried a 1 into the next digit.

Add each of the following base 2 numbers.

1011012

+ 101102

2

110

1

0001

11 1

010

1

111

1

1

Subtraction also has the same idea as that of base 10 (decimal). In subtraction we will either need to borrow or we won't. Here are the basic subtraction problems.

0- 0

0

1- 0

1

1- 1

0

No Borrowing needed.

10- 1

1

100- 1

11

Carried a 1 into the next digit.

1 111

Here are the steps needed to do the subtraction problem below.

1010112

- 101012

2

1. Subtract 1's digits.

2. Subtract 2's digits.

3. Borrow from 8's digit to 4's digit.

4. Subtract 4's digits.

5. Subtract 8's digits.

6. Borrow from 32's digit to 16's digit.

7. Subtract 16's digits.

10

1 1

110