Addition & Subtraction in Other Bases

Worked Example 2.3.1:

Solve the following:

a. 3042_five   + 2103_five
b. 2001_three   + 1022_three
c. 151.24_six +   12.34_six  + 20.11_six
d. 5346₈ – 637₈
e. 3002_four – 2213_four

Solution

a. 3042_five   + 2103_five

Explanation:

1st Column (from right to left):  2 + 3 = 5. How many fives are there in 5? \( \frac{5}{5} \)= 1 remainder 0. Write 0 and carry 1 to column 2.

2nd Column: 1 + 4 + 0 = 5. How many fives are there in 5? \( \frac{5}{5} \)= 1 remainder 0. Write 0 and carry 1 to column 3.

3rd Column: 1 + 0 + 1 = 2. 2 is less than 5 so we write 2 down.

4th column: 3 + 2 = 5. How many fives are there in 5? \( \frac{5}{5} \)= 1 remainder 0. Write 0 and carry 1 to column 5.

5th column: We add 1 to the 5th column.

Answer = 10200₅

b. 2001_three   + 1022_three

Explanation:

1st Column (from right to left): 1 + 2 = 3. How many threes are there in 3? \( \frac{3}{3} \)= 1 remainder 0. Write 0 and carry 1 to column 2.

2nd Column: 1 + 0 + 2 = 3. How many threes are there in 3? \( \frac{3}{3} \)= 1 remainder 0. Write 0 and carry 1 to column 3.

3rd Column: 1 + 0 + 0 = 1. 1 is less than 3 so we write 1 down.

4th column: 2 + 1 = 3. How many threes are there in 3? \( \frac{3}{3} \)= 1 remainder 0. Write 0 and carry 1 to column 5.

5th column: We add 1 to the 5th column.

Answer = 10100₅

c. 151.24_six +   12.34_six  + 20.11_six

Explanation:

1st Column (from right to left):  4 + 4 + 1 = 9. How many six’s are there in 9? \( \frac{9}{6} \) = 1 remainder 3. Write 3 and carry 1 to column 2.

2nd Column: 1 + 2 + 3 + 1 = 7. How many six’s are there in 7? \( \frac{7}{6} \) = 1 remainder 1. Write 1 and carry 1 to column 3.

3rd Column: 1 + 1 + 2 + 0 = 4. 4 is less than 6 so we write 4 down.

4th column: 5 + 1 + 2 = 8. How many six’s are there in 8? \( \frac{8}{6} \) = 1 remainder 2. Write 2 and carry 1 to column 5.

5th column: 1 + 1 = 2. 2 is less than 6 so we write 6 down.

Answer = 224.13₆

d. 5346₈ – 637₈

1st Column (from right to left):  6 is less than 7 so we will have to borrow from the 2nd column. Since it is in base 8 any number borrowed is 8. So borrowing 8 from the second column will give (8 + 6) – 7 = 14 – 7 = 7

2nd Column: This reduces 4 to 3 in the second column. 3 – 3 = 0. We write 0 down.

3rd Column: 3 is less than 6 so we will have to borrow from the 4th column. Since it is in base 8 any number borrowed is 8. So borrowing 8 from the fourth column will give (8 + 3) – 6 = 11 – 6 = 5

4th Column: This reduces 5 to 4 in the fourth column. 4 – 0 = 4. We write 4 down.

Answer = 4507₈

e. 3002_four – 2213_four

Explanation:

1st Column (from right to left): 2 is less than 3 so we will have to borrow from the 2nd column. Since it is in base 4 any number borrowed is 4. So borrowing 4 from the second column will give (4 + 2) – 3 = 6 – 3 = 3

2nd Column: This reduces 0 to -1 in the second column. We will have to borrow (4) from the third column. 4 – 1 – 1 = 2. We write 2 down.

3rd Column: This reduces 0 to -1 in the third column. We will have to borrow (4) from the 4th column. 4 – 1 – 2 = 1. We write down 1.

4th Column: This reduces 3 to 2 in the fourth column. 2 – 2 = 0.

Answer = 123₄