Addition in Other Bases

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Addition in Other Bases

When adding binary numbers some simple rules can be used.

The rules of binary addition are:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 with a Carry of 1
1 + 1 + 1 = 1 with a Carry of 1

But we can also use another technique for the addition of binary and other bases, as shown below.

Example 1.4.1:

Evaluate the following:

1. 111100_two + 10111_two + 10011_two
2. 110.011_two + 10.111_two+ 1.01_two
3. 1304₇ + 46₇+ 1426₇
4. 5D6C₁₆ + 6ABF₁₆ + 3D2₁₆

1. 111100_two + 10111_two + 10011_two

Solution

Explanation

1^st Column (from right to left): 0 + 1 + 1 = 2. How many twos are there in 2? The answer is 1 remainder 0. Write 0 and carry 1 to column 2.

2^nd Column: 1 + 0 + 1 + 1 = 3. How many twos are there in 3? The answer is 1 remainder 1. Write 1 and carry 1 to column 3.

3^rdColumn: 1 + 1 + 1 + 0 = 3. How many twos are there in 3? The answer is 1 remainder 1. Write 1 and carry 1 to column 4.

4^th Column: 1 + 1 + 0 + 0 = 2. How many twos are there in 2? The answer is 1 remainder 0. Write 0 and carry 1 to column 5.

5^th Column: 1 + 1 + 1 + 1 = 4. How many twos are there in 4? The answer is 2 remainder 0. Write 0 and carry 2 to column 6.

6^th Column: 2 + 1 = 3. How many twos are there in 3? The answer is 1 remainder 1. Write 1 and carry 1 to the next (7^th)column.

This yields 1100110_two

2. 110.011_two + 10.111_two+ 1.01_two

Solution

Explanation

1st Column (from right to left): 1 + 1 + 0 = 2. How many twos are there in 2? The answer is 1 remainder 0. Write 0 and carry 1 to column 2.

2nd Column: 1 + 1 + 1 + 1 = 4. How many twos are there in 4? The answer is 2 remainder 0. Write 0 and carry 2 to column 3.

3rd Column: 2 + 0 + 1 + 0 = 3. How many twos are there in 3? The answer is 1 remainder 1. Write 1 and carry 1 to column 4.

4th Column: 1 + 0 + 0 + 1 = 2. How many twos are there in 2? The answer is 1 remainder 0. Write 0 and carry 1 to column 5.

5th Column: 1 + 1 + 1 = 3. How many twos are there in 3? The answer is 1 remainder 1. Write 0 and carry 1 to column 6.

6th Column: 1 + 1 = 2. How many twos are there in 2? The answer is 1 remainder 0. Write 0 and carry 1 to column 7.

This yields 1010.100_two

3. 1304₇ + 46₇+ 1426₇

Solution

Explanation

1st Column (from right to left): 4 + 6 + 6 = 16. How many sevens are there in 16? The answer is 2 remainder 2. Write 2 and carry 2 to column 2.

2nd Column: 2 + 0 + 4 + 2 = 8. How many sevens are there in 8? The answer is 1 remainder 1. Write 1 and carry 1 to column 3.

3rd Column: 1 + 3 + 4 = 8. How many sevens are there in 8? The answer is 1 remainder 1. Write 1 and carry 1 to column 4.

4th Column: 1 + 1 + 1 = 3. How many sevens are there in 3? The answer is 3 is not up to 7. Finally, Write 3.

This yields 3112₇

4. 5D6C₁₆ + 6ABF₁₆ + 3D2₁₆

Solution | Hex Addition

Explanation

Hex addition follows the same rules as decimal addition with the only difference being the added numerals A, B, C, D, E, and F. It may be convenient to have the decimal equivalent values of A through F handy when performing hex operations if the values have not yet been committed to memory. Below is an example of hex addition. Work through the example, and refer to the text below it for further details.

Remember A = 10, B = 11, C= 12, D = 13, E = 14, F = 15

1st Column (from right to left): C + F + 2 = 13 + 15 + 2 = 29. How many sixteens are there in 29? The answer is 1 remainder 13. But D = 13, so write D and carry 1 to column 2.

2nd Column: 1 + 6 + B + D = 1 + 6 + 11 + 13 = 31. How many sixteens are there in 31? The answer is 1 remainder 15. But F = 15, so write F and carry 1 to column 3.

3rd Column: 1 + D + A + 3 = 1 + 13 + 10 + 3 = 27. How many sixteens are there in 27? The answer is 1 remainder 11. But B = 11, so write B and carry 1 to column 4.

4th Column: 1 + 5 + 6 = 12. 12 is not up to 16. So write 12 in this column which is also C.

This yields CBFD₁₆