The addition of 10101 and 00111, when interpreted as binary numbers, results in 11100.
Understanding Binary Addition
Binary numbers, or base-2 numbers, are fundamental in computing and digital electronics, using only two digits: 0 and 1. Performing addition with binary numbers follows a set of rules distinct from decimal (base-10) addition, primarily due to the limited set of digits and the way carries are handled.
Core Rules for Binary Addition
The basic principles for adding two binary digits are straightforward:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (This means the sum is 0, with a carry-over of 1 to the next more significant bit position.)
- 1 + 1 + 1 = 11 (This scenario occurs when two digits are 1, and there's also a carry-in from the previous column. The sum is 1, with a carry-over of 1 to the next position.)
Step-by-Step Calculation: 10101 + 00111
Let's break down the addition of 10101 and 00111 column by column, starting from the rightmost digit (the least significant bit).
| Position (from Right) | First Number (10101) | Second Number (00111) | Carry-in | Sum Digit | Carry-out |
|---|---|---|---|---|---|
| 1st (Rightmost) | 1 | 1 | 0 | 0 | 1 |
| 2nd | 0 | 1 | 1 | 0 | 1 |
| 3rd | 1 | 1 | 1 | 1 | 1 |
| 4th | 0 | 0 | 1 | 1 | 0 |
| 5th (Leftmost) | 1 | 0 | 0 | 1 | 0 |
Here’s a detailed walkthrough of each step:
-
Rightmost Column (1 + 1):
- 1 plus 1 in binary equals 10.
- We write down 0 in the result for this column and carry-over 1 to the next column on the left.
-
Second Column from Right (0 + 1 + carry-over 1):
- First, 0 plus 1 equals 1.
- Then, this result (1) plus the carry-over (1) equals 10 in binary.
- We write down 0 in the result for this column and carry-over 1 to the next column.
-
Third Column from Right (1 + 1 + carry-over 1):
- First, 1 plus 1 equals 10 in binary.
- Then, this result (10) plus the carry-over (1) equals 11 in binary.
- We write down 1 in the result for this column and carry-over 1 to the next column.
-
Fourth Column from Right (0 + 0 + carry-over 1):
- First, 0 plus 0 equals 0.
- Then, this result (0) plus the carry-over (1) equals 1.
- We write down 1 in the result for this column. There is no carry-over.
-
Fifth Column from Right (1 + 0):
- 1 plus 0 equals 1.
- We write down 1 in the result for this column. There is no carry-over.
By assembling the digits from the leftmost column to the rightmost column, the final sum obtained is 11100. This demonstrates how the binary addition of 10101 and 00111 correctly yields 11100.
