Sure, let's add the binary numbers [tex]\(1001_2\)[/tex] and [tex]\(1100_2\)[/tex] step by step.
### Step-by-Step Binary Addition:
1. Align the two binary numbers:
```
1001
+ 1100
```
2. Add the binary numbers starting from the rightmost bit.
- Rightmost column: [tex]\(1 + 0 = 1\)[/tex] (no carry)
- Next column: [tex]\(0 + 0 = 0\)[/tex] (no carry)
- Next column: [tex]\(0 + 1 = 1\)[/tex] (no carry)
- Leftmost column: [tex]\(1 + 1 = 10_2\)[/tex], which means 0 with a carry of 1.
3. Since there is a carry of 1 to the leftmost bit:
```
101
1001
+ 1100
------
10101
```
So, the result of the addition in binary is [tex]\(10101_2\)[/tex].
### Conversion to Decimal (to verify):
4. Convert [tex]\(1001_2\)[/tex] to decimal:
- [tex]\(1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\)[/tex]
- [tex]\(8 + 0 + 0 + 1 = 9\)[/tex]
- Hence, [tex]\(1001_2 = 9_{10}\)[/tex]
5. Convert [tex]\(1100_2\)[/tex] to decimal:
- [tex]\(1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0\)[/tex]
- [tex]\(8 + 4 + 0 + 0 = 12\)[/tex]
- Hence, [tex]\(1100_2 = 12_{10}\)[/tex]
6. Add the decimal numbers:
- [tex]\(9 + 12 = 21\)[/tex]
7. Convert the result [tex]\(21_{10}\)[/tex] back to binary:
- [tex]\(21 = 16 + 4 + 1 = 2^4 + 2^2 + 2^0\)[/tex]
- [tex]\(21_{10} = 10101_2\)[/tex]
By confirming both the binary and decimal additions, we see that the binary addition result [tex]\(10101_2\)[/tex] is correct, and it matches the decimal addition confirmation of 21.
