Let's solve the equation [tex]\( 110_{\text{two}} + X_{\text{two}} = 1000_{\text{two}} \)[/tex] step-by-step.
1. Convert the binary numbers to decimal:
- Binary [tex]\(110_{\text{two}}\)[/tex] to decimal:
[tex]\( 110_{\text{two}} \)[/tex] can be written as:
[tex]\[
1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 4 + 2 + 0 = 6
\][/tex]
So, [tex]\( 110_{\text{two}} = 6_{\text{ten}} \)[/tex].
- Binary [tex]\(1000_{\text{two}}\)[/tex] to decimal:
[tex]\( 1000_{\text{two}} \)[/tex] can be written as:
[tex]\[
1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0 = 8 + 0 + 0 + 0 = 8
\][/tex]
So, [tex]\( 1000_{\text{two}} = 8_{\text{ten}} \)[/tex].
2. Set up the equation in decimal:
[tex]\[
6 + X = 8
\][/tex]
3. Solve for [tex]\(X\)[/tex]:
[tex]\[
X = 8 - 6 = 2
\][/tex]
4. Convert the decimal result back to binary:
- Decimal [tex]\(2_{\text{ten}}\)[/tex] to binary:
[tex]\[
2 \div 2 = 1 \text{ remainder } 0 \quad \rightarrow \quad 1 \div 2 = 0 \text{ remainder } 1
\][/tex]
Therefore, [tex]\(2_{\text{ten}} = 10_{\text{two}}\)[/tex].
Hence, the value of [tex]\(X_{\text{two}}\)[/tex] is [tex]\(10_{\text{two}}\)[/tex].
