To simplify the binary subtraction [tex]\(1110_2 - 111_2\)[/tex], follow these steps:
1. Convert the binary numbers to decimal:
- For [tex]\(1110_2\)[/tex]:
[tex]\[
1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 8 + 4 + 2 + 0 = 14
\][/tex]
Therefore, [tex]\(1110_2 = 14_{10}\)[/tex].
- For [tex]\(111_2\)[/tex]:
[tex]\[
1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 4 + 2 + 1 = 7
\][/tex]
Therefore, [tex]\(111_2 = 7_{10}\)[/tex].
2. Perform the subtraction in decimal form:
[tex]\[
14_{10} - 7_{10} = 7_{10}
\][/tex]
3. Convert the result back to binary:
- Convert [tex]\(7_{10}\)[/tex] to binary:
[tex]\[
7 \div 2 = 3 \text{ remainder } 1 \\
3 \div 2 = 1 \text{ remainder } 1 \\
1 \div 2 = 0 \text{ remainder } 1
\][/tex]
Reading the remainders from bottom to top, we get [tex]\(111_2\)[/tex].
Thus, the simplified form of [tex]\(1110_2 - 111_2\)[/tex] is [tex]\(111_2\)[/tex].
