Question

Simplity 101011 base 2+ 10111-10101 base 2

Answer 1

To simplify the binary expression \( 101011_2 + 10111_2 - 10101_2 \), we will first convert each binary number to decimal, perform the calculations in decimal, and then convert the result back to binary if necessary.

1. **Convert each binary number to decimal:**

\[
101011_2 = 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 0 + 8 + 0 + 2 + 1 = 43_{10}
\]

\[
10111_2 = 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 0 + 4 + 2 + 1 = 23_{10}
\]

\[
10101_2 = 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 16 + 0 + 4 + 0 + 1 = 21_{10}
\]

2. **Perform the operations in decimal:**
\[
43 + 23 - 21 = 65
\]

3. **Convert the result back to binary:**
To convert \( 65_{10} \) to binary, we divide by 2 and keep track of the remainders.

- \( 65 \div 2 = 32 \) remainder 1
- \( 32 \div 2 = 16 \) remainder 0
- \( 16 \div 2 = 8 \) remainder 0
- \( 8 \div 2 = 4 \) remainder 0
- \( 4 \div 2 = 2 \) remainder 0
- \( 2 \div 2 = 1 \) remainder 0
- \( 1 \div 2 = 0 \) remainder 1

Reading the remainders from bottom to top gives us \( 1000001_2 \).

Thus, the final simplified result is:

\[
1000001_2
\]