To subtract the binary number [tex]\(10110_2\)[/tex] from [tex]\(101001_2\)[/tex], follow these detailed steps:
1. Convert the binary numbers to decimal equivalents:
- The binary number [tex]\(101001_2\)[/tex]:
- Start from the right, each digit represents a power of 2.
- [tex]\(1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\)[/tex]
- [tex]\(1 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 1 \cdot 1\)[/tex]
- Adding these, you get [tex]\(32 + 0 + 8 + 0 + 0 + 1 = 41\)[/tex].
- So, [tex]\(101001_2 = 41_{10}\)[/tex].
- The binary number [tex]\(10110_2\)[/tex]:
- Again, starting from the right.
- [tex]\(1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0\)[/tex]
- [tex]\(1 \cdot 16 + 0 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 0 \cdot 1\)[/tex]
- Adding these, you get [tex]\(16 + 0 + 4 + 2 + 0 = 22\)[/tex].
- So, [tex]\(10110_2 = 22_{10}\)[/tex].
2. Perform the subtraction in the decimal system:
- Subtract [tex]\(22\)[/tex] from [tex]\(41\)[/tex].
- [tex]\(41 - 22 = 19\)[/tex].
3. Convert the decimal result back to binary:
- To convert [tex]\(19_{10}\)[/tex] to binary:
- Divide [tex]\(19\)[/tex] by [tex]\(2\)[/tex] which gives quotient [tex]\(9\)[/tex] and remainder [tex]\(1\)[/tex].
- Divide [tex]\(9\)[/tex] by [tex]\(2\)[/tex] which gives quotient [tex]\(4\)[/tex] and remainder [tex]\(1\)[/tex].
- Divide [tex]\(4\)[/tex] by [tex]\(2\)[/tex] which gives quotient [tex]\(2\)[/tex] and remainder [tex]\(0\)[/tex].
- Divide [tex]\(2\)[/tex] by [tex]\(2\)[/tex] which gives quotient [tex]\(1\)[/tex] and remainder [tex]\(0\)[/tex].
- Divide [tex]\(1\)[/tex] by [tex]\(2\)[/tex] which gives quotient [tex]\(0\)[/tex] and remainder [tex]\(1\)[/tex].
- Writing the remainders from last to first gives [tex]\(10011_2\)[/tex].
So, the binary subtraction result of [tex]\(10110_2\)[/tex] from [tex]\(101001_2\)[/tex] is:
[tex]\[ 101001_2 - 10110_2 = 10011_2 \][/tex]
In decimal, the subtraction [tex]\(41 - 22\)[/tex] gives [tex]\(19\)[/tex] and the binary equivalent of [tex]\(19\)[/tex] is [tex]\(10011_2\)[/tex].
