To compute the sum of the binary numbers [tex]\(1010_2\)[/tex] and [tex]\(1010_2\)[/tex], follow these steps:
1. Convert the binary numbers to decimal:
- For the binary number [tex]\(1010_2\)[/tex]:
- The rightmost digit represents [tex]\(2^0 = 1\)[/tex] and its value is 0.
- The next digit to the left represents [tex]\(2^1 = 2\)[/tex] and its value is 1.
- The next digit to the left represents [tex]\(2^2 = 4\)[/tex] and its value is 0.
- The leftmost digit represents [tex]\(2^3 = 8\)[/tex] and its value is 1.
- Adding up these values:
[tex]\[
1 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 0 \cdot 1 = 8 + 0 + 2 + 0 = 10
\][/tex]
- Therefore, [tex]\(1010_2\)[/tex] converts to [tex]\(10_{10}\)[/tex] in decimal.
2. Repeat the conversion for the second binary number [tex]\(1010_2\)[/tex]:
- Using the same process as above, [tex]\(1010_2\)[/tex] also converts to [tex]\(10_{10}\)[/tex] in decimal.
3. Compute the sum of the decimal numbers:
- Add the two decimal numbers obtained:
[tex]\[
10 + 10 = 20
\][/tex]
- Therefore, the sum of [tex]\(1010_2\)[/tex] and [tex]\(1010_2\)[/tex] in decimal is [tex]\(20_{10}\)[/tex].
4. Convert the decimal sum back to binary:
- To convert [tex]\(20_{10}\)[/tex] to binary:
- Divide 20 by 2, quotient is 10, remainder is 0.
- Divide 10 by 2, quotient is 5, remainder is 0.
- Divide 5 by 2, quotient is 2, remainder is 1.
- Divide 2 by 2, quotient is 1, remainder is 0.
- Divide 1 by 2, quotient is 0, remainder is 1.
- Reading the remainders from bottom to top, the binary equivalent is [tex]\(10100_2\)[/tex].
Therefore, the sum of the binary numbers [tex]\(1010_2\)[/tex] and [tex]\(1010_2\)[/tex] is [tex]\(10100_2\)[/tex].
