1. Problem statement: Add the binary numbers $100100111$ and $111010010$. 2. Formula and important rules: Binary addition uses base 2 rules where each column sums bits and any value 2 or greater produces a carry to the next column. 3. Basic binary addition facts to remember: $0+0=0$ with carry 0. 4. Basic binary addition facts to remember: $0+1=1$ with carry 0. 5. Basic binary addition facts to remember: $1+1=0$ with carry 1 because $1+1=10_2$. 6. Basic binary addition facts to remember: $1+1+1=1$ with carry 1 because $1+1+1=11_2$. 7. Align the numbers by place value (right-aligned) and add from right to left. 8. Column-by-column addition with carries follows. 9. Position 0 (rightmost): $1 + 0 + 0 = 1$, so write $1$ and carry $0$. 10. Position 1: $1 + 1 + 0 = 10_2$, so write $0$ and carry $1$. 11. Position 2: $1 + 0 + 1 = 10_2$, so write $0$ and carry $1$. 12. Position 3: $0 + 0 + 1 = 1$, so write $1$ and carry $0$. 13. Position 4: $0 + 1 + 0 = 1$, so write $1$ and carry $0$. 14. Position 5: $1 + 0 + 0 = 1$, so write $1$ and carry $0$. 15. Position 6: $0 + 1 + 0 = 1$, so write $1$ and carry $0$. 16. Position 7: $0 + 1 + 0 = 1$, so write $1$ and carry $0$. 17. Position 8 (leftmost of inputs): $1 + 1 + 0 = 10_2$, so write $0$ and carry $1$. 18. After the leftmost column there is a carry $1$, so prepend $1$ to the result. 19. Putting the result bits from left to right gives $1011111001$. 20. Display of the addition in binary is: $$100100111 + 111010010 = 1011111001$$ 21. Verification in decimal: $100100111_2 = 295$. 22. Verification in decimal: $111010010_2 = 466$. 23. The decimal sum is $295 + 466 = 761$, which equals $1011111001_2$. 24. Final answer: $1011111001$.