To convert a decimal number to a binary number, we need to repeatedly divide the decimal number by 2 and take note of the remainder. The binary number is obtained by arranging the remainders in the reverse order. Here's the process for converting 89\(_{10}\) to binary:
1. Divide 89 by 2. The quotient is 44 and the remainder is 1.
2. Divide 44 by 2. The quotient is 22 and the remainder is 0.
3. Divide 22 by 2. The quotient is 11 and the remainder is 0.
4. Divide 11 by 2. The quotient is 5 and the remainder is 1.
5. Divide 5 by 2. The quotient is 2 and the remainder is 1.
6. Divide 2 by 2. The quotient is 1 and the remainder is 0.
7. Divide 1 by 2. The quotient is 0 and the remainder is 1.
The remainders, taken in reverse order, are 1011001, which is the binary equivalent of 89\(_{10}\). Therefore, the correct answer is option (B) 1011001.