Convert the decimal number 99 into its equivalent binary (base 2) representation.

Introduction / Context:
This question is from the number systems and base conversion topic. You are asked to convert a decimal number into its binary equivalent. Binary numbers are base 2 numbers and use only the digits 0 and 1. Understanding how to convert between decimal and binary is important in computer science, digital electronics, and competitive exams that include basic computer awareness.

Given Data / Assumptions:
- The decimal number to convert is 99.
- The target base is binary (base 2).
- We must express 99 as a sum of powers of 2 and then write the corresponding binary digits.

Concept / Approach:
To convert a decimal number to binary, we repeatedly divide the number by 2 and record the remainders, or we decompose the number into a sum of distinct powers of 2. The presence of a power of 2 in the sum corresponds to a binary 1 in that position, and its absence corresponds to a binary 0. Powers of 2 near 99 include 64, 32, 16, 8, 4, 2, and 1.

Step-by-Step Solution:
Step 1: Identify the largest power of 2 less than or equal to 99. That power is 64, which is 2^6.Step 2: Subtract 64 from 99 to get the remaining value: 99 - 64 = 35.Step 3: The next lower power of 2 is 32, which is 2^5. Subtract 32 from 35 to get 35 - 32 = 3.Step 4: The next power of 2 below 32 is 16 (2^4), but 3 is less than 16, so there is a 0 in the 2^4 position.Step 5: Similarly, for 8 (2^3) and 4 (2^2), the remaining value 3 is less than both, so there are 0s in the 2^3 and 2^2 positions.Step 6: The next power is 2 (2^1). Since 3 is at least 2, include 2 in the sum and subtract: 3 - 2 = 1. This means the 2^1 position is 1.Step 7: Finally, the remaining 1 corresponds to 2^0, so there is a 1 in the 2^0 position.Step 8: Putting these together from 2^6 to 2^0, we have 1 (for 64), 1 (for 32), 0 (for 16), 0 (for 8), 0 (for 4), 1 (for 2), and 1 (for 1).Step 9: Therefore, the binary representation is 1100011.

Verification / Alternative check:
Convert 1100011 back to decimal to confirm. The binary 1100011 means 1 * 64 + 1 * 32 + 0 * 16 + 0 * 8 + 0 * 4 + 1 * 2 + 1 * 1. Calculate this as 64 + 32 + 2 + 1 = 99. This matches the original decimal number, confirming the correctness of the conversion.

Why Other Options Are Wrong:
Option 1100101 corresponds to 64 + 32 + 4 + 1 = 101. Option 1101001 represents 64 + 32 + 8 + 1 = 105. Option 11100011 is an eight bit number equal to 128 + 64 + 32 + 2 + 1 = 227. Option 1011111 is 64 + 16 + 8 + 4 + 2 + 1 = 95. None of these equal 99, so they are not valid binary equivalents of 99.

Common Pitfalls:
One common mistake is to miscalculate powers of 2 or their sums. Another is to arrange the binary digits in the wrong order when writing the final answer. Some learners also confuse this conversion with the repeated division method but forget to reverse the sequence of remainders. A careful stepwise decomposition into powers of 2 avoids these errors.

Final Answer:
The decimal number 99 in binary is 1100011.