Circular Logic: The Modulo Operator

We previously established the core index-wrapping formula, (Current Index + 1) % MAX_SIZE, which is essential for creating the continuous loop effect required by a Circular Queue. This mechanism relies entirely on the mathematical properties of the Modulo Operator (%) to handle index constraint and wrap-around behavior.

The modulo operator is fundamentally a way to map an infinite number line onto a finite, fixed range. For any index calculation (I + 1) % N, where N is MAX_SIZE:

• Normal Advance (I < N - 1): When the index plus one is less than MAX_SIZE, the division returns a remainder equal to I + 1. Pointers (front or rear) advance normally, preserving the $O(1)$ time complexity.
• Wrap Around (I = N - 1): When the index reaches the last valid position, I + 1 equals MAX_SIZE. Dividing MAX_SIZE by itself always yields a remainder of 0. This mathematically forces the pointer back to the beginning of the array.

Key Takeaway: The use of modulo guarantees the resulting index will always be in the valid range of [0, MAX_SIZE - 1], preventing out-of-bounds access and solving the False Overflow problem.