Maths: When Compression Ratios Become Irrational

Compression ratios sit at the intersection of the real number system and practical digital audio, and they reveal something surprising: not every compression outcome is a tidy fraction. When a WAV file is compared with its MP3 counterpart, the ratio of MP3 size to WAV size simplifies to a rational number such as 1/10, since both sizes are integers. This rational result reflects a clean, measurable reduction in data. The concept matters because compression ratios determine whether audio information survives intact. A lossless format preserves every sample, while a lossy format discards data permanently. Introducing an irrational ratio, such as 1/√10, changes the picture: multiplying a rational file size by this ratio yields an irrational result like 3√10 MB, since √10 cannot be written as p/q. This distinction between rational and irrational outcomes mirrors the deeper split between exact preservation and irreversible loss, shaping how producers archive recordings for the future.