Modern deep learning models are often trained in parallel over a collection of distributed machines to reduce training time. In such settings, communication of model updates among machines becomes a significant performance bottleneck and various lossy update compression techniques have been proposed to alleviate this problem. In this work, we introduce a new, simple yet theoretically and practically effective compression technique: natural compression (Cnat). Our technique is applied individually to all entries of the to-be-compressed update vector and works by randomized rounding to the nearest (negative or positive) power of two, which can be computed in a natural way by ignoring the mantissa. We show that compared to no compression, Cnat increases the second moment of the compressed vector by not more than the tiny factor 9/8, which means that the effect of Cnat on the convergence speed of popular training algorithms, such as distributed SGD, is negligible. However, the communications savings enabled by Cnat are substantial, leading to 3-4x improvement in overall theoretical running time. For applications requiring more aggressive compression, we generalize Cnat to natural dithering, which we prove is exponentially better than the common random dithering technique. Our compression operators can be used on their own or in combination with existing operators for a more aggressive combined effect, and offer new state-of-the-art both in theory and practice.