This book constitutes the refereed post-conference proceedings of the 5th International Conference for Next Generation Arithmetic, CoNGA 2024, held in Sydney, NSW, Australia, during February 20-21, 2024. The 5 revised full papers presented were carefully selected from 9 submissions. CoNGA is the leading conference on emerging technologies for computer arithmetic. The demands of both AI and HPC have led the community to realize that something better than traditional floating-point arithmetic is needed to reach the speed, accuracy, and energy-efficiency that are needed for today's most challenging workloads. In particular, posit arithmetic is achieving rapid adoption as a non-proprietary format, but CoNGA welcomes papers about any arithmetic format that breaks from the past and shows merit and promise.