We design a systolic coprocessor for the addition of signed normalized rational numbers. This is the most complicated rational operation: it involves GCD, exact division, multiplication and addition/subtraction. In particular the implementation of GCD and exact division improve significantly (2 to 4 times) previously known solutions. In contrast to the traditional approach, all operations are performed least-significant digits first. This allows bit-pipelining between partial operations at reduced area-cost. An Atmel FPGA design for 8-bit operands consumes 730 cells (3,500 equivalent gates) and runs at 25 MHz (5 MHz after layout). For 32-bit operands this would be in the same timing range as the software solutions, however a significant speed-up can be expected for longer operands because the linear time-complexity of the hardware algorithms.