The faster gamma case drives both the longer-time and shorter-length ledgers. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The faster triangle still gives exact time

Gamma 5/3 turns proper time 3 s into lab time 5 s.

353=53\cdot\frac{5}{3}=5
Time dilationGamma scales a moving clock interval in the lab.speedquiet leglimitbetagammaproper timelab time

The same gamma shortens length

The same gamma turns rest length 5 m into moving length 3 m.

5÷53=35\div\frac{5}{3}=3
Length contractionGamma divides the rest length along motion.speedquiet leglimitbetagammarest lengthmoving length