The special relativity equations
Lorentz factortime dilation formulalength contractionE = mc2relativistic equations
The Lorentz factor and the three consequences that follow from it: time dilation, length contraction and mass-energy equivalence.
Lorentz Factor
The relativistic stretch factor, written in terms of β = v/c.
Time Dilation
A moving clock's proper time Δt₀ stretches to Δt for a stationary observer; β = v/c.
Length Contraction
A moving object's rest length L₀ contracts to L along its direction of motion; β = v/c.
Mass-Energy Equivalence (E = mc²)
The rest energy locked in mass: multiply by the speed of light squared.
How they fit together
Compute γ first and the rest is arithmetic. The Lorentz factor is the whole of special relativity's kinematics compressed into one number: at 10% of light speed it is 1.005 and nothing much happens, at 87% it is 2, at 99.9% it is about 22. Time dilation multiplies by γ, length contraction divides by it, and E = mc² is the energy budget that makes γ matter — the rest energy a particle carries before it moves at all.
The trap is not the algebra, it is knowing whose clock and whose ruler. The proper time is measured by the clock that is present at both events — the one riding along — and every other observer measures a longer interval. The proper length is measured in the frame where the object is at rest, and everyone else measures it shorter, along the direction of motion only. Get those backwards and you will confidently produce a moving clock that runs fast. Note also that these are for constant relative velocity; acceleration and gravity are general relativity's business, which is why GPS satellites need both corrections and they pull in opposite directions.