I recently started self-learning special relativity. I am having lots of trouble convincing myself about the Lorentz transformations. I realize that I must be doing something really stupid, but I just can't pinpoint it. I get that the equation for x' follows from the fact that "rulers" used in the reference frame of x' is shorter than the rulers used in x by a factor of sqrt(1-(v/c)^2), so that the x' coordinate must be equal to the x-coordinate shifted by how far the two frames have moved relatively and then divided by sqrt(1-(v/c)^2). But for the time equation, I tried using a similar reasoning. The person in the x' frame measures time with a clock which, according to the person in the x frame, is slower. That means that if in x, an event happened t seconds after some time coordinate, the event happened after a shorter interval of time the x' frame. Doesn't that mean that t' should be divided by the time dilation factor? So basically, my understanding is that clocks are sort of like time-rulers. But length is contracted, while time is dilated. How can it be that the equations for t' and x' have the save factor in the denominator? Thanks in advance for any help!