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Time as the 4^th Dimension

Ok, so we now know about the Lorentz Transformation Equations. So why is time the 4^th dimension?

Here's a simple answer:

1). From the onset of John Harrison's H-4 clock-watch [ 1761-62 and 1764 ] where

 "Time of longitude, adjusted for latitude, and hence converted to longitudinal distance traveled " and " ... John Harrison's H-4 clock-watch [ ... ] relates the idea of time with space-distance" 

already there had been made a direct connection between space-distance and time. [ see: "Distance and the John Harrison time clocks which measure it" ]

2). Also we know that

from before. 

It's obvious that in order to describe the position of an event E in space-time S that it essentially takes 4 coordinates, namely (x, y, z, t). 

So there you have it: time as the 4^th dimension. It's really a mathematical thing. 

But how do advanced mathematicians actually write all this down?

This then gives us the Lorentz Transformation Equations [ see above ] using some advanced matrix multiplication. Actually we are also verging upon Tensor Analysis which is an advanced form of matrix analysis whose usage Einstein made salient in Special Relativity  mathematics.

In other words, all of the Lorentz Transformation Equations can be written as

which is always something to be admired for its mathematical elegance and simplicity!!

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