For the record, 4k does not mean "4x 1080p". It means approximately 4,000 horizontal pixels.
Up to the present, we've always measured resolution in shorthand by the number of vertical pixels, e.g. 720p, 1080i, 1080p. But the true number of vertical pixels changes based on aspect ratio: if a movie is 16:9, it truly can have 1080 vertical pixels, but if it is 2.35:1 or a different aspect ratio, it can still be 1920 pixels horizontally, but it is no longer 1080 vertical pixels.
To avoid confusion over aspect ratios incorrectly representing resolutions, the decision has been made to switch to tracking the horizontal resolution. Also, they decided to round it for convenience. Next-gen is 4k, followed by 8k. Current, by the same system, is 2k. Each time, they are doubling the width (and height follows suit), which results in 4x the resolution per step-up.
What I don't know is if we'll always use numbers under the rounded figure, like 1920 under 2k or 7680 under 8k. Some have attempted to establish exactly 4,000 pixels horizontal, and others 4,096 pixels horizontal. It probably doesn't make that big of a difference visually, but as resolutions increase, our specifications stray further and further from the actual measurements, just like selling hard drives listing the capacity assuming a TB is 1,000,000,000,000 bytes, rather than the actual space which comes out less, since our computers see each TB as 1,099,511,627,776. Not a big deal with the old MB hard drives, but when we start seeing petabyte and exabyte arrays it is increasingly deceiving, since the rounding errors are compounded with each larger prefix.
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