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Leap seconds are seconds added to "official" time to account for changes in the Earth's rotation. These changes are largely unpredictable, including changes from melting glaciers and earthquakes, as well as the predictable (very minor) slow of the Earth's orbit. That's about it, really.
 Creationist claim
Young Earth creationists have claimed that leap seconds have had to be inserted into the year 22 times between 1970 and 1999, showing that the Earth is slowing 0.77 seconds per year. At this rate, the Earth would have slowed to a stop if it were billions of years old.
There are several issues with this claim:
- So-called "leap seconds" have nothing to do with any change in the Earth's rotation. Rather, leap seconds are a consequence of the fact that human timekeeping systems are arbitrary human constructs which don't match up exactly against the Earth's rotation. Just as those every-four-years leap days are not a sign that the Earth is slowing down in its orbit around the Sun, so it is that leap seconds are not a sign that the Earth's rotation is slowing down.
- If the Earth's movement around the Sun were slowing down, we wouldn't need a constant rate of leap seconds but an increasing rate.
- The earth's rotation (not to be confused with its movement around the Sun) is slowing down, but not at such a great rate and not in this direction. The length of a day now is very slightly more than 24 hours. Therefore the year will have fewer days in the far future (because the days get longer and the year stays the same), and one will have to remove more and more leap seconds.
- The claim grossly exaggerates the rate of change of Earth's day. In reality, it only decreases by 1.7 milliseconds per century, whereas the claim states it decreases by 77,000 milliseconds per century. This is wrong by four orders of magnitude and a factor of 4.
- Creationists who use this claim fail to, or refuse to appreciate how slow the rate of slowing actually is… 450 million years ago, days in the Ordovician period were around 22 hours long.
 Real-life problems
Computers tend to choke on them. Linux (which large chunks of the Internet runs on) implements leap seconds by winding back a second; this caused a pile of problems during the 23:59:60 30 June 2012 leap second, as things expecting the time to always increase and never go backwards got horribly confused and fell over, causing sysadmins much, ah, entertainment early on a Sunday morning. This was fixed for the 2015 leap second.
- See the Wikipedia article on Leap second.
- Robinson, B.A., 2002. A failed attempt to dialog with creation scientists, 
- Thwaites, William M. and Frank T. Awbrey, Summer 1982. As the World Turns: Can Creationists Keep Time? Creation/Evolution IX: 18-22, 
- NIST, updated monthly. NIST Time Scale Data Archive.