The Babylonians liked using base 60, or sexagesimal, favouring it nearly four thousand years ago, taking inspiration from the Sumerians for their system. From an outsider's perspective, this may seem peculiar, compared to the seemingly more simple base 10 (decimal) system used by most today. However, 60 is perhaps a more useful number than 10 when dividing numbers.
60 has a wide range of factors - 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 itself, compared to the measly 1, 2, 5 and 10 that 10 offers. Not just that, but you can count up to 60 using your hands, too - count each phalanx on your fingers on one hand (not including your thumb). There are twelve in all, and by counting how many times you reach twelve, you end up at sixty (5x12 - five fingers in all on one hand). As well as this, the system was likely to be practical to the Babylonians, with a theory suggesting that sexagesimal would have eased trading between those who used a base 5 system and those using a base 12 system.
To be fair, you can also count up to different numbers by using different systems, such as 27 in the body-tallying system in many languages across New Guinea and Australia. Thus it might be said that 60 isn't the all perfect number for a base counting system - the Babylonians didn't use zero (though they eventually adopted a symbol to represent the concept of nothing), and the way how numbers would be represented in Cuneiform itself uses a sub-base 10 system (as in the image on this Wikipedia link), which at least means you don't have sixty symbols completely independent of each other.
Either way, the legacy of sexagesimal lives on to this day - there are 360° in a circle, and there are sixty seconds in a minute and sixty minutes in an hour. And let's not forget that the Babylonians also came up with a good approximation of √2 (info at bottom of page) using base 60 as well. Why shouldn't we take joy in it?
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