Algorithms are much, much older than computers (the word comes from the name of a 9th-century Persian mathematician). Simply put, an algorithm is just a set sequence of instructions to solve a problem or perform some kind of computation. Lots of the pen-and-paper math done in school, like long division, could be described as algorithms.
These days the “algorithms” most people are talking about are the recommendation systems used by online stores, social media, internet marketers, etc. E.g. YouTube takes your viewing history and puts that into an algorithm that recommends other videos for you to watch.
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