As a programmer are you suppose to have all methods of java API along with its implementation in your memory?
Like there are so many of them and even just for a single class in the library one can just look at the java documentation or the api?
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I always use an IDE and all the methods are shown to me by the IDE. So, is this good or bad.
You do not have to memorize all of them, as you use them you will become more familiar. It is however a good idea to look at the API documentation to get a better idea of what is out there so when you start a programming project you do not have to guess at what potential methods you might need.
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There is nothing wrong with using an IDE. They are good tools, but it is good to at least be familiar with the most common methods and classes. I don't generally use an IDE for Java (although I have tried and liked Eclipse: http://www.eclipse.org/downloads/), so on the occassions where I find myself lacking the knowledge of an existing class I either a) create my own, or b) use Google to find and existing solution. I have also found a good number of great API and classes at Apache Commons: http://commons.apache.org/