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to determine the value of a number
A value of a number? What do you mean by that?
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whether a number is equal to or greater than one
Like a>=1 ? I don't think that that is a problem suitable for neural networks to solve. It is simply too simple, and doesn't really involve or require any AI - it's pure mathematics. You should probably try something else, like a very simple pattern recognition.
Another thing is: usually a neuron takes more than one input.
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It also doesn’t help that I am a poor programmer and worse mathematician :D
No, it certainly doesn't. I suggest that you bring your programming skills up to a level that makes you confident of what you are doing when you are coding. Maths is really important, because many aspects of programming involve a thorough knowledge of different mathematical concepts. AI programming is no different. So before you continue, you probably should increase your mathematical understanding.
Besides that, Neural Networks (and AI in general) are not the subject you usually want to study when you are a beginner. They are rather complex. So why not try something easier?
But, if you want to continue anyway, you should get yourself a good book about AI. The tutorials you get online are handy, but do not give you the insight you really need. I would suggest for example "AI Techniques for Game Programming" by Mat Buckland. And you must probably study genetic algorithms before you even begin to look into neural nets. These two subjects are very often connected.
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I was told that they were random, is this true?
Well, yes the wights are random. But after a while they are altered (or calibrated, if you wish) so that they "learn" how to solve the problem more efficiently. That is usually done with good help from genetic algorithms.
The threshold, though, does not change. It is constant and is usually added as an additional negative weight (which you will learn if you study NN closely). But it might be that the threshold in some problems is random. Though I personally don't see purpose of that approach.