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Nov3-11, 09:05 PM
P: 800
Quote Quote by weetabixharry View Post
I was browsing through the Notation section of my favourite book and noticed that symbols were defined for the "sets" of real, integer and natural numbers... but for the "field" of complex numbers.

Is the terminology different because a complex number comprises two components (real and imaginary)?

I don't want a cripplingly in-depth definition - Wikipedia tells me a field is a "commutative ring whose nonzero elements form a group under multiplication". This means almost nothing to me.
Ok here's a cripplingly easy explanation.

Say you had the familiar integers: 0, 1, 2, 3, 4, 5, ... and their negatives. But you forgot how to add or multiply. So you can recognize that, say, 47 is different than 169. You can identify each of the numbers. But you can't DO anything with them.

That's a set.

Now if someone hands you the addition and multiplication tables so that you can do arithmetic -- that's a commutative ring. In other words it's a set of objects along with some rules for doing operations on them.

If besides multiplying you can also divide, then its a field. For example you can't divide 3 by 4, so the natural numbers aren't a field.

But the complex numbers are a field. You can in fact divide one complex number by any other nonzero complex number and get a unique answer.

So the complex numbers, along with the usual addition and multiplication, are a field.