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does complex number have anything like consicutive numbers? . Like 2,3,4 are consicutive integers...
There is one similarity between natural numbers and complex numbers which is that two concecutive integers multiplied together then multiplied by 4 is one less than a perfect square. In other words if n is an integer then 4n(n+1)+1 is a perfect square. Likewise if z is a Gausian integer then 4z(z+1) + 1 is a perfect complex square. This leads me to define two consecutive Gausian Integers to be z and z plus 1.does complex number have anything like consicutive numbers? . Like 2,3,4 are consicutive integers...
I think Halls was referring to the property of being dense(http://en.wikipedia.org/wiki/Dense_set).Sure they do. Every set can be well-ordered.
Give the complex numbers a well-ordering. For each complex number z, consider the minimal element of the set of complex numbers larger than z. This will be the least number larger than z, in other words, a consecutive element.
Now this of course applies to R and Q as well, but note that this cannot correspond to their usual order. The complex numbers however don't have any natural ordering (which satisfies e.g. a < b --> a+c < b+c), so I don't see how the order properties of the real numbers extends to the complex numbers.
no, which is a bit frustrating. we can't "totally order" the complex numbers in a way that corresponds to our notion of "bigger".does complex number have anything like consicutive numbers? . Like 2,3,4 are consicutive integers...
But you can never make an "ordered field" out of them: http://en.wikipedia.org/wiki/Ordered_fieldsuppose for numbers with the same norm, we arbitrarily list numbers counter clockwise from zero degrees. Then we can definitively say that one complex number comes before or after another in our catalog.
the trouble with this, is that even though we keep going "bigger" (counter-clockwise), we eventually "come full circle" (as we approach 360 degrees). this is the same kind of trouble with have with ordering finite fields: since p-1 = -1, is this number < 0 or > 0 (apparently it's BOTH).suppose for numbers with the same norm, we arbitrarily list numbers counter clockwise from zero degrees. Then we can definitively say that one complex number comes before or after another in our catalog.
http://en.wikipedia.org/wiki/Ordered_fielddoes complex number have anything like consicutive numbers? . Like 2,3,4 are consecutive integers...
yes but you are using numbers less than 0.Consider the integers. We have -1< 0, -2< 0 and (-1)*(-2)>0. So the integers aren't an "orderred field" either but they do have an order to them. So back to OP's question: Can we order the complex numbers? yes. Does that order mean anything? I don't know.