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Complex numbers can help visualize physical effects like the electromagnetic fields

around wires carrying current. Refer to the link below.

Q: Why do you use complex numbers?

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Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.

The complex numbers are a field.

All of them. Real numbers are a subset of complex numbers.

Gerolamo Cardano is an Italian mathematician who introduced complex numbers. Complex numbers are those that can be expressed in the form of a+bi where a and b represent real numbers.

Engineering (especially electrical engineering), Math, Physics mainly.

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Complex numbers are the square roots of negative numbers. i.e. root -1 = i

yes it does

electrical engineers and quantum mechanics use them.

Yes, many

I suggest you read the Wikipedia article con complex numbers, specifically the section "Applications". One example is electrical engineering: in the case of AC, it helps to express all voltages, currents and impedances (equivalent of resistances) as complex numbers.

Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.

I'm not sure about how to use complex numbers to do this, but I've posted a link to a pretty neat website about Synthetic Division.

when you multiply it with another polynomial

using contraction and expansion

Complex numbers are a proper superset of real numbers. That is to say, real numbers are a proper subset of complex numbers.

No. Complex numbers is the highest set of numbers you can go, and there are no sets outside of complex numbers.

Yes. Integers, Real Numbers, and even Complex Numbers can as well.