The real number system is a mathematical field.
To start with, the Real number system is a Group. This means that it is a set of elements (numbers) with a binary operation (addition) that combines any two elements in the set to form a third element which is also in the set. The Group satisfies four axioms: closure, associativity, identity and invertibility.
In addition, it is a Ring. A ring is an Abelian group (that is, addition is commutative) and it has a second binary operation (multiplication) that is defined on its elements. This second operation is distributive over the first.
And finally, a Field is a Ring over which division - by non-zero numbers - is defined.
There are several mathematical terms above which have been left undefined to keep the answer to a manageable size. All these algebraic structures are more than a term's worth of studying. You can find out more about them using Wikipedia but be sure to select the hit that has "mathematical" in it!
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A complex number must have a real and imaginary part. It can be in the form: a + bi i is an imaginary number and a and b are real numbers
Yes, all natural numbers are real numbers. Natural numbers are a subset of real numbers, so not all real numbers are natural numbers.
All rational numbers are real numbers.
Complex numbers are a proper superset of real numbers. That is to say, real numbers are a proper subset of complex numbers.
Real Numbers cannot be the square root of a negative number. Real Numbers are not divided by zero. Basically, Real Numbers cannot be anything that is undefined.