Journey from Natural Numbers to Complex Numbers

5Complex Numbers

We have constructed the set of real numbers from rational numbers using Dedekind cuts. Sets of rational numbers and real numbers with addition and multiplication defined on them are ordered fields. An interesting thing about both the fields is that between any two distinct members of the set, we can always find a new number in the respective field, that is, between the two numbers. The middle number is different from both the given numbers. Rational numbers have a unique solution for single variable linear equations ( a x + b = 0 ; a ≠ 0 ; a , b ∈ Z ) for unknown x. Some of the polynomials with rational coefficients have zeroes but not all. Equations like x ² = 4 have rational solutions, whereas equations like x ² = 2 ...