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Complex Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
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 (ax+b=0;a≠0;a,b∈Z) for unknown x. Some of the polynomials with rational coefficients have zeroes but not all. Equations like x2 = 4 have rational solutions, whereas equations like x2 = 2 do not have a rational solution. By the introduction of irrational numbers (numbers for gaps in rational numbers!), we could get the solution of x2 = 2 in a set of real numbers. Even after inclusion of irrational numbers, simple polynomial equations like x2 + a = 0, where a is a positive rational number, do not have a solution. In order to obtain the solution for such problems, we are going to introduce complex numbers.
Improvement of constructing real numbers by Dedekind cuts
Published in International Journal of Mathematical Education in Science and Technology, 2023
Dedekind cuts are often used to construct real numbers in textbooks of mathematical analysis. However, the definition of a Dedekind cut is always involved in the whole process; see two classic textbooks by Fichtenholz (1948) and Rudin (1964), for example. Whenever a new real number is introduced, one needs to check if it is really a Dedekind cut. Sometimes, it should be careful in defining correctly an expected cut, such as the negative element of a cut α. In addition, all discussions on real numbers are restricted in treating Dedekind cuts – sets of rational numbers. For beginners, this process is quite difficult to understand. Nevertheless, it seems that recent textbooks still follow this process.