Least-upper-bound property

Real Numbers

Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020

We have extended the set of rational numbers to have the least upper bound property. This makes it a complete field with the total ordering. Even after such big achievement, equations like x2 + 1 = 0 do not have the solution in it. We will be extending real numbers to complex numbers in the next chapter.

Improvement of constructing real numbers by Dedekind cuts

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2023

Hongwei Lou

In this paper, we will try to do this. We first use Dedekind cuts to construct the set of real numbers. Then, we define the order in it and establish the density properties of rational cuts in the set of real numbers and the least-upper-bound property as usual. After establishing the above properties, the rest discussions will be not concerned with the definition of the cut again.

Least-upper-bound property

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Real Numbers

Improvement of constructing real numbers by Dedekind cuts