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The best option here is to read Dedekind's original "Continuity and irrational numbers" or its exposition in Hardy's "A Course of Pure Mathematics".

The best option here is to read Dedekind's original Continuity and irrational numbers or its exposition in Hardy's A Course of Pure Mathematics.

Expansion of numbers systems can be seen driven by algebraic needs as one moves along the path $\mathbb {N}\to\mathbb{Z} \to\mathbb {Q} $. But the next step to $\mathbb {R} $ is totally non-algebraic and not based on finding solutions to polynomial equations. Rather the need is to enhance the order relations. When one tries to analyze the structure of set $\mathbb {Q} $ in terms of order relations $<, >$ a different kind of inadequacy presents us. The idea first popularized by Dedekind is not difficult to grasp and its a wonder why the issue is not dealt with in high school curriculum.

The writing of Dedekind shows how all this is developed from scratch and gives lot of intuitive explanations. IMHO understanding the construction of real numbers from scratch (ideally before you have heard of any calculus related terms like limits) is essential for a thorough study of calculus/rreal-analysis and the effort is very rewarding.

Expansion of numbers systems can be seen driven by algebraic needs as one moves along the path $\mathbb {N}\to\mathbb{Z} \to\mathbb {Q} $. But the next step to $\mathbb {R} $ is totally non-algebraic and not based on finding solutions to polynomial equations. Rather the need is to enhance the order relations. When one tries to analyze the structure of set $\mathbb {Q} $ in terms of order relations $<, >$ a different kind of inadequacy presents us. The idea first popularized by Dedekind is not difficult to grasp and it's a wonder why the issue is not dealt with in high school curriculum.

The writing of Dedekind shows how all this is developed from scratch and gives lot of intuitive explanations. IMHO understanding the construction of real numbers from scratch (ideally before you have heard of any calculus related terms like limits) is essential for a thorough study of calculus/real-analysis and the effort is very rewarding.

For example we can try to imagine the existence of a point $a$ such that $a^3=2$. Such a number is not available in $\mathbb {Q} $. But instead of solving $a^3=2$ we can look at inequations $a^3<2$ and $a^3>2$. This leads us to study the partition of $\mathbb {Q} $ into two non-empty disjoint subsets $A$ and $B$ each corresponding to numbers satisfying these inequalities. Dedekind's idea is that as we try to take larger and larger numbers in $A$ and smaller and smaller numbers in $B$ their cubes get closer and closer to $2$. And then Dedekind realizes that the key here is not the algebraic equations and the related inequalities but rather the partitioning $\mathbb {Q} $ into two sets $A, B$ such that they are non empty, disjoint and exhaustive and further element member of $A$ is less than every member of $B$.

Most modern presentations of Dedekind's approach (especially those which appear in real-analysis textbooks, they are written as if author is highly disinterested and is doing so only as a formality) are totally unmotivated because they present an already developed idea. The writing of Dedekind shows how all this is developed from scratch and gives lot of intuitive explanations.

For example we can try to imagine the existence of a point $a$ such that $a^3=2$. Such a number is not available in $\mathbb {Q} $. But instead of solving $a^3=2$ we can look at inequations $a^3<2$ and $a^3>2$. This leads us to study the partition of $\mathbb {Q} $ into two non-empty disjoint subsets $A$ and $B$ each corresponding to numbers satisfying these inequalities. Dedekind's idea is that as we try to take larger and larger numbers in $A$ and smaller and smaller numbers in $B$ their cubes get closer and closer to $2$. And then Dedekind realizes that the key here is not the algebraic equations and the related inequalities but rather the partitioning of $\mathbb {Q} $ into two sets $A, B$ such that they are non empty, disjoint and exhaustive and further every member of $A$ is less than every member of $B$.

Most modern presentations of Dedekind's approach (especially those which appear in real-analysis textbooks) are totally unmotivated and are written as if author is highly disinterested and is doing so only as a formality. 

The writing of Dedekind shows how all this is developed from scratch and gives lot of intuitive explanations. IMHO understanding the construction of real numbers from scratch (ideally before you have heard of any calculus related terms like limits) is essential for a thorough study of calculus/rreal-analysis and the effort is very rewarding.