Origin of the terminology "Completeness" in R When we study Real Analysis first, we usually analyse R equipped with the usual metric space. We intuitively make a picture of R as a line on which all the points are laid. We say that the set Q is not ordered complete. One reason for this is that the set {$x\in $ Q: $0\leq x^2 \leq 2$} has no supremum in Q. I got stuck in the word "completeness". Why is it so called? I thought that it could be explained in a way as follows:
There is no rational number whose square is 2. By the order completeness property of R, we show the existence of a positive real number whose square is 2. We represent it by $\sqrt{2}$. Now if all the rational numbers are plotted on a line, then there is a "hole" at the position of $\sqrt{2}.$ To make the line  "complete", we have to remove the "holes", we have to plot all the real numbers on the line.
I had satisfied myself with this justification. But one day, when I was studying the concept of metric spaces, there I saw the definition of the completeness of metric spaces. It is defined as:
A metric space is said to be complete if all Cauchy sequences in it converge in it.
I again got stuck. But again I justified it, correctly or not. A standard example is given in relation to the completeness of spaces:
Consider a metric space (0, 1) with the usual distance. Then the sequence {$\frac{1}{n}: n \in $N} is Cauchy but not convergent. It happens because of "a hole" at the position of $0.$ 
I want to share this with you guys. But it might be an incomplete motivation for the origin of the word "Completeness" of R
 A: It's in the word "complete". This word can mean different things in different circumstances. The general idea is as follows: We have some algebraic or geometric structure $X$. Certain things we'd like to make use of all the time are only partially available in $X$.
Examples: In ${\mathbb N}$ we can always add two numbers, but subtraction is not always possible. So ${\mathbb N}$ is "incomplete" with respect to subtraction. In the same way ${\mathbb Z}$ is "incomplete" with respect to division. ${\mathbb Q}$ is incomplete with respect to inverses of strictly monotone functions: ${\rm sq}(x):=x^2$ maps ${\mathbb Q}_{\geq0}$ injectively into ${\mathbb Q}_{\geq0}$, but ${\rm sqrt}:={\rm sq}^{-1}$ is not defined on all of ${\mathbb Q}_{\geq0}$.
In such cases one tries to enlarge the structure $X$ in a "minimal way" to a structure $\tilde X$ which then possesses the desired property without restriction. This process is called a completion of $X$ with respect to the property in question.
It so happens that the completion of ${\mathbb Q}$ with respect to the order relation and the completion with respect to the metric $d(x,y):=|x-y|$ result in the same structure ${\mathbb R}$. By the way, there are other "completions" of ${\mathbb Q}$ which are of a completely different nature.
A: From my experience the completeness of it can be motivated by how it is constructed in for example Commutative Algebra. Normally using cauchy sequences you can construct from, for example $\mathbb{Q}$ a new ring with new properties, even though the cauchy sequence contains nothing but elements from $\mathbb{Q}$. It becomes quite natural to ask, if we repeat the process is there any new ring we can construct? Or what about the method used to construct $\mathbb{Q}$ itself?
In the instance of real numbers we cannot, we only get a ring that is isomorphic to the real numbers again so they are really the one and the same ring still. It is complete because we cannot do much more to construct new rings from it.
A: Fréchet's 1906 thesis speaks of spaces that "admit a generalization of Cauchy's theorem". Hausdorff's 1914 monograph on set theory calls such a space vollständig, meaning "complete" or "full". (German voll is a cognate of English full.) Kuratowski's 1930 Sur les espaces complets translates this as complet, a French word from which we get the English complete, and which in turn derives from Latin verb pleō, "to fill".
Long story short, you're right: we remove the holes by filling them in.
A: The definition you saw for completeness in terms of Cauchy sequences is in perfect alignment with your understanding of completeness as including irrational numbers. Consider the sequence 3, 3.1, 3.14, 3.141 3.1415, 3.14159, ... where each additional term adds another digit of $\pi$. This is a Cauchy sequence and it represents the irrational number $\pi$. A similar Cauchy sequence can be formed for each of the irrational numbers.
