what makes the number continuum a necessity in Calculus? I know that if we have the number continuum(the system
of real numbers) , then there is no restriction on  using a number to represent the length of a line segment, but what makes the number continuum a necessity in Calculus ?  for measuring?
 A: The continuum nature of the real numbers - the fact that they possess the least upper bound property - is what makes true the kind of basic facts from analysis we'd like to be true.  Without it intuitively attractive and simple properties of real functions and sequences (intermediate value property, bounded + monotonic = convergent, Cauchy iff convergent, all the things you do in a first course in real analysis) fail.
You can't do better than to read Dedekind's Continuity and Irrational Numbers (here, for example). Dedekind explains that while teaching calculus in 1858 he wanted to present the result that a monotonic bounded sequence converges, but was only able to give a non-rigorous, "geometric" proof.  He adds:

...that this form of introduction into the differential calculus
  can make no claim to being scientific, no one will deny. For myself this feeling
  of dissatisfaction was so overpowering that I made the fixed resolve to keep
  meditating on the question till I should find a purely arithmetic and perfectly
  rigorous foundation for the principles of infinitesimal analysis.

His construction of the reals using Dedekind cuts and demonstration that they have the nature of a continuum is what allows him to put the kind of analysis he wants to do on a rigorous footing.

The statement is
  so frequently made that the differential calculus deals with continuous magnitude,
  and yet an explanation of this continuity is nowhere given; even the most
  rigorous expositions of the differential calculus do not base their proofs upon
  continuity but, with more or less consciousness of the fact, they either appeal
  to geometric notions or those suggested by geometry, or depend upon theorems
  which are never established in a purely arithmetic manner. Among these, for example,
  belongs the above-mentioned theorem, and a more careful investigation
  convinced me that this theorem, or any one equivalent to it, can be regarded in
  some way as a sufficient basis for infinitesimal analysis

A: I think to summarize why $\Bbb R$ (the continuum) is seen as superior to $\Bbb Q$ (not the continuum) is because of its so called completeness, something which the rationals simply lack.
It is nice to have square roots and logarithms, etc. but all of this is just a special case of having a complete number field. What does completeness say? Informally: There are no "small holes" in the number line. When you are moving and you move less and less over time (in some sense) then there is a point to which you are approaching to. Formally: given a sequence $(x_n)$ with $x_n-x_m\to0$ for $n,m\to\infty$ (a so called Cauchy sequence), then $x_n$ converges to some $x^*$.
So why is completeness so nice? Maybe someone working in analysis can explain this better. But I think that it models reality very nicely. Some examples that only work with completeness:


*

*When you are walking down a hill, the top is above sea level, the bottom is below it, then $-$ at some point $-$ you must have been at sea level. (Intermediate value theorem)

*When you zoom into the night sky using your telescope, and you zoom in deeper and deeper (without moving the telescope left and right, just going depper into the sky), you are observing a smaller and smaller portion of the night sky. Imagine going on for ever with the zoom, then there is some point on the sky to which you are zooming in (and you can point to it with your finger). Without completeness, there might be no such point. (Banach's fixed point theorem)

*Then you have a disc and you rotate it, then there is an axis of rotation. This does not hold without completeness. You also may not be abled to rotate the disc by some common angles dependent on how many numbers your are missing from $\Bbb R$.

A: Here is a step-by-step summary of why one takes each step to produce the real numbers.


*

*The natural numbers $\mathbb{N}=\{0,1,2,\dotsc\}$ arise out of counting things. Everyone is happy with this.

*Rational numbers give you a way to divide an integer $p$ evenly into $q$ parts. Anyone cutting a cake agrees these are a good idea, but the Greeks didn't use rational numbers in geometry, because their system of proportion (invented by Eudoxus) was different: one says that $ad:bc$ instead of $a/b=c/d$; two quantities can be compared if one can be multiplied by an integer enough times to make it bigger than the other. So lengths can be compared, areas can be compared, but you can't compare blue to a chair, for example.

*Negative numbers are useful as a way to indicate position in the opposite direction from the positive scale (debt, for example, or distance west instead of distance east, or a force in the opposite direction to motion such as friction).

*To measure distances in plane figures constructed using ruler and compass, one only needs numbers formed by successive application addition, subtraction, multiplication, division and square roots: these are the only possible magnitudes constructible using these instruments, and so are called constructible numbers. So, for example, $\sqrt{2}$ is not a ratio of two integers, but it is the diagonal of a square with side $1$.

*The ratio of a circle's diameter to its circumference is not constructible: given a circle, one cannot use ruler and compass to construct a straight line with the same length as the circumference. Equally, the ratio of area of the circle to its radius is a non-constructible number. It turns out these ratios are the same for every circle, and we call this number $\pi$.


For geometry, this is all one needs. We now move into the algebraic world.


*

*It is very useful to work in a system where every polynomial function 
$$p(x) = a_nx^n+a_{n-1}x^{n-1}+\dotsb+a_0$$
has a root. A number $\alpha$ for which there is a polynomial $p(x)$ of degree $n$ with integer coefficients $a_n$ so that $p(\alpha)=0$ is called an algebraic number of degree $n$. This is the same as saying that $\alpha$ is a solution to the equation
$$ a_nx^n+a_{n-1}x^{n-1}+\dotsb+a_0 = 0. $$
One needs complex numbers to talk about this in general, but we can demand that every odd-degree polynomial with integer coefficients has a root, and that every even polynomial with integer coefficients which has at least one point where it is positive and one point where it is negative (note that this is quite artificial-looking: the algebraic complex numbers are a much nicer object).

*There are various extensions of the algebraic numbers that are interesting objects, such as periods or computable numbers, but let's jump to the real issue (pun not originally intended).

*We care about functions other than polynomials. Obvious examples include the trigonometric functions, logarithms and exponential functions. We would like a condition that guarantees a root exists for a very general class of functions. An obvious suggestion is the continuous functions: those that only change a small amount when we change the argument a little bit. We adopt the least-upper bound property: given any set of numbers that is bounded above, there is a least upper bound. If we take $f$ to be a continuous function that is negative at $a$ and positive at $b>a$, then $\{ x \in [a,b] : f(x)<0 \}$ is a set of real numbers bounded above by $b$, so it has a least upper bound $c$. Then the continuity implies that $f(c)=0$. I think this is the simplest answer to your question: we need this condition (called the Intermediate Value Theorem) to do any calculus.

A: Galileo and others, up to, but before Newton, found that by the method of "indivisibles" (infinitesimals) they could make new discoveries.
Essentially they were doing integration without the fundamental theorem of calculus. And in every instance where a result, whether new or old, was 
obtained this way and could be done without this method, it was found to be correct.   Most of the attempts at explaining or justifying this new method were vague or even illogical.
Newton, with the fundamental theorem of calculus, along with the generalized binomial theorem, and other power series methods, applied it to celestial mechanics (astronomy). And it was soon found to have unlimited applications in navigation, engineering,  etc.
None of the math of it makes literal sense without  the continuum. Much of it can be re-stated  to apply only  to rationals (or the rationals along some other numbers, like some or all of the algebraic numbers), if you don't mind 2-page definitions and  5 pages of work to do what you can do in 2 lines and 1 continuum.... & Don't read Newton to understand the continuum. 
It was only the the latter 19th century that a rigorous foundation was built, after of hundreds of years of successful use. 
