Table of Contents

- 1 Why are rationals not complete?
- 2 Why is the set of rationals not closed?
- 3 Why are the rationals closed?
- 4 Why is Q dense in R?
- 5 Is the rationals open or closed?
- 6 Is RA closed interval?
- 7 Is 3.141414 A irrational number?
- 8 Is 9.68 repeating rational or irrational?
- 9 Why are the rational numbers not complete in the sense that?
- 10 Why are rational numbers written as integer fractions?

## Why are rationals not complete?

The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. Consider for instance the sequence defined by x1 = 1 and. The open interval (0,1), again with the absolute value metric, is not complete either.

### Why is the set of rationals not closed?

The set of rational numbers Q ⊂ R is neither open nor closed. It isn’t open because every neighborhood of a rational number contains irrational numbers, and its complement isn’t open because every neighborhood of an irrational number contains rational numbers.

#### Why are the rationals closed?

The interior of Q is empty (any nonempty interval contains irrationals, so no nonempty open set can be contained in Q). Since Q does not equal its interior, Q is not open. The closure of Q is all of R: every real number is the limit of a sequence of rationals, so every real number lies in the closure of Q.

**Is set of irrational numbers complete?**

Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable.

**Are the rationals a complete metric space?**

In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete. Thus, some bounded complete metric spaces are not compact. The rational numbers Q are not complete.

## Why is Q dense in R?

Theorem (Q is dense in R). For every x, y ∈ R such that x, there exists a rational number r such that x

### Is the rationals open or closed?

The set of rational numbers are determined to be neither an open set nor a closed set . The set of rational numbers is not considered open since each…

#### Is RA closed interval?

Note R not a closed interval, that is R≠[−∞,∞]. If you define open sets in Rn with a help of open balls then it can be proved that set is open if and only if its complement is closed. Then, it happens that any union (finite or infinite) of open sets is an open set and that any finite intersection of open sets is open.

**What is the closure of rationals?**

The closure of the set of rationals is all of R because ev- ery real number is a limit of a sequence of rationals. For example, 3,3.1,3.14,3.141,3.1415,… converges to π. The closure of the set of irrationals is also R.

**Why is R both open and closed?**

R is open because any of its points have at least one neighborhood (in fact all) included in it; R is closed because any of its points have every neighborhood having non-empty intersection with R (equivalently punctured neighborhood instead of neighborhood).

## Is 3.141414 A irrational number?

Option (d) 3.141141114 is an irrational number.

### Is 9.68 repeating rational or irrational?

Ans : Yes, number x = 9.688888…… is rational number.

#### Why are the rational numbers not complete in the sense that?

The real numbers are complete in the sense that every set of reals which is bounded above has a least upper bound and every set bounded below has a greatest lower bound. The rationals do not have this property because there is a “gap” at every irrational number.

**Is there a lower bound of rationals greater than 1?**

Similarly, there is no greatest lower bound of the rationals greater than although 1 is a perfectly good example of a lower bound. The easiest way to see this is to construct an increasing sequence of rationals, all of which are lower bounds of that set of rationals greater than .

**Is the set of rational numbers satisfy the Archimedian property?**

Prove that the set of rational numbers satisfies the Archimedian property but not the least-upper-bound property. This shows that the Archimedian property does not imply the least-upper-bound axiom. I don’t get what I have to do, and what it means — to satisfythe least-upper-bound-property.

## Why are rational numbers written as integer fractions?

A number that can be written as an integer fraction. Integers are the whole numbers and their negative counter parts . So the rational numbers are . Now a fraction is fully simplefied when the greatest common divisior of and is . So is even, but if is even so is because an odd number squared would be odd again.