For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.

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31
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15answers
7k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
0
votes
1answer
36 views

Rational number, dense but measure zero

When calculating the measure of Q in real number interval [0, 1], an interval $ (q_n-\epsilon, q_n + \epsilon)$ around each rational number $ q_n $ is defined to show the measure of Q is zero. Is ...
3
votes
0answers
44 views

Rational analysis

I found myself thinking about how much of real analysis that can also be developed within the rational numbers. Of course, $\Bbb Q$ is lacking what is perhaps the most important property of the real ...
7
votes
3answers
854 views

Is it possible to find square root using only rational numbers and elementary arithmetic operators

Suppose I have a number a How can I find it's square root using only +, -, /, ...
1
vote
2answers
53 views

Definition of Rational/ Irrational Numbers reguarding denominators

The definition of a Irrational number is "Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern." So ...
1
vote
1answer
38 views

Finding rational points at rational distance in the plane

Take any point $p$ in the real plane. Does there always exist a rational point at a rational distance from $p$? (A rational point is a point $(q,r)$ where $q$ and $r$ are rational.)
2
votes
0answers
31 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
12
votes
1answer
145 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
0
votes
3answers
40 views

Help solve rational expression

I need help solving this rational expression. Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$ How do you solve this problem? Where do I start?
0
votes
3answers
43 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
9
votes
4answers
985 views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
0
votes
1answer
38 views

Zero vs Infinity relation type

I'm not sure it should be asked here or in philosophy. Bertrand Russell in his book "Introduction to Mathematical Philosophy" in chapter 7 when discussing rational numbers on page 66 says: "It will ...
0
votes
1answer
24 views

Prove that a number is rational if and only if it has a finite or periodic representation on every base

How do we go about proving: $q$ is rational $\iff$ $q$ has a finite or periodic representation on every natural base $n>1$? In other words, we need to prove each of the following statements: ...
4
votes
3answers
1k views

If $x$ is a rational number, then $1/x$ is a rational number

Why is this statement false? If $x$ is a rational number, i.e. $\frac{p}{q}$, then shouldn't it be obvious that $\frac{q}{p}$ is also a rational number, by definition of rational numbers?
11
votes
1answer
105 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} ...
0
votes
0answers
16 views

Question on rounding off rule

There is a specific rule(Banker's Rule I think) for rounding of numbers that end in 5. The rule is that we add 1 to the preceding digit of it's odd but keep it as it is if it's even. It's always ...
0
votes
0answers
14 views

Using LLL to get approximate rational representations of numbers

Does anyone understand how is the LLL algorithm implemented to obtained the values of $(x,y)$ for approximating $\pi$ in this portion of text?
2
votes
1answer
49 views

A question on the equation $^qx=2$

Given the equation $$^qx=2$$ with $q\gt3$ where $^qx$ means the 'tetration' operation on $x$, my question is: is it possible to find a value for $q$ for which the solution $x$ of the equation is a ...
2
votes
2answers
49 views

Question about the density of Q in R

So I was looking over a density that shows that the rational numbers are dense in the real numbers. If $0< a <b$, with with $a,b$ real numbers, then I understood why we can chose n such that: ...
0
votes
4answers
46 views

How to multiply two different numbers with different powers

How do you multiply and simplify: $\left(\frac{2}{3}\right)^{1/6}\cdot 18^{1/3}$? Simplify in surd form.
0
votes
0answers
50 views

Related to $\pi$ and $\tau$ constants, are they transcendental, irrational, or rational numbers?

Below are three OEIS constant sequences and values. Are they transcendental, irrational, or rational numbers? Note: $\tau = 2*\pi$ and the last two values are in radians. A233700. Decimal ...
-6
votes
3answers
277 views

Why integers are not enough? [closed]

My Maths teacher asks me this in my homework and i cant find it anywhere. Thanks for your help. P.S.: she asks another thing... ''why rational numbers are not enough?'' could you help me with that ...
2
votes
1answer
63 views

How do I work out the aspect ratio from the resolution by hand?

For $1024 \times 768$ I can see that $768/1024 = 0.75$, i.e. $\frac34$, so $4:3$ makes sense. How do I do it for other resolutions like $1920 \times 1080$ though?
0
votes
0answers
21 views

Adding a natural number to a normalized fraction

I am currently writing yet another rational number class where the fraction should always be normalized. When adding a natural number to a normalized fraction, it possible to get a non-normalized ...
0
votes
2answers
60 views

Subfields of $\mathbb{Q}$

How to prove that $\mathbb{Q}$ doesn't have any proper subfields? I have no idea how to prove it.
10
votes
3answers
175 views

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

If $\begin{cases}a^4+b^4\in\mathbb Q\\ a^3+b^3\in\mathbb Q\\ a^2+b^2\in\mathbb Q\end{cases}$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. It is given that $a,b\in\mathbb R$. The proof of ...
1
vote
0answers
46 views

Rational approximation bound for real numbers in (0,1)

I am working on a problem that related to rational approximation of real numbers. I am looking for a bound of the form: Given a positive real number, $\alpha \in (0,1)$, there exist positive ...
0
votes
2answers
56 views

Integers, rationals and reals as sets? [duplicate]

Natural numbers can be represented as pure sets by defining them to contain every number that is smaller than them. Arithmetic can be performed on them using the Peano axioms. Are there any similar ...
1
vote
2answers
56 views

Isn't this wrong?

This worksheet This question: $$w^2 - w \leq 0$$ This answer: $$(-\infty, -1] \cup [0, 1]$$ Isn't this wrong ? At $w = -2$, it becomes: $(-2)^2 - (-2)$, which is $4 + 2$, which is $\geq 0$. But ...
1
vote
3answers
41 views

How to solve this inequality question without manual checking?

Question: Find the maximum integral value which satisfies: $$\frac{x-2}{x^2-9}<0$$ I know that this means either of the following: #1. $x-2<0$ and $x^2-9>0$. Implies that $x \in (3, ...
3
votes
3answers
230 views

Which is greater: $1000^{1000}$ or $1001^{999}$

Question: Find the greater number: $1000^{1000}$ or $1001^{999}$ My Attempt: I know that: $(a+b)^n \geq a^n + a^{n-1}bn$. Thus, $(1+999)^{1000} \geq 999001$ And $(1+1000)^{999} \geq ...
0
votes
5answers
43 views

Converting Repeating Decimal Numbers to Fractions

Is it possible to write any decimal number, with a repeating decimal part, and be able to convert it into the form n/d (where both n and d are natural numbers)? I know rational numbers that are ...
0
votes
2answers
24 views

Inequalities giving incorrect solution

Question: Find the solution set for:$$\frac{|x|-1}{|x|-2} \geq 0$$ $x\not=\pm2$ My attempt: Let $|x| = y$, then inequality becomes $(y-1)(y-2)>=0$ Implies that: #1. $y-1\geq0$ and ...
0
votes
0answers
24 views

Non-trivial solutions for $(x;\sin{x})\in\mathbb{Q}^2$? [duplicate]

Is it possible? For $\cos{x}$ were analogous solutions also okay. $$(x;\sin{x})\in\mathbb{Q}^2$$
1
vote
4answers
70 views

A triangle has to find its third side.

Problem: (Euclid had a triangle in mind - I am including this line so that future googles come across this question) The triangles longest side is $20$ and another side is $10$. Its area is $80$. ...
0
votes
1answer
18 views

Proving congruency of triangles

Question: Given $AB$ is diameter, $C$ and $D$ lie on circumference, $AB = 15cm$, $AC = 12cm$, $BD = 9cm$, find area of quadrilateral ABCD. Note that the points $O$ and $Q$ were not in the ...
2
votes
1answer
38 views

Horses grazing in a circle.

Question: Diagram: Note that The circle with center $C$ is touching the arc of semi-circle $AB$ also; I couldn't draw it. The figure wasn't drawn on cartesian planes; so, though it may seem ...
1
vote
1answer
66 views

Why a decimal fraction is not expressing exactly what a rational number is in base 2?

I am currently using rational numbers to express currency and math operations with currency, while dealing with rational numbers has provided a great convenience in over coming the limitations of ...
4
votes
2answers
81 views

Prove that if $a$ is a rational number and $a^2$ is an integer then $a$ is an integer.

Question on a proof's review: Proof by contradiction: Suppose $a$ is not an integer. Then $a=p/q$ where $p$ and $q$ are coprime, $q$ is not 0, and $q$ is not 1. Then $a^2 = p^2/q^2$. This is ...
1
vote
1answer
32 views

Solving this equation

Question: Solve: $$3^{2x^2}-2\cdot3^{x^2+x+6}+3^{2(x+6)}=0$$ I thought that we can take $a=3^{x^2}$ and $b = 3^{x+6}$. Then equation becomes $a^2-2ab+b^2=0$, which obviously means $a-b=0$. ...
3
votes
2answers
35 views

Which of the following is the highest value?

Question: Find the highest value among $12^9$, $10^{11}$ and $11^{10}$. I have seen problems like this, but they had surds, these are integers. Also, the LCM of $10$, $11$, $9$ $(990)$ is fairly ...
2
votes
1answer
32 views

Confusing sum of fractions

Question is to find the sum of: $$(\frac{1}{2^2-1})+(\frac{1}{4^2-1})+(\frac{1}{6^2-1})+(\frac{1}{20^2-1})$$ I know that $a^2-b^2=(a+b)(a-b)$, and that with this I can find the LCM to be 1995, ...
2
votes
1answer
26 views

Rational Numbers and farey fractions

How can I go about proving the following problem: Prove that a number a is rational if and only if there exists a positive integer k such that $[ka]=ka$. Prove that a number is rational if and only ...
0
votes
1answer
37 views

Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
2
votes
2answers
66 views

show that if $|x-\frac{m}{n}| \leq \epsilon$ then $n$ is very large

I am working on my calculus homework currently, and in order to solve a question, I need to prove this more simple statement: if $|x-\frac{m}{n}| \leq \epsilon$ for all $\epsilon>0$ then $n$ has ...
0
votes
0answers
26 views

Rational doubt ( doubt in rational number) [duplicate]

If there is a prime number x, if we reciprocate it we will get 1/x. Reciprocal of prime number will be a rational number , Except 1/2 and 1/5 , every number which is reciprocal of prime number is a ...
5
votes
2answers
406 views

Is the Nested Radical Constant rational or irrational?

Given the sequence $A_n=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{\dots+\sqrt{n}}}}}$: Are there any known rational elements in $A_n$, or has it been proved that all are irrational? Is there any proof for ...
6
votes
3answers
1k views

number between 17 and 18, and has a rational square root

"number between 17 and 18, and has a rational square root" Is there even one? They all keep coming up irrational for me
1
vote
1answer
45 views

Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$ (where $T$ = transcendental numbers)

Question: Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$. This is Exercise 2.7.13(a) from Mark E. Watkins, Jeffrey L. Meyer: Passage to Abstract Mathematics. I'm currently ...
6
votes
3answers
112 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...