Irrational Definition and 353 Threads

I have the folloring problem: Given the following flow on the torus (θ_1)' = ω_1 and (θ_2)' = ω_2, where ω_1 /ω_2 is irrational then I am asked to show that each trajectory is DENSE. So I need to prove that Given any point p on the torus, any initial condition q, and any ε > 0, then there...

What kind of number is sqrt(2)^sqrt(2)? I have noted sqrt(2)^sqrt(2) = 2^(sqrt(2)/2) = 2^(1/sqrt(2)), i.e. a rational number to an irrational power. Now, 1/sqrt(2) is less than 1, but greater than zero. So, given that 2^x is an increasing function, 2^(1/sqrt(2)) is less than 2^1, but...

Homework Statement Let F = ( -y/(x2+y2) , x/(x2+y2) ) Show that this vector field is irrotational on ℝ2 - {0}, the real plane less the origin. Then calculate directly the line integral of F around a circle of radius 1.Homework Equations The Attempt at a Solution To show F is irrotational we...

Homework Statement Suppose \mathbb{Q},\mathbb{R} are the set of all rational numbers and the set of all real numbers, respectively. Then what is |\mathbb{R} \backslash \mathbb{Q}|?Homework Equations |\mathbb{Q}| = |\mathbb{Z^{+}}| < |P(\mathbb{Z^{+}})| = |\mathbb{R}|The Attempt at a Solution I...

Homework Statement Suppose a is irrational, prove√(1+a) is irrational. Homework Equations A number is rational if it can be expressed as p/q, p,q integers with q≠0 The Attempt at a Solution I can reason through it intuitively but not sure how to demonstrate it formally. Any...

Homework Statement Let a be a positive real number. Prove that if a is irrational, then √a is irrational. Is the converse true? Homework Equations So, an irrational number is one in which m=q/p does not exist. I understand that part, but then trying to show that the square root of an...

Hi, So I proved that 71/3 is irrational, and I know 21/2 is irrational, but how can I show if (21/2 + 71/3)1/2 is irrational? Hint of where to start?

someome please help me with this problem: "Any real numbers x and y with 0 < x < y, there exist positive integers p and q such that the irrational number s =( p√2)/q is in the interval (x; y)."

Homework Statement This is taken from an answer book that I have. I don't understand the bolded step. Can someone explain it to me?Suppose x = p/q where p and q are natural numbers with no common factor. Then: pn/qn + an-1pn-1/qn-1 + ... + ao = 0 and multiplying both sides by...

Non-repeating patterns in decimal expansions of irrational numbers seem to have two forms. I am wondering if there is any theory about the two. First - the decimal expansion is ultimately random - unpredictable Second - The decimal expansion follows an algorithm e.g. .01001000100001 ...

I'm confused with a question and wondered if anyone could help explain where I need to go... let x ε R. Prove that x is irrational thenI'm confused with a question and wondered if anyone could help explain where I need to go... let x ε R. Prove that x is irrational then ((5*x^(1/3))-2)/7)...

Homework Statement Prove that \sqrt{6} is irrational. The Attempt at a Solution Would I just do a proof by contradiction and assume that \sqrt{6} is rational and then get that 6q^2=p^2 which would imply that p is even so I put in p=2r and then multiply it out. then this would imply...

Homework Statement Prove that \sqrt{3} is irrational. The Attempt at a Solution SO I will start by assuming that \sqrt{3} is rational and i can represent this as 3=\frac{b^2}{a^2} and I assume that a and b have no common factors. so now I have 3b^2=a^2 but this is not possible...

Homework Statement Prove If x^2 is irrational then x is irrational. I can find for example π^2 which is irrational and then π is irrational but I don't know how to approach the proof. Any hint?

Homework Statement Prove that log_2(3) is irrational. The Attempt at a Solution This is also equivalent to 2^x=3 from the definition of logs. Proof: For the sake of contradiction let's assume that x is rational and that their exists integers P and Q such that x=P/Q . so now we have...

Homework Statement I need to evaluate this particular integral and I'm confused on what method to use. I'm currently learning integration calculus and I tried doing some introduction on electromagnetic field. Quite unexpectedly the integral turned to be heavy. \int_{-a}^a...

Is there a good way to do this? I am trying to figure out a good way to calculate a^b mod m, but the problem is that b is huge and a is irrational, and therefore I am getting inaccurate values because too much precision is required. I'm trying to find a smaller, "equivalent" ^b mod m to use...

Is 2.71771177711177771111... irrational? Homework Statement I'm student teaching 8th graders Numbers and Operations. This is from an 8th grade activity I inherited with no "answer key." Is this decimal (with a pattern but not a repeating pattern) irrational? I am guessing it is, but I want to...

Let n>2. Where n is integer show that sqrt(n!) is irrational. I am supposed to use the Chebyshev theorem that for n>2. There is a prime p such that n<p<2n. So far I am up to inductive hypothesis. Assume it holds for k then show it holds for k+1. Well if k! is irrational==> k!=...

Heres two problems from an A Level related paper: prove that if pq is irrational then atleast one of p or q is irrational. Also prove that if if p + q is irrational then atleast one of p or q is irrational. These two proofs are trivial proof by contradiction problems but it got me thinking more...

Homework Statement log418 rational numbers are in form x/y Homework Equations logab = logcb / logca The Attempt at a Solution log218 / log218 = x/y (b) log218 = (a) log218 log218b = log218a Then I am stuck.

Homework Statement F(x) = x if x is rational, 0 if x is irrational. Use the δ, ε definition of the limit to prove that lim(x→0)f(x)=0. Use the δ, ε definition of the limit to prove that lim(x→a)f(x) does not exist for any a≠0. Homework Equations lim(x→a)f(x)=L 0<|x-a|<δ...

Prove that if x satisfies 'xn +an-1xn-1+ ... a0=0' for some integers an-1,..., a0, then x is irrational unless x is an integer. My main question is that I don't quite understand what the question is trying to ask me prove. I'm fairly new with this so pardon me if this question is really basic...

Homework Statement Determine if the statement is true or false. Prove those that are true and give a counterexample for those that are false. If r is any rational number and if s is any irrational number, then r/s is irrational. Homework Equations A rational number is equal to the...

Homework Statement Prove by contradiction. Your proof should be based only on properties of the integers, simple algebra, and the definition of rational and irrational. If a and b are rational numbers, b does not equal 0, and r is an irrational number, then a+br is irrational. Homework...

Homework Statement Let f be the function defined on the real line by f(x)= \begin{cases} \frac{x}{3} & \text{if $x$ is rational } \\ \frac{x}{4} &\text{if $x$ is irrational.} \end{cases} Let D be the set of points of discontinuities of f. What is D? Homework Equations None...

Hi, All: This is an old problem I never solved, and I recently saw somewhere else: We are given an enumeration {q_1,q_2,..,q_n,...} of the rationals in the real line. We construct the union : S:=\/ (q_i+[e/2^(i+1)] , q_i-[e/2^(i+1)] ) for i=1,2..,n,.. i.e., we want the...

This has been aggravating me for years. Call it "IDP" as a placeholder name for now, if you will. How come irrational numbers keep propelling forward for particular divisions? My inquiry applies for both repeating and non-repeating irrational numbers. "Just is" or "You're thinking too much into...

I'd like to know if this indeed proves that between any 2 reals is an irrational. Choose two reals A and B, B>A. There are two cases of B: B is irrational or B is rational. Assume B is irrational. Then B- \frac{1}{n} (n is a natural number) is irrational. You can get as close as you like...

I think this needs it's own thread. e and pi are transcendental numbers: http://en.wikipedia.org/wiki/Transcendental_number The square root of 2 is n irrational number: http://en.wikipedia.org/wiki/Irrational_number 1/3 is a rational number...

I have been seeing a few during in my practice questions which leaves me worrying. If it is a quadratic function, the irrational numbers can be easily obtained using the equation. However, I got a question today which eventually took this form: 28D3+36D2-41D2+4 = 0 (I reevaluated...

Homework Statement For z complex: a.) is z\sqrt{2} a multi-valued function, if so how many values does it have? b.) Claim: z\sqrt{2}=e\sqrt{2}ln(z)=e\sqrt{2}eln(z)=ze\sqrt{2} Since \sqrt{2} has 2 values, z\sqrt{2} is 2 valued. Is this correct? If not, correct it. Homework...

Hi. I found some rational sequences that converge to irrational limits, but am not having any luck going the other direction, i.e., an irrational sequence that converges to a rational limit. Any suggestions?

Prove that: 1-If n^2 (n is a natural number) is even then n is even too . 2-Product of infinit number of primes bigger than 2 is not even. Please do not "google it for me" :biggrin: .

In this field, computer algorithms may produce false continued fraction expansions because of the limited accuracy in the floating point arithmetic used. Who knows more?

How do you deduce that a1 - \sqrt{N} b1 = a2 - \sqrt{N}b2 to be a1=a2 and b1=b2?

I came across this question. How do you show that √N is irrational when N is a nonsquare integer? Cheers.

It would be foolish to try and predict how employers will react -- having few job experiences, currently. So I’m here to ask: How much trouble will I have attempting to be hired as a Software Developer, after graduating with a BS in mathematics, and a minor in computer science? Are interviews...

It's my understanding that algebraic numbers are the roots of polynomials with rational (or equivalently integer) coefficients. I know all surds have a simple repeating continued fraction representation Is it also the case that all simple repeating continued fractions are algebraic numbers...

So I was thinking about numbers like pi. If you were to measure the area or circumference of a sphere in real life, you would get a never ending decimal. How can this exist in real life? How can an actual physical object have a circumference that is an irrational number?

I am playing around with the set {e^(2ki)|k=intiger}. all of these numbers when raised to the pi power give one (at least as one possible value). In other words, they can all be thought of as values of 1^(1/pi). There are a countable infinity of them, and I believe that these numbers are...

Rational numbers are those that can be represented as a/b. It is simple (I think) to demonstrate that the series of rationals is continuous, since, for any two rational numbers, X=a/b, and Y=c/d, you can always find at least one rational number between them. \frac{X+Y}{2} = \frac{ad+bc}{2bd}...

Is it possible and is there a general method?

1. For n ≥2, n^(1/n) is irrational. Hint provided: Use the fact that 2^n > n2. This is probably familiar to many. By contradiction, n = a^n/b^n --> a^n = n(b^n) --> n|a^n --> n|a Am I trying to force the same contradiction as with 2^1/2 is rational, that is, that a/b are not in lowest terms? Or...

Any number c in the real numbers has the form x.{c_1}{c_2}...{c_n}, in which x is an integer and 0 \le {c_n} \le 9 is a natural number. From the way that we have enumerated the decimal places, clearly number of decimal places is countable. Then there is a bijection from the indexes of the...

It is clear that 10^2 can be simplified to 10*10=100. But what about say, 10^0.5? I have been thinking about this for days and can't figure out how it simplifies. 10^1 is 10, 10^0 is 1, so 10^0.5 should be under 1, but it is 3.16, so I don't get it. Same with 10^-1 is 0.1. How exactly are...

How to prove that root n is irrational, if n is not a perfect square. Also, if n is a perfect square then how does it affect the proof.

Prove that cube root of 6 irrational. Solution: I am trying to prove by contradiction. Assume cube root 6 is rational. Then let cube root 6 = a/b ( a & b are co-prime and b not = 0) Cubing both sides : 6=a^3/b^3 a^3 = 6b^3 a^3 =...

Homework Statement So, I'm trying to prove that the square root of 3 is irrational. 2. Attempt at a Solution 2x can be any even number and 2x+1 can be any odd number. Since an irrational number is any number that can't be expressed as a ratio of two integers, I just have to show that...

They can fit into number lines but not marked on a sewing thread ? I love to think of between 2 infinity small rational numbers there is a infinity deep hole that you can always pick a different irrational number out of it. (Is it a safe idea? )