Irrational Definition and 349 Threads

The value of irrational number π, correct to ten decimal places (without rounding), is 3.1415926535. By using your calculator, determine to how many decimal places the following quantity (22/7) agrees with π. Extra notes from textbook: Archimedes (287-212 B.C.) showed that (223/71) < π <...

The value of irrational number π, correct to ten decimal places (without rounding), is 3.1415926535. By using your calculator, determine to how many decimal places the following quantity [(4/3)^4] agrees with π. The value used for π in the Rhind papyrus, an ancient Babylonian text written...

I have authored documents of 40 years of computer software development with a mind to collect them into a publication at some point. They have been built around several software topics but mathemetics is a favorite of mine. I find a point of inspiration and write a piece of software around it...

Homework Statement True or false and why: If a and b are irrational, then ##a^b## is irrational. Homework Equations None, but the relevant example provided in the text is the proof of irrationality of ##\sqrt{2}## The Attempt at a Solution Attempt proof by contradiction. Say ##a^b## is...

Most than a question, I'd like to show you what I've got to understand and I want you to tell me what do you think about it. I'm not a math expert, I just beginning to study maths, and I'm reading Elements by Euclids, and I've been doing some research on immeasurable numbers. My statement is...

Homework Statement Let a = (1+(3)^1/2)^1/2. Find the minimal polynomial of a over Q. Homework EquationsThe Attempt at a Solution Maybe the first thing to realize is that Q(a):Q is probably going to be 4, in order to get rid of both of the square roots in the expression. I also suspect that...

So I am trying to solve a simple rational inequality: ##\sqrt{x} < 2x##. Now, why can't I just square the inequality and go on my way solving what results? What precisely is the reason that I need to be careful when squaring the square root?

I have been looking at various proofs of this statement, for example Proof 1 on this page : http://www.cut-the-knot.org/proofs/sq_root.shtml I'd like to know if the following can be considered as a valid and rigorous proof: Given ##y \in \mathbb{Z}##, we are looking for integers m and n ##\in...

Homework Statement Find the integral \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx 2. The attempt at a solution I can't find a useful substitution to solve this integral. I tried x-2=\frac{1}{u},x=\frac{1}{u}+2,dx=-\frac{1}{u^2}du that gives \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm...

Hello Aren't all irrational numbers having an infinitely long decimal component? If so, how can any measure of a physical quantity be irrational? the decimal component is infinitely long..but the magnitude of the physical quantity surely isnt?

Isaac0427 submitted a new PF Insights post Complex and Irrational Exponents for the Layman Continue reading the Original PF Insights Post.

Homework Statement Let a and n be positive integers. Prove that a^(1/n) is either an integer or is irrational. Homework EquationsThe Attempt at a Solution Proof: If a^(1/n) = x/y where y divides x, then we have an integer. If a^(1/n) = x/y where y does not divide x, then a = (a^(1/n))^n =...

Homework Statement Express the following using existential and universal quantifiers restricted to the sets of Real numbers and natural numbers Homework EquationsThe Attempt at a Solution I believe the existence of rational numbers can be stated as: ##(\forall n \in \Re)(\exists p,q \in...

Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...

Homework Statement Find [(3 - 51/2)/2]1/2 Homework EquationsThe Attempt at a Solution My calculator says (-1 + √5)/2 I have no idea how. Rationalising doesn't really do much good. Just tell me where to start.

Am using Spivak. Spivak elegantly proves that √2 is irrational. The proof is convincing. For that he takes 2 natural numbers, p and q ( p, q> 0)...and proves it. He defines irrational number which can't be expressed in m/n form (n is not zero). Here he defines m and n as integers. But in the...

Would it be possible to write an equation utilizing a summation of a modular function of a Cartesian function, whose degree is dependent upon the index of the root, in that it predicts the digits less than 1 of the root, that when summed equals the computed value sqrt( n )? I already have what...

Mod note: Thread moved from Precalc section Homework Statement F(x)=sqrt(-2x^2 +2x+4) 1.discuss variation of f and draw (c) 2.find the equation of tangent line to (c) that passes through point A(-2,0) The Attempt at a Solution I solved first part I found the domain of definition and f'(x) and...

1. The problem statement, all variables and given/known dat F(x)=x+1-3sqrt((x-1)/(ax+1)) For which value of a ,(c) has no asymptote? Homework EquationsThe Attempt at a Solution I know if a>0 then (c) will have 2 asymptote And if a<o then (c) will have 1 vertical asymptote. But I can't find...

Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found...

Homework Statement Determine a positive rational number whose square differs from 7 by less than 0.000001 (10^(-6)) Homework Equations - The Attempt at a Solution Let p/q be the required rational number. So, 7> (p/q)^(2) > 7-(0.000001) ⇒ √(7) > p/q > √(7-.000001) ⇒√(7) q> p >...

Is it sensible to consider a base pi number system? Can one make an irrational number rational by defining it as the unit of a counting system? I don't know what constitutes an mathematically consistent 'number line' - this question might not make sense. I'm just thinking that if I use pi as...

I have the irrational equation ##\sqrt{x - 1} + \sqrt{2 - x} = 0##, which has no real solutions. However, when I try to solve the equation, I get a real solution, that is: ##\sqrt{x - 1} + \sqrt{2 - x} = 0## ##\sqrt{x - 1} = -\sqrt{2 - x}## ##(\sqrt{x - 1})^{2} = (-\sqrt{2 - x})^{2}##...

[Mentor's note: this was originally posted in the Quantum Physics forum, so that is what "this section" means below.] ---------------------------------------------------- I wasn't sure whether to post this question in this section or the general math section, so I just decided to do it here...

Is $\sin\left(10^{\circ}\right)$ rational or not? Prove it.

Prove that $\cos \dfrac{\pi}{100}$ is irrational.

I am reading Chapter 1:"Real Numbers" of Charles Chapman Pugh's book "Real Mathematical Analysis. I need help with the proof of Theorem 7 on pages 19-20. Theorem 7 (Chapter 1) reads as follows: In the above proof, Pugh writes: " ... ... The fact that $$a \lt b$$ implies the set B \ A...

Solve the irrational equation $x^4-9x^3+16x^2+15x+26=\dfrac{7}{\sqrt{x^2-10x+26}+\sqrt{x^2-10x+29}+\sqrt{x^2-10x+41}}$.

I'm just curious as to how a calculator does the following operation: ##5^{1/\pi}## I mean, it has to look for the number that raised to the power of pi, gives me 5. I think that's insane. How does it do that? How does a calculator store the value of pi? -- I guess that's a more boring...

In the classic proof of irrationality of SQRT(2) we assume that it can be represented by a rational number,a/b where a, b are integers. This assumption after a few mathematical steps leads to a contradiction, namely that both a, b are even numbers. Why is that a contradiction? Well you can...

Homework Statement Prove that √2 is irrational as follows. Assume for a contradiction that there exist integers a, b with b nonzero such that (a/b)2=2. 1. Show that we may assume a, b>0. 2. Observe that if such an expression exists, then there must be one in which b is as small as...

$$g(x)=\begin{cases}x^2, & \text{ if x is rational} \\[3pt] 0, & \text{ if x is irrational} \\ \end{cases}$$ a) Prove that $\lim_{{x}\to{0}}g(x)=0$ b) Prove also that $\lim_{{x}\to{1}}g(x) \text{ D.N.E}$ I've never seen a piecewise function defined that way...hints?

Find all irrational numbers $k$ such that $k^3-17k$ and $k^2+4k$ are both rational numbers.

Here it is, for you to critique. This is a proof by contradiction. This is a good example of how I usually go about doing proofs, so if you give me tips on how to improve this particular proof, I'll be able to improve all my other proofs. I just learned how to do proof by contradiction...

An irrational number is any real number which cannot be expressed as the ratio of two real numbers. Then is 3.62566 is also an irrational number? I thought all irrational numbers are uncountable. I am not sure that the above is an irrational number :confused:

This question specifically relates to a numerator of '1'. So if I had the irrational number √75: 1/(x*√75) Could I have some irrational non-transindental value x that would yield a non '1', positive integer while the x value is also less than 1/√75? Caviat being x also can't just be a division...

Homework Statement Prove that \sqrt{6} is irrational.Homework Equations The Attempt at a Solution \sqrt{6} = \sqrt{2}*\sqrt{3} We know that \sqrt{2} is an irrational number (common knowledge) and also this was shown in the textbook. So, let's assume \sqrt{6} and \sqrt{3} are both rational...

In the comum sense, the number 10 is the base of the decimal system and is the more intuitive basis for make counts (certainly because the human being have 10 fingers). But, in the math, the number 10 is an horrible basis when compared with the constant e. You already thought if an irrational...

Homework Statement A user on math.se wanted to prove that any Mersenne number m = 2^n - 1 has an irrational square root for n > 1. So, it can be proved rather easily that any non-perfect square has an irrational root, and that all of the numbers to be considered are not perfect squares...

The methods of proving irrational have always been bothering me in my study of proof. It seems that for each case a new method has to be invented out of the blue. I understand only the proof that ##\sqrt{k}## is irrational. But what will happen if I want to prove ##\sqrt{2}+\sqrt{5}## or...

Okay, so this is a problem I've been pondering for a while. I've heard from many people that pi doesn't repeat. Nor does e, or √2, or any other irrational or transcendental number. But what I'm wondering is, how do we know? If there truly is an infinite amount of digits, isn't it bound to...

prove that $\sqrt{3}$ is irrational. this is what I tried $\sqrt{3}=\frac{p}{q}$ whee p and q are integers in lowest terms. common factor of +\-1 only. squaring both sides $\frac{p^2}{q^2}=3$ $p^2=3q^2$ assuming that $3q^2$ is even then $p^2$ is even hence p is also even. $(3k)^2=3q^2$...

Is this proof that e is an irrational number valid? e = ∑^{∞}_{n=0} 1/n! = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n! +... Let e = a + b where a = Sn = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n! b= 1/(n+1)! + 1/(n+2)! + 1/(n+3)! +... Multiply both sides by (n!) giving e(n!) = a(n!) + b(n!)...

I am trying to prove that √2 is irrational using proof by contradiction. Here is my work so far: √2 = p/q where p & q are in their lowest terms. Where q is non-zero. 2=p2/q2 2q2 = p2 Which tells me that p2 is an even number, using the definition of an even number. We can use this definition...

Other than Bernouilli, Euler, and Lagrange, who else discovered an irrational number in which transcendental operators have been developed to simplify physics and geometry?

Prove that if p and q are positive distinct primes,then $\log_p(q)$ is irrational. Attempt: Proof by contradiction: Assume $\log_p(q)$ is rational.Suppose $\log_p(q) = \dfrac{m}{n}$ where $m,n \in \mathbb{Z}$ and $\gcd(m,n) = 1$. Then, $p^{\frac{m}{n}} = q$ which implies $p^m = q^n$.

Homework Statement Use the rational roots theorem to prove 31/2-21/3is irrational. The Attempt at a Solution My teacher strongly hinted to us that this problem had something to do with the fact that complex roots come in conjugate pairs, and all we had to do was, "flip the sign"...

I'm aware of the standard proof. What I'm wondering is why we can't just do the following. Given, I haven't slept well and I'm currently out of caffeine, so this one might be trivial for you guys. Suppose, by way of contradiction, that ##\sqrt{p}=\frac{m}{n}##, for ##m,n\in\mathbb{Z}##...

Homework Statement Im trying to prove that if p is prime, then its square root is irrational. The Attempt at a Solution Is a proof by contradiction a good way to do this? All i can think of is suppose p is prime and √p is a/b, p= (a^2)/ (b^2) Is there any property i can...

Greetings , Im taking an online course on mathematical thinking, and this question has me stumped. r is irrational: Show that r+3 is irrational Show that 5r is irrational Show that the square root of r is irrational. Im sorry if i posted this in the wrong forum, but I am not sure...