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rationalized

ˈrationalized, ppl. a. [f. rationalize v. + -ed1.] a. That has been rendered rational.1855 Sir G. C. Lewis Credib. Rom. Hist. xi. I. 426 According to another, and probably a rationalized, version. 1896 Spectator 11 Apr. 519 Swift's grim conceptions of animalized man and rationalized animals. b. spec... Oxford English Dictionary

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Does rationalizing the denominator lead to more or less round-off error? I evaluated $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{2}$ in Matlab, and got a slight difference: $0.707106781186547$ and $0.707106781186548$, r...

In this case, the value is $0.707106781186547524400844362104849039284835937688474036588339...$, so the second is (marginally) closer. I wouldn't expect there to be a uniform rule on which is more accurate, depending on what expressions you are considering. The point about division by $2$ being exact...

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how to rationalize $\frac{1}{\sqrt a+\sqrt b+\sqrt c+\sqrt d+e}$ let say $a, b, c, d, e \in \Bbb Z^+ $ and $\sqrt a, \sqrt b, \sqrt c, \sqrt d \notin \Bbb Z^+ $ My problem is how to rationalize the denominator of $\f...

In general you have that, $$\Pi_{(i_1,\dots, i_n)\in C_2^n}((-1)^{i_1}\sqrt{a_1}+\dots+(-1)^{i_n}\sqrt{a_n})$$ is an integer, where $a_i$ is an integer $\forall i$. Can you prove this? * * * I think an elementary proof may be this: The polynomial $$\Pi_{(i_1,\dots, i_n)\in C_2^n} ((-1)^{i_1}X_1+\dot...

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Unrationalize Denominator What is the method to unrationalize or reverse a rationalized fraction? For example: How do you simplify $\frac{1}{2\sqrt\frac{1}{2}}$ = $\frac{1}{\sqrt{2}}$

$$\frac{1}{2\sqrt{\frac{1}{2}}}=\frac{1}{\sqrt{4}\sqrt{\frac{1}{2}}}=\frac{1}{\sqrt{\frac{4}{2}}}=\frac{1}{\sqrt{2}}$$ Now you can multiply by "1": $$\frac{1}{\sqrt{2}}\frac{1}{1}=\frac{1}{\sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{(\sqrt{2})^2}=\frac{\sqrt{2}}{2}$$

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Find $\lim_{x \to 1}\frac{\sqrt{x^2+35}-6}{x-1}$ I've rationalized the numerator to $x^2+35-36$ $$= \frac{x^2+35-36}{(x-1)\left(\sqrt{x^2+35}+6\right)} = \frac00$$ (when I substitute $x=1$) I don't know what to do t...

**Hint 1:** $$35 - 36 = -1$$ **Hint 2:** You know that the numerator is $0$ when you substitute $x=1$. That means that $(x-1)$ is a factor of the numerator: $$x^2+35-36 = (x-1)(\ \ldots\ )$$ You can then cancel out $(x-1)$ from top and bottom. (But simplify first using Hint 1.)

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Limit at negative infinity of a function with a radical $$\lim_{x \to -∞}{\sqrt{x^2 -x} + x}.$$ First I rationalized, to get $$\frac{x^2-x-x^2}{\sqrt{x^2-x}-x}$$ Then I wanted to factor the whole thing by the largest ...

> Where did I go wrong? The following is **wrong** : $$\frac{\sqrt{x^2-x}}{x}=\sqrt{1-\frac 1x}$$ Note that for $x\lt 0$ $$\frac{\sqrt{x^2-x}}{x}=\frac{\sqrt{x^2(1-\frac 1x)}}{x}=\frac{\sqrt{x^2}\sqrt{1-\frac 1x}}{x}=\frac{|x|\sqrt{1-\frac 1x}}{x}=\frac{\color{red}{-}x\sqrt{1-\frac 1x}}{x}=\color{re...

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Why does the domain of a function such as sqrt(x-5) /sqrt(x+2) change when rationalizing the denominator? I was tutoring a student the other day and the above function in the title came up. I initially showed her how ...

Note that $\sqrt{-2}/\sqrt{-1}$ does not exist in the reals but that $\sqrt{(-2)/(-1)}=\sqrt2$ does. Likewise, the domain of $\sqrt{x-5}/\sqrt{x+2}$ is where $x-5\geqslant0$ and $x+2\gt0$ while the domain of $\sqrt{(x-5)/(x+2)}$ is where $(x-5)(x+2)\geqslant0$ and $x+2\ne0$.

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Confusing rational numbers Question: > If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the ...

$$\frac x2=\frac{2\sqrt2}{\sqrt2+1}$$ Applying Componendo & Dividendo, $$\frac{x+2}{x-2}=\frac{2\sqrt2+(\sqrt2+1)}{2\sqrt2-(\sqrt2+1)}=\frac{3\sqrt2+1}{\sqrt2-1}=\frac{(3\sqrt2+1)(\sqrt2+1)}{2-1}$$ $$=7+4\sqrt2$$ Similarly, $$\frac x{2\sqrt2}=\frac2{\sqrt2+1}$$ Apply Componendo & Dividendo again

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Express the following complex quantity in the form $a+bi$ I'm currently working on a problem and I need to simplify the following complex quantity in the form of $a+bi$. Assuming that $a$ and $b$ are real numbers, her...

You have $z=a+bi$ and want to compute $$\frac{z}{\bar z}-\frac{\bar z } {z}$$ Where $\bar z= a-bi$. Combine denominators to get $$\frac{z^2-\bar z^2}{z\bar z}$$ From which you get $$\frac{4iab}{a^2+b^2}$$ So if you wanted to put this in the shape of a complex number $A+Bi$ you get that $A=0$ and $B=...

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Trigonometric limit: $(1-\sqrt{\cos x})/x^2$ as $x\to 0$, without using L'Hopital I have to evaluate this limit without using L'Hopital. Could you help me $$\lim_{x \to 0} {1-\sqrt{\cos(x)}\over x^2}$$ I already rat...

Note that $$ \frac{1-\cos(x)}{x^2(1+\sqrt{\cos x})}=\frac{1-\cos^2 x}{x^2(1+\sqrt{\cos x})(1+\cos x)}=\left(\frac{\sin x}{x}\right)^2\frac{1}{(1+\sqrt{\cos x})(1+\cos x)} $$ then you can use $\lim_{x\to 0}\sin(x)/x=1$. Arguments for this latter fact can be seen here. Some of them do not employ L'Hop...

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Value of $\lim\limits_{x\to-\infty}{(4x^2-x)^{1/2} +2x}$ The value of the $$\lim\limits_{x\to-\infty}{(4x^2-x)^{1/2} +2x}$$ is? The answer given is $1/4$. I rationalized and got $$\lim\limits_{x\to-\infty} \frac {-x...

The limit is equivalent to $$\begin{align} \lim_{x\to-\infty}\left(2|x|\left(1-\frac1{4x}\right)^{1/2}+2x\right) &=\lim_{x\to-\infty}\left(-2x\left(1-\frac1{4x}\right)^{1/2}+2x\right)\\\ &=\lim_{x\to-\infty}-2x\left(\left(1-\frac1{4x}\right)^{1/2}-1\right)\\\ \end{align}$$ Then using the generalized...

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First and second derivative of $|x|^3$ I need to prove that $|x|^3$ is twice differentiable, by showing that the first and second derivatives exist using the definition. I've tried several ways, this is what I've got:...

At $x\ne0$, you can take $h$ so small that $x$ and $x+h$ have the same sign. Then the function can be processed as two polynomial pieces which derive seamlessly: $$|x^3|=\pm x^3$$ gives the derivatives $$\pm3x^2=3x|x|$$ and $$\pm6x=6|x|$$ (as the $\pm$ signs match that of $x$). At $x=0$, the first d...

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How to solve the sequence limit The sequence limit is: $$\lim_{n\to \infty}\left[\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n+1}-\sqrt{n}}\right]$$ I rationalized and got: $$\lim_{n\to \infty}\left[\frac{\sqrt{n+1}+\sqrt{n}}...

Hint. It should be $$\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n+1}-\sqrt{n}}=\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n}+\sqrt{n-1}}=\frac{\sqrt{1+\frac{1}{n}}+1}{1+\sqrt{1-\frac{1}{n}}}.$$ Can you take it from here?

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Why is the double factorial $(-1)!! = 1$, by definition? By definition, the double factorial $(-1)!! = 1$. How can this be rationalized?

This is a "double factorial": > The product of all odd integers up to some odd positive integer n is often called the double factorial of n (even though it only involves about half the factors of the ordinary factorial, and its value is therefore closer to the square root of the factorial). It is de...

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Finding the transformation I am trying to find the transformation of $x = 2$ under $T(z) = \frac{z+1}{z-1}$ My approach was as follows, I put $u +iv = \frac{z+1}{z-1} $ and then rationalized the RHS. I get $u = \frac...

Let $$w=u+iv=\frac{z+1}{z-1}\implies z=\frac{w+1}{w-1}$$ Hence $$z=\frac{1+u+iv}{u-1+iv}\cdot\frac{u-1-iv}{u-1-iv}$$ The real part is $$\frac{u^2+v^2-1}{(u-1)^2+v^2}=2$$ And this simplifies to $$(u-2)^2+v^2=1$$

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rationalized

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rationalized

Does rationalizing the denominator lead to more or less round-off error? I evaluated $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{2}$ in Matlab, and got a slight difference: $0.707106781186547$ and $0.707106781186548$, r...

how to rationalize $\frac{1}{\sqrt a+\sqrt b+\sqrt c+\sqrt d+e}$ let say $a, b, c, d, e \in \Bbb Z^+ $ and $\sqrt a, \sqrt b, \sqrt c, \sqrt d \notin \Bbb Z^+ $ My problem is how to rationalize the denominator of $\f...

Unrationalize Denominator What is the method to unrationalize or reverse a rationalized fraction? For example: How do you simplify $\frac{1}{2\sqrt\frac{1}{2}}$ = $\frac{1}{\sqrt{2}}$

Find $\lim_{x \to 1}\frac{\sqrt{x^2+35}-6}{x-1}$ I've rationalized the numerator to $x^2+35-36$ $$= \frac{x^2+35-36}{(x-1)\left(\sqrt{x^2+35}+6\right)} = \frac00$$ (when I substitute $x=1$) I don't know what to do t...

Limit at negative infinity of a function with a radical $$\lim_{x \to -∞}{\sqrt{x^2 -x} + x}.$$ First I rationalized, to get $$\frac{x^2-x-x^2}{\sqrt{x^2-x}-x}$$ Then I wanted to factor the whole thing by the largest ...

Why does the domain of a function such as sqrt(x-5) /sqrt(x+2) change when rationalizing the denominator? I was tutoring a student the other day and the above function in the title came up. I initially showed her how ...

Confusing rational numbers Question: > If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the ...

Express the following complex quantity in the form $a+bi$ I'm currently working on a problem and I need to simplify the following complex quantity in the form of $a+bi$. Assuming that $a$ and $b$ are real numbers, her...

Trigonometric limit: $(1-\sqrt{\cos x})/x^2$ as $x\to 0$, without using L'Hopital I have to evaluate this limit without using L'Hopital. Could you help me $$\lim_{x \to 0} {1-\sqrt{\cos(x)}\over x^2}$$ I already rat...

Value of $\lim\limits_{x\to-\infty}{(4x^2-x)^{1/2} +2x}$ The value of the $$\lim\limits_{x\to-\infty}{(4x^2-x)^{1/2} +2x}$$ is? The answer given is $1/4$. I rationalized and got $$\lim\limits_{x\to-\infty} \frac {-x...

First and second derivative of $|x|^3$ I need to prove that $|x|^3$ is twice differentiable, by showing that the first and second derivatives exist using the definition. I've tried several ways, this is what I've got:...

How to solve the sequence limit The sequence limit is: $$\lim_{n\to \infty}\left[\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n+1}-\sqrt{n}}\right]$$ I rationalized and got: $$\lim_{n\to \infty}\left[\frac{\sqrt{n+1}+\sqrt{n}}...

Why is the double factorial $(-1)!! = 1$, by definition? By definition, the double factorial $(-1)!! = 1$. How can this be rationalized?

Finding the transformation I am trying to find the transformation of $x = 2$ under $T(z) = \frac{z+1}{z-1}$ My approach was as follows, I put $u +iv = \frac{z+1}{z-1} $ and then rationalized the RHS. I get $u = \frac...