Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [approximation]

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

Approximations are representations of numbers that aren't exact, for example $\sqrt{2}\approx 1.41$. Such representations may be obtained using differentials (more generally, Taylor's formula), linear interpolation, etc.

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Any simplification or approximation of sqrt

As you know, we could write $(a+b+c)^2$ as $a^2+b^2+c^2+2ab+2ac+2bc$. what about $(a+b+c+\cdots)^{1/2}$? is there any expansion for $(a+b+c+\cdots)^{1/2}$? Any simplification or approximation of $(a+b+c+\cdots)^{1/2}$ could help me. Thanks in…
user1936314
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Adding 3000 (an approximate number) and 5 (an exact number)–would it still be 3005?

There is an old anecdote. A group of tourists visits a museum. A tourist asks the museum worker: 'How old is this statue?' The worker responds: 'It is 3005 years old.' The tourist expresses his astonishment: 'Wow, it can't be! It must be some kind…
Alexander
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Approximating two-variable function with a product of one-variable functions

Suppose I have an integrable non-const function $f(x,y)$ defined on $V$. I'd like to approximate it with a product $a(x)b(y)$. How can I find such $a(x)$ and $b(y)$ that $$\tag1 \iint_V\left(f(x,y)-a(x)b(y)\right)^2\,dxdy=min$$ ? Are $a$ and $b$…
Ruslan
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Showing $\frac{1-\cos \left((n-1)\pi/n\right)}{1-\cos \left(\pi/n\right)} \approx 4n^2/\pi^2$

Problem Show $\frac{1-\cos \left((n-1)\pi/n\right)}{1-\cos \left(\pi/n\right)} \approx 4n^2/\pi^2$ Try I have noticed that the numerator can be approximated $$ 1-\cos \left((n-1)\pi/n\right) \approx 2 $$ and the denominator can be…
Moreblue
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How do terms like $x\sqrt{1-x^2}$ enhance the ability to approximate analytical functions?

In this paper Quantum Circuit Learning it wrote that the ability of a quantum circuit to approximate a function can be enhanced by terms like $x\sqrt{1-x^2}$ ($x\in[-1,1])$. Given inputs {x,f(x)}, assume we are going to approximate an analytical…
raycosine
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Does there exist infinitely many $\mu$ which satisfy this:

Let $\Bbb S$ be the set of rational numbers $r$ with the property that $\sqrt{r}$ is rational as well, in other words a number $\frac{a^2}{b^2}$ with integers $a,b$ and $b\ne 0$ Example is $\frac{81}{121}=(\frac{9}{11})^2$ My question is this:…
Rhys Hughes
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Approximation of numbers: Am I using ~ correctly?

I saw this Difference between "≈", "≃", and "≅" but it doesn't answer about the single ~. There are also no examples for people who do not major in Math (such as myself). I learned that if I wish to have a number which is "round about" the correct…
Karl Morrison
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What is the proof of the rules of significant figures?

I wanted to know how do we know that the rules that we follow when doing arithmetic with significant figures are correct? Like why when adding or subtracting we keep the same number of decimal places as the original number with the least decimal…
Alraxite
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Smooth approximation of absolute value inequalities

Is there an analytic approximation to the inequality: $$\sum_{i=1}^{n} |x_i| \leq \delta ? $$ I would like to replace the above inequality with a smooth inequality that is "valid" in the sense that if the approximate smooth inequality is satisfied…
Ravi
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Is there a section of mathematics that studies near-integer equations.

When I saw: $$e^\pi-\pi \approx 20$$ I thought it was pretty cool. And : $$\pi^3 \approx 31$$ So now the thought comes to me is what positive integer value of $n$ will make the expression: $$\pi^n$$ As close to a nearest integer as possible.…
Ahmed S. Attaalla
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Gamma function and Stirling's approximation

I am interested in strong upper and lower bounds on $\frac{\Gamma(n+\alpha)}{\Gamma(n)},$ where $n$ is a large non-integral number and $\alpha$ is a small constant like $3.5.$ I know the answer is approximately $n^\alpha$ but I want multiplicative…
Hedonist
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Approximation rules?

Let's say I need to approximate the expression $$\frac{1}{2}mv^2\left(\frac{M}{M+m}+1\right)$$ when $m<
user135842
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Approximation of $\frac{1+a}{1+b}$

I've found the following assertion on an economics book: For $r$ and $g$ small enough, $\frac{1+r}{1+g}\approx 1+r-g$ (where $r$ is the interest rate and $g$ is the growth rate of the economy) I would like to know why this is true. I've tried to…
Luigi
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Curve fitting a power law to a linear fractional transformation

Good Morning ! I have a function $y = \frac{ax + b}{cx + d}$. I want to fit the curve $y_f = c_1 + c_2 x^{c_3}$, so that $||y_f - y||$ is minimized in some norm (say $L^2$), by varying $c_1, c_2$ and $c_3$. I can do the curve fit manually, but I…
user89699
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How to approximate this expression that involve trigamma functions: $\psi ^{(1)}(n+1)-\psi ^{(1)}(n+m+2)$

I have come up with this expression for the standard deviation for a certain distribution: $$\psi ^{(1)}(n+1)-\psi ^{(1)}(n+m+2)$$ where $n$ and $m$ are integers and $\psi^{(1)}(x)$ is a trigamma function. For the software engineering project I am…
Adam Ryczkowski
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