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mesa
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I am trying to find a general expression for radicals, for example,
√k=f(k)
Does anyone know of any?
Thanks!
√k=f(k)
Does anyone know of any?
Thanks!
HallsofIvy said:I don't know what you mean by a "general expression". The square root, that you have there, can be written [itex]\sqrt{k}= k^{1/2}[/itex]. A general radical, the "nth root", can be written [tex]\sqrt[n]{k}= k^{1/n}[/tex]. Is that what you mean?
disregardthat said:By phi, I assume you mean [itex]\frac{1+\sqrt{5}}{2}[/itex], so what you've done is just rewriting [itex]\sqrt{5}[/itex] in terms of this. I don't see any general about this. What exactly is it what you want, when you say you want a general expression?
disregardthat said:[itex]\sqrt{x}[/itex] is an expression of x. It is also unclear why you would accept [itex]\sqrt{5} = 2 \cdot \frac{1+\sqrt{5}}{2} -1[/itex] as an 'expression' of [itex]\sqrt{5}[/itex]. Why not accept [itex]\sqrt{x} = 2 \cdot \frac{1+\sqrt{x}}{2} -1[/itex] as an 'expression' of x?
mesa said:It is correct but [itex]\sqrt{x} = 2 \cdot \frac{1+\sqrt{x}}{2} -1[/itex] is nothing more than √x=√x.
Do you know of a form like √5=2phi-1 for say √3=?
disregardthat said:√5=2phi-1 is also nothing but √5=√5.
I think you need to explain more clearly what you mean by a 'general expression'.
mesa said:Sorry about that, let's try it this way, can you write √3 as an expression with phi?
disregardthat said:Without using √ you mean? And why would you want that? And what is an expression with phi? Is √3=√3 +phi -phi an expression with phi? Just explain what you are after in a clear and precise manner, it's no fun guessing.
Hertz said:Hi mesa,
You can approximate the square root of a number using this formula:
$$
f(x)=\sqrt{x}\approx1+\frac{1}{2}(x-1)+(-\frac{1}{4})(x-1)^2+\frac{3}{8}(x-1)^3+...=\sum_{n=0}^{...}f_n(x-1)^n
$$
where the coefficients ##f_n## can be computed by
$$
f_n=\frac{1}{n!}\prod_{m=0}^{n-1}(\frac{1}{2}-m)
$$
I haven't checked what values of x this is valid for though
Yes. A guy named Taylor.mesa said:I stand corrected, that is a very nice approximation! I wrote a continued fraction but it is ugly :P
Any chance you know who put this together?
mesa said:I stand corrected, that is a very nice approximation! I wrote a continued fraction but it is ugly :P
Any chance you know who put this together?
Hertz said:It's a Taylor Series expansion of ##f(x)=x^{1/2}## around the point ##x=1##. Originally I did it for ##f(x)=x^m## for arbitrary m. For that one, the expansion is this: $$
f(x)=x^m=\sum_{n=0}^{\infty}f_n(x-1)^n$$ where ##f_0=1## and for ##n\neq 0## $$f_n=\frac{1}{n!}\prod_{k=0}^{n-1}(m-k)$$ So
$$f(x)=1+\sum_{n=1}^{\infty}\frac{1}{n!}\Bigg(\prod_{k=0}^{n-1}(m-k)\Bigg)(x-1)^n$$
Kind of a messy formula lol
If the engineering curriculum includes calculus, it's a safe bet that Taylor's series (and Maclaurin series) will be covered pretty extensively, usually after the course where integration is taught.mesa said:I should really look into Taylor series more... The schools engineering program hardly touches on it :P
mesa said:Anyway, I don't think it is all that messy, thanks for sharing! (it is always good to know the 'gaps' in order to correct them)
Mark44 said:If the engineering curriculum includes calculus, it's a safe bet that Taylor's series (and Maclaurin series) will be covered pretty extensively, usually after the course where integration is taught.
A radical is a mathematical expression that includes a root, such as a square root, cube root, etc. It is denoted by the √ symbol and is used to find the value that, when multiplied by itself a certain number of times, will result in the given number.
Looking for expressions for radicals allows us to simplify and solve equations that involve roots. It also helps us to find the exact value of a root without using a calculator.
To express a radical in terms of exponents, we use the index of the root as the exponent. For example, the square root of a number can be written as that number raised to the power of 1/2.
A radical is a mathematical expression that includes a root, while a rational exponent is a fraction in which the numerator is the power and the denominator is the root. For example, √x can be written as x^(1/2).
To simplify expressions with radicals, we can use the rules of exponents and properties of radicals. This involves factoring the number under the radical, canceling out any perfect squares, and simplifying any remaining radicals.