\(\text { Is } \frac{1}{\infty}=0 ?\)
Answer 1:
\(\frac{1}{\infty}\) does not equal \(0\), but brings it relatively as close to it as possible, which some will just accept it as 0.
Explanation:
\(\frac{1}{\infty}\) is equals not 0. but a number relatively close to it.
\(∞\) is a number not capable of being defined so are things like \(\frac{\infty}{\infty}\) and \(\frac{-\infty}{-\infty}\) is not possible to understand.
Ex:
\(\begin{aligned}
&\frac{1}{10}=0.1 \\
&\frac{1}{100}=0.01 \\
&\# 1 / 10000000=0.0000001
\end{aligned}\)
Now you can these number are not 1 and are not even relatively close to 0, but if you divide by a relatively huge number you will get a extremely small number which will nowhere be 0, but close to it.
As for anything times \(0\) is just \(0\).
Answer 2:
See below.
Explanation:
As others have pointed out, it is an error to say that \(\frac{1}{\infty}=0\).
If a fraction is formed by the ratio of \(1\) with a function (like \(x\) or \(x^{4}\) or \(\tan (x)\), then it is true to say that as the denominator increases without bound, the ratio approaches \(0\).
Another way of saying this is:
If we divide 1 by numbers that get bigger and bigger, then the result gets closer and closer to zero. (Notice that no mention of “infinity” is needed here.) The precise mathematical definition takes a bit of work to understand:
we say that \(f(x)\) approaches \(0\) as \(x\) increases without bound (as \(x\) tends to infinity) it and only if
For any positive number \(\varepsilon\), there is a number \(M\) that satisfies:
for all \(x>M\), we have \(|f(x)|<\varepsilon\).
Notice that in the part after “if and only if” there is no mention made of infinity. We do not treat infinity as a number. (In spite of the phrase “as \(x\) tends to infinity”).
We do use the notation
\(\lim _{x \rightarrow \infty} f(x)=0\)and read it “the limit as \(x\) approaches infinity of \(f\) of \(x\) is zero”.
The phrase “x approaches infinty” is easy to misunderstand and is best replaced by “x increases without bound”.
Answer 3:
I would like to use polar coordinates.
In polar form, \(θ = α\) represents the radial line from pole, in the direction making an angle \(α\) with the initial line \(θ = 0.\)
Upon this line \(r \in[0, \infty)\).
Let \(f(r, \theta)=\frac{\theta}{r}\).
Let us make \(θ\) an arbitrary constant \(α\) so that
\(f(r, \theta)=\frac{\alpha}{r}\),
Now, \(f \rightarrow \pm \infty, \text { as } r \rightarrow 0\) and \(f \rightarrow 0, \text { as } r \rightarrow \infty\).
Cross multiply and take the limits.
\(\alpha=\lim r \rightarrow 0(r f)=\lim r \rightarrow \infty(r f)\)Is it not understandable that, in either case, the indeterminate form
of (rf) is \(\pm 0 X \infty ?\) the limit could be any \(\alpha \in(-\infty, \infty)\),
covering both clockwise and anticlockwise rotations, for \(θ\)?
Remember that the direction \(\theta=\alpha\) is at your choice.
Answer 4:
\(∞\) is not a number. In calculus it is a shorthand used when describing limit processes.In calculus it is used as shorthand notation for limits. In complex analysis and other interesting areas of mathematics it can have other meanings…
Consider a sphere with equation \(x^{2}+y^{2}+\left(z-\frac{1}{2}\right)^{2}=\frac{1}{4}\)
This has centre \(\left(0,0, \frac{1}{2}\right)\) and radius \(\frac{1}{2}\)
It “sits on” the XY plane at the origin, with the top point of the sphere being \(\) (0,0,1)
Now consider lines passing through (0,0,1)
Any such line will either intersect the plane at one point (x,y) and the sphere at one point in addition to (0,0,1) or will be parallel to the plane and tangential to the sphere, only touching it at (0,0,1)
Now suppose the XY plane represents the Complex numbers. Each \(x+i y \in \mathbb{C}\) corresponds to a unique line through (0,0,1) that intersects the sphere at one other point. Hence every Complex number is represented by a unique point on the sphere.
The unit circle corresponds to the equator of the sphere. We now label the top of the sphere \(∞\)
This spherical representation of \(\mathbb{C} \cup\{\infty\}\) is called the Riemann sphere and is sometimes designated \(\mathbb{C}_{\infty}\)
We can define a few arithmetical operations involving \(∞\), but several are undefined:
With these restrictions \(\mathbb{C}_{\infty}\) is not a field, but it does have some nice properties:
A Möbius transformation is a function of the form:
for some \(a, b, c, d \in \mathbb{C} \text { with } a \neq 0 \text { or } c \neq 0\) (or both).
Such a transform maps circles on \(\mathbb{C}_{\infty}\) to circles – regardless of whether they pass through \(∞.\)