Is $\frac 10 = \infty$?

Question

In many places I have heard that $\frac 10 = \infty$. While I do believe this to be a flawed concept, and there are many posts on this, I wanted to investigate some properties of $\frac 10$. First I would like to state some properties of infinity:
$$1)\infty + 1 = \infty$$
$$2)k* \infty = \infty$$
These can be hard to prove with most other expressions for $\infty$ but with this very flawed description, both these properties can be shown!
$1) LHS= \frac 10 + 1= \frac{1+1*0}{0} = \frac 10 = RHS$
$2) LHS= k*\frac 10 = \frac k0 = \frac{1}{0/k} = \frac 10 = RHS$
However sometimes it is simply wrong. For example:
$\frac 10 = \infty$
$1 = \infty * 0$
$1 = 0$

So I would like to know if it actually is a plausible solution or not. I know there are many posts about this but I hope no one minds if these properties are investigated.
Thanks in advance!

Answer 1

This is not true. Period. Why you ask? Because $0$ doesn't have a multiplicative inverse. You yourself have noted that it produces contradictions. $\infty$ is more of an upper bound for reals than a number itself. While there are fields of algebra where it's treated as a number, most of the time, it's not. BUT, in some fields outside mathematics e.g. Physics, for all intents and purposes ${1 \over 0}=\infty$ as they only need approximations for real life applications. But, in mathematics, $$\lim_{\alpha \to 0+}{1 \over \alpha}=\infty$$ This is as much as you can say.

Answer 2

The "properties" you mentioned

• $\infty+1=\infty$
• $k\cdot\infty=\infty$ (for $k>0$)

are called arithmetic operations for the extended real numbers. They are true by definition. With such definition, lots of theorems in real analysis can be stated in a neat way. If one is talking about the set $\overline{\mathbb R}$ of extended real numbers, then there two different "infinities": $\pm\infty$. Note that $\overline{\mathbb R}$ is defined as $$ \overline{\mathbb R}=\{\mathbb R\}\cup\{-\infty,+\infty\} $$ where ${\mathbb R}$ denotes the set of real numbers. In this case, the plus sign in $+\infty$ is usually not omitted. On the other hand, one could also talk about the extended nonnegative real numbers $[0,\infty]$, which appears a lot especially in integration theory.

However, one should be careful that the arithmetic operations of ${\mathbb R}$ can be only partially extended to $\overline{\mathbb R}$ or $[0,\infty]$. For instance $\frac{1}{0}$ is not defined in $\overline{\mathbb R}$: it is neither $-\infty$ nor $+\infty$. On the other hand, $\frac{1}{0}=\infty$ (as well as $\frac{1}{\infty}=0$) is used in the statement of the Cauchy-Hadamard Theorem since with such definition, the radius of convergence of any power series $\sum a_nz^n$ can be written as $$ R=\frac{1}{\limsup_{n}|a_n|^{1/n}}. $$

See also a discussion on the extended reals in this set of lecture notes by Terry Tao. Here is an excerpt:

Most of the laws of algebra for addition, multiplication, and order continue to hold in this extended number system; for instance addition and multiplication are commutative and associative, with the latter distributing over the former, and an order relation ${x \leq y}$ is preserved under addition or multiplication of both sides of that relation by the same quantity. However, we caution that the laws of cancellation do not apply once some of the variables are allowed to be infinite; for instance, we cannot deduce ${x=y}$ from ${+\infty+x=+\infty+y}$ or from ${+\infty \cdot x = +\infty \cdot y}$. This is related to the fact that the forms ${+\infty - +\infty}$ and ${+\infty/+\infty}$ are indeterminate (one cannot assign a value to them without breaking a lot of the rules of algebra). A general rule of thumb is that if one wishes to use cancellation (or proxies for cancellation, such as subtraction or division), this is only safe if one can guarantee that all quantities involved are finite (and in the case of multiplicative cancellation, the quantity being cancelled also needs to be non-zero, of course). However, as long as one avoids using cancellation and works exclusively with non-negative quantities, there is little danger in working in the extended real number system.

Answer 3

I have always known infinity as a concept not as a no. Dealing with infinity the way we deal with no. Is wrong ,we can't really do that. 1/0 is not equal to infinity just because it's a pattern that it goes larger consider lim $1/x$ as $x$ goes to zero but if you see from what direction we are going that matters a lot left hand limit comes to be different from the right hand limit hence limit actually doesn't exist.

Answer 4

The appropriate setting for $\frac{1}{0}=\infty$ is the complex numbers. Here the field $\mathbb C$ is extended to the Riemann sphere $\hat{\mathbb C}=\mathbb C \cup \{\infty\}$. Such an extension is extremely useful in complex analysis when one works with holomorphic functions, and is a first step toward the modern approach to Riemann surfaces.