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Why does Michio Kaku say that $\frac{1}{0} = \infty$?

Instead of $\frac{1}{0}$ that's not defined, so we don't know.

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He's just being sloppy for the lay person viewer, really he is taking a limit. – Ragib Zaman Jun 2 '13 at 15:53
He is just a famous physicist. Just trust in Mathematicians. :D – Babak S. Jun 2 '13 at 15:53
@RagibZaman I get that, but I'll pick on physicists any chance I get. – Git Gud Jun 2 '13 at 16:10
@RagibZaman Did you mistakenly invert "mathematicians" and "physicist" in your comment? – Did Jun 2 '13 at 16:18
Dear @Babak, Kaku is notorious for hyping his science-fiction books disguised as physics. Journalists take him seriously, not physicists (just browse Woit's blog "Not Even Wrong"). Anybody capable of writing a sentence as idiotic as "To a mathematician $\infty$ is just a number without limit" (limit of a number! The guy doesn't even understand elementary calculus...) or of scornfully talking of a "fundamental flaw" in Einstein's theory does not deserve to be taken seriously. – Georges Elencwajg Jun 2 '13 at 17:26

This is context-dependent. For some purposes, in particular in projective geometry, in trigonometry, in dealing with rational funtions, it makes sense to have a single object called $\infty$ that's at both ends of the real line, so that the line is topologically a circle. In other contexts it makes sense to distinguish between two objects, $\pm\infty$. Any of these three things can in some instances be the limit of a function.

I don't agree with his statement that to mathematicians, infinity is simply a number without limit. A variety of different concepts of infinity exist in mathematics. There are some things that must be considered infinite numbers, including (1) cardinalities of infinite sets and (2) infinite nonstandard real numbers and (3) some other things. (1) and (2) in this list are definitely not the same thing. There are also the infinities involved in things like the Dirac delta function $\delta$, where, loosely speaking, one says $\delta(0)=\infty$, but notice that $2.3\delta$ is different from $\delta$, so this "$\infty$" is not "simply a number . . . . . .". There is the $\infty$ of measure theory, satisfying the identity $0\cdot\infty=0$, and thre are the $\infty$s of calculus, in which $0\cdot\infty$ is a indeterminate form. This is far from a complete enumeration . . . . . .

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He was talking about the 'singularity' in physics. (Not to be confused with technological singularity) – user8005 Jun 2 '13 at 16:59

The existing comments and answer don't seem to mention the pertinent fact that in complex analysis it is indeed correct to assert that $\frac{1}{0}=\infty$ because the complex line $\mathbb{C}$ is completed to the Riemann sphere $\mathbb{C}\cup \{ \infty \}$ by means of adding a single point at infinity, which is the reciprocal of $0$.

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