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

http://youtu.be/AJ4zlvqOtE8?t=4m43s

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

Jeel Shah
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user8005
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    He's just being sloppy for the lay person viewer, really he is taking a limit. – Ragib Zaman Jun 02 '13 at 15:53
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    He is just a famous physicist. Just trust in Mathematicians. :D – Mikasa Jun 02 '13 at 15:53
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    @BabakS. Wise words. – Git Gud Jun 02 '13 at 15:59
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    "To a mathematician $\infty$ is just a number without limit". I didn't know he was a comedian too. Maybe trying to compete with Neil deGrasse? – Git Gud Jun 02 '13 at 16:01
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    @GitGud He's clearly not trying to be precise for any mathematicians who happen to be watching. It was simply the quickest way to convey his message that mathematicians don't find infinity intimidating and it's not really a problem, while to a physicist it's a huge problem. – Ragib Zaman Jun 02 '13 at 16:08
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    @RagibZaman I get that, but I'll pick on physicists any chance I get. – Git Gud Jun 02 '13 at 16:10
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    @RagibZaman Did you mistakenly invert "mathematicians" and "physicist" in your comment? – Did Jun 02 '13 at 16:18
  • @Did No but I can see how it could be read differently to how I intended it. I meant "He's not worried about being precise just to please any mathematicians who happen to be watching". – Ragib Zaman Jun 02 '13 at 16:26
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    @Ragib: I think that Did was suggesting that mathematicians tend to struggle more with infinity than do physicists, largely because physicists are more likely to use "handwavery" such as that mentioned by the OP, in the rare cases that they are obliged to deal with the infinite at all. Some mathematicians deal with the infinite all the time, and know how frustrating it can be when it isn't behaving nicely. – Cameron Buie Jun 02 '13 at 17:07
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    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 02 '13 at 17:26
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    Babak's comment is right on the money. I thought Michio was dumbing down something to sell programs for TV, but that infinity thing is a nonsense the size of the whole last Bush's administration: you can't dumb that down something without being called dumb yourself. – DonAntonio Jun 02 '13 at 19:59
  • @GeorgesElencwajg: Did you mean http://www.math.columbia.edu/~woit/wordpress/?p=1520? =) – user21820 Jan 25 '14 at 13:29
  • @Georges Elencwajg: I thought he was using terms like "limit" in colloquial, non-technical senses (note that he's doing pop sci here, not technical papers or textbooks), so what he would mean would be a "number without any [strict] bound [in at least one direction]", or a "number which has no number greater [or lesser] than it". This is what $+\infty$ is on the extended real line, if you call that a "number" (and $-\infty$ too, for "less than it" instead of "greater than it"). – The_Sympathizer Oct 15 '14 at 02:12
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    Additionally I've heard Kaku say that the mathematics behind physics is "just bookkeeping." This is a guy who works in string theory, mind you. – Tim kinsella Jun 05 '16 at 10:50

2 Answers2

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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 . . . . . .

Michael Hardy
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  • He was talking about the 'singularity' in physics. (Not to be confused with technological singularity) – user8005 Jun 02 '13 at 16:59
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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 $0$.

Mikhail Katz
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