Today I Learned

[etotheipi]

$\sin^4 x +\cos^4 x$ can never be equal to 2. So, we cannot just blindly add two inequalities.

Well no it can't be 2, but of course $0 \leq \sin^4{x} + \cos^4{x} \leq 2$ is still true. That is the relation that follows from your first two inequalities, in fact in order to justify changing the right one to a "<" you need to do some additional working.

 

[Adesh]

Well no it can't be 2, but we of course $0 \leq \sin^4{x} + \cos^4{x} \leq 2$ is still true. That is the relation that follows from your first two inequalities, in fact in order to justify changing the right one to a "<" you need to do some additional working.

That’s why said inequalities are disloyal, you cannot trust them.

 

[Adesh]

course 0≤sin4⁡x+cos4⁡x≤2 is still true.

No, you have to have strict inequality for 2, that expression cannot be equal to 2.

 

[etotheipi]

No, you have to have strict inequality for 2, that expression cannot be equal to 2.

If I wanted to I could write $\sin^4{x} + \cos^4{x} \leq 10000000000007846789$.

When you add inequalities you get out something weaker, but it's still a valid statement. It would actually be invalid without further justification/proof to switch a non-strict inequality to a strict inequality, as you are suggesting.

 

[DrGreg]

No, you have to have strict inequality for 2, that expression cannot be equal to 2.

Do you think the statement$$ 1 \leq 2 $$is true or false?

 

[Adesh]

Do you think the statement$$ 1 \leq 2 $$is true or false?

Okay, it says “1 is less than 2 or equal to 2” so if any of them is true then the statement is true, therefore your statement is true from the logic rules, but that’s really not a good way to say 1 is less than 2, in my opinion.

 

[Adesh]

If I wanted to I could write $\sin^4{x} + \cos^4{x} \leq 10000000000007846789$.

When you add inequalities you get out something weaker, but it's still a valid statement. It would actually be invalid without further justification/proof to switch a non-strict inequality to a strict inequality, as you are suggesting.

When I write $-1\leq \sin x\leq 1$ I mean that inf of sin x is -1 and sup of sin x is 1. Non-strict inequality does imply a inf/sup, while a strict equality means just a lower/upper bound.

 

[Adesh]

If I wanted to I could write $\sin^4{x} + \cos^4{x} \leq 10000000000007846789$.

When you add inequalities you get out something weaker, but it's still a valid statement. It would actually be invalid without further justification/proof to switch a non-strict inequality to a strict inequality, as you are suggesting.

If that’s true then I don’t think strict inequality have any use, for me non-strict inequality does always imply an equality and inequality.

 

[etotheipi]

for me non-strict inequality does always imply an equality and inequality.

This doesn't make any sense...

 

[Adesh]

This doesn't make any sense...

It does make sense

 

[DrGreg]

When I write $-1\leq \sin x\leq 1$ I mean that inf of sin x is -1 and sup of sin x is 1.

You might mean that, but that's an incorrect interpretation.

$-2\leq \sin x\leq 2$ is a true statement.

By the way, I don't think you meant "inf" and "sup", you meant "min" and "max".

 

[fresh_42]

I think you should stop this discussion in this thread.
1.) It was a pun, not a theorem: $0\leq \sin^4x\, , \,\cos^4x\leq 1$ but $0 < \sin^4x+\cos^4x<2$. Nothing wrong here.
2.)

for me non-strict inequality does always imply an equality and inequality.

was meant as $\leq$ implies the possibilities $<$ and $=$. O.k. an OR would have been better. but it was clear what has been meant.

You are riding dead horses here.

 

[etotheipi]

1.) It was a pun, not a theorem

OK well in that case I think it's better off in Collection of Lame Jokes

 

[jbriggs444]

Non-strict inequality does imply a inf/sup, while a strict equality means just a lower/upper bound.

It may imply, but it would be strictly incorrect to infer.

 

[Adesh]

We are asked to stop this discussion, I too have many things to say but I’m following what I have been asked to do.

 

[Adesh]

It may imply, but it would be strictly incorrect to infer.

Well I don’t see any reason for writing $\sin x \leq 2020$. What’s the use of strict inequality then if all of our job is implied in non-strict inequality?

 

[Swamp Thing]

At one time it was pretty common to start certain engines with a kick from a shotgun type blank cartridge.=================================
See also:
https://en.wikipedia.org/wiki/Coffman_engine_starter
www.youtube.com/watch?v=vb0-br7hTw0
=================================

 

[Drakkith]

Today I learned that over 160 people have been fitted with a plutonium fueled nuclear powered artificial pacemaker, and that one woman's pacemaker was still functional after 34 years: https://uk.reuters.com/article/heal...nergized-after-34-years-idUKN1960427320071219

Article is from 2007, so it's possible the woman is still alive and her pacemaker still functioning for more than 45 years after first being implanted.

 

[atyy]

Today I learned that both are these are sung by the same singer. I've long known the Jessye Norman's recording of Beim Schlafengehen. I first thought it was something like an April Fool's joke when I came across the rendition of When the Saints, but I've now learned that it is also sung by Norman.

 

[fresh_42]

TIL that the world (virtual) speed record is 25c (by kinematic effect).

 

[Buzz Bloom]

TIL that the world (virtual) speed record is 25c (by kinematic effect).

Hi fresh:

Please explain this. Is it a joke? I was unable to find anything abut this on the internet.

I am guessing that it refers to a SF film in which a spaceship travels at 25 times the speed of light. If that is correct, what is the film? I would also calculate that Star Trek, Star Wars, and The Last Starfighter beat that speed easily.

Regards,
Buzz

 

[fresh_42]

No, it was an astronomy professor who said this. His explanation was poor to non existent, he just mentioned it. It is of course no real velocity, only a virtual one. He talked about a very specific type of stars, which he called plasars. He said that if we look directly into their jets, it looks like we measured FTL velocities. I think it has to do with the relativistic version of the Doppler effect. As it was a tv show, we couldn't expect him to provide further details.

 

[jbriggs444]

No, it was an astronomy professor who said this. His explanation was poor to non existent, he just mentioned it. It is of course no real velocity, only a virtual one. He talked about a very specific type of stars, which he called plasars. He said that if we look directly into their jets, it looks like we measured FTL velocities. I think it has to do with the relativistic version of the Doppler effect. As it was a tv show, we couldn't expect him to provide further details.

As I recall, a jet that is emitted at relativistic velocities at an angle almost but not quite directly toward the viewer can give the appearance of faster-than-light speeds.

The direct signal and the tip of the jet are nearly side-by-side and can be separated by a far greater distance "horizontally" than one might expect based on the time lag of their signals as picked up by the receiver.

 

[Ibix]

Please explain this. Is it a joke? I was unable to find anything abut this on the internet.

https://en.wikipedia.org/wiki/Superluminal_motion

 

[Drakkith]

Today I learned what it's like to have a tooth fall apart in your mouth. I go see a dentist tomorrow. Wish I had dental insurance.

 

[Drakkith]

Today I learned that the JunoCam, the camera aboard the Juno spacecraft orbiting Jupiter, uses the exact same camera sensor that my own astrophotography camera uses: the Kodak KAI-2020 sensor.

 

[Swamp Thing]

Today I learned that no one knows if $\pi + e$ is rational or irrational.

youtu.be/1LoSV1sjZFI?t=364

 

[Keith_McClary]

what it's like to have a tooth fall apart in your mouth

The first time that happens is most upsetting. After a few you get used to it.

 

[Fewmet]

Today I learned that an exonym is a word applied to a "a geographical place, group of people, individual person, or a language/dialect" by an outsider.

I also learned that "calque" (a great Scrabble word) is a word or phrase that is taken as a literal translation from another language. Wikipedia gives a list. (It is not always clear to me, though, why some of those examples are "borrowed" and not simple translations.

 

[mfb]

(It is not always clear to me, though, why some of those examples are "borrowed" and not simple translations.

These lists tend to grow by everyone adding their favorite words, fitting or not. Obvious nonsense is filtered out by Wikipedia's control mechanisms, but "it's not clear how fitting this is" has a good chance to stay in the article.