शैडोइंग अभ्यास: Mathematician Collapses All Functions to One Weird Formula - वीडियो के साथ अंग्रेजी बोलना सीखें

plus, minus, x times y, x to the power of y, sine and cosine.

Maths is full with complicated operations that you do to numbers.

An ambitious new paper now claims that actually you can do everything with only one operation.

That's quite a claim.

Maybe that finally explains why the Americans have only one math, where the Brits have maths.

I've had a look.

The new paper's a preprint that appeared on the archive the other week and comes from a mathematician in Poland.

The one operation

that he says can give everything is taking the exponential function of a number x minus the logarithm of another number y.

He calls that ML, short for exponential minus logarithm.

For example, if you put x and 1 into the ML function, you get e to the x minus log of 1.

Log of 1 is 0, so this gives you just the exponential function.

You can also get the logarithm itself back out.

For this, you first take ml of 1 and x, that gives you e minus log x.

Then you put that back into the first slot with another one.

So ml of ml 1 x 1, that gives e to the e over x.

And then you put that yet again into the second slot of ml with a 1 in the first.

If you work through the algebra, all the extra pieces cancel and you're left with just log of x.

So the same function gives you exponentials and logarithms indirectly.

But how do we get the standard operations, addition, subtraction, multiplication, etc.? They're all in there.

Take subtraction.

We just saw that you can express the exponential function and the logarithm through ML operations.

If you put those back into the ML function, what do you get?

You get x minus y.

You can also get a plus just by entering minus y.

And now you see you can put the sum of two logarithms into the exponent of the exponential function.

That'll give you multiplication.

And once you have multiplication, you can define powers and so on.

If you want to get trigonometric functions, you first need imaginary numbers.

You can get those from the logarithm at negative numbers, though there's a branch cut involved there.

I know that sounds like a gardening accident, but it just defines the logarithm for a negative number.

But once you have that, you get complex numbers, and you get sine and cosine and even the number pi.

What you do not get, for all I can see, is all real numbers starting from just 1 and 0, not without taking infinite limits.

But the author doesn't make that claim.

He says one can get all mathematical operations.

The paper hasn't yet been peer -reviewed, but I'm reasonably sure it's correct.

I saw a lot of people online worry about the arbitrary branch cut of the logarithm, but I don't think this matters.

Yes, you could choose it differently, and that it'd give you a different definition of complex numbers, but it'd still work.

Now, you may be wondering whether this is actually useful.

The answer is, for normal people, probably not.

No one's going to replace a normal calculator with one

that has a single deranged button that expects you to rebuild cosine by nesting 15 different operations.

The interesting part is structural.

In computer logic, there's a logical operation called NAND, which is enough on its own to build all the usual logical gates in a computer.

This paper's trying to find the analog for ordinary continuous maths.

The claim is that EML is kind of a NAND gate for scientific calculators.

And there is a possible practical angle.

If every ordinary formula can be rewritten as a tree made from one repeated building block,

then that gives you a very uniform way to search for formulas with a computer.

The author suggests this could be useful for symbolic regression, which means trying to guess the equation behind a set of data.

So this may be less about helping humans do maths

and more about giving machines a simpler language in which to search for equations,

which is nice because if the machines are going to take over science, then at least they will do it in an orderly manner.

I give this paper a 0 out of 10 on the bullshit meter.

It's a cute paper.

I think that the point isn't super surprising to anyone, but it is surprising that no one has done it before.

There are two ways one can look at this result.

One is that mathematics appears more complicated than it is.

The other way to look at it is that what we think of as complicated is very subjective.

You could rightfully argue that maybe using 10 or

so different operations is easier than nesting just one operation 20 times in itself.

In the end, it depends on just how you quantify simplicity.

This is also an issue that haunts the foundations of physics, because what really do we mean when we say

that simple explanations are better if we don't agree on how to measure simplicity?

Or maybe it just tells us that mathematicians have too much time.

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इस पाठ की शब्दावली और बोलने के सुझाव

इस वीडियो में शैडोइंग के लिए 81 वाक्य और 1038 शब्द हैं। बोला गया हिस्सा 6:24 लंबा है। वक्ता लगभग 162 शब्द प्रति मिनट की स्वाभाविक गति से बोलता है, जो रोज़मर्रा की बातचीत के करीब है। केवल 79% शब्द अंग्रेज़ी के 3,000 सबसे आम शब्दों में से हैं, इसलिए शब्दावली कठिन है।

इस वीडियो की मुख्य शब्दावली

वीडियो के 15 सीखने लायक शब्द, उच्चारण और अर्थ के साथ:

शब्द	उच्चारण	अर्थ
minus	/ˈmaɪ.nəs/	कम, घटाकर
mathematics संज्ञा	/mæθ(.ə)ˈmæt.ɪks/	गणित, गणित-विधा
complicated विशेषण	/ˈkɑm.plɪˌkeɪ.tɪd/	जटिल, पेचीदा
earn क्रिया	/ɝn/	कमाना
practical विशेषण	/ˈpɹæk.tɪ.kəl/	प्रायोगिक
gate संज्ञा	/ˈɡeɪ̯t/	फाटक, द्वार
formula संज्ञा	/ˈfɔː.mjʊ.lə/	सूत्र, फ़ॉरमूला
ordinary विशेषण	/ˈɔɹdɪnɛɹi/	साधारण, मामूली
logical विशेषण	/ˈlɑd͡ʒɪkəl/	तार्किक
nest संज्ञा	/nɛst/	घोसला, घोंसला
equation संज्ञा	/ɪˈkweɪ.ʒən/	समीकरण
calculator संज्ञा	/kæl.kjə.leɪ.tɚ/	परिकलक, कैलकुलेटर
explanation संज्ञा	/ˌɛkspləˈneɪʃən/	स्पष्टीकरण, समझाना
physics संज्ञा	/ˈfɪz.ɪks/	भौतिक शास्त्र, भौतिकी
angle संज्ञा	/ˈæŋ.ɡəl/	कोण

इस वीडियो का व्याकरण

वक्ता जिन संरचनाओं का सबसे अधिक प्रयोग करता है, वीडियो के असली शब्दों के साथ:

संरचना	वीडियो में
Passive voice be + past participle — ज़ोर इस पर कि क्या होता है, न कि कौन करता है	is rated · be prepared · are limited
Relative clauses who / which + उपवाक्य — व्यक्ति या वस्तु के बारे में अतिरिक्त जानकारी	NAND, which is · regression, which means · those who don
Present perfect have/has + past participle — बीता हुआ काम जिसका असर अभी भी है	has done · has already built

ध्यान देने योग्य उच्चारण

वक्ता 14 बार संक्षिप्त रूपों का प्रयोग करता है, जैसे don't, it'd, doesn't। इन्हें वैसे ही छोटा बोलें जैसा आप सुनते हैं।

“th” की ध्वनियाँ: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
“sh” और “zh” की ध्वनियाँ: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
लंबे शब्द — बलाघात सही रखें: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/

इस वीडियो से अभ्यास कैसे करें

पहले बिना बोले पूरा वीडियो एक बार सुनें और जो शब्द नहीं आते उन्हें लिख लें।
0.75× गति से शुरू करें, एक-एक वाक्य की शैडोइंग करें, और सहज लगने पर सामान्य गति पर लौटें।
अपनी आवाज़ रिकॉर्ड करें और मूल से तुलना करें, ख़ासकर minus, mathematics, complicated जैसे शब्दों पर ध्यान दें।

शैडोइंग तकनीक क्या है?

शैडोइंग (Shadowing) एक विज्ञान-समर्थित भाषा सीखने की तकनीक है जो मूल रूप से पेशेवर दुभाषिया प्रशिक्षण के लिए विकसित की गई थी। विधि सरल लेकिन शक्तिशाली है: आप मूल अंग्रेज़ी ऑडियो सुनते हैं और तुरंत इसे ज़ोर से दोहराते हैं — जैसे वक्ता की छाया 1-2 सेकंड की देरी से। शोध से पता चलता है कि यह उच्चारण सटीकता, स्वर, लय, जुड़ी हुई ध्वनियाँ, सुनने की समझ और बोलने की प्रवाहशीलता में काफ़ी सुधार करता है।

शैडोइंग तकनीक: पूरी चरण-दर-चरण गाइड पढ़ें →

शैडोइंग अभ्यास: Mathematician Collapses All Functions to One Weird Formula - वीडियो के साथ अंग्रेजी बोलना सीखें

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इस वीडियो की मुख्य शब्दावली

इस वीडियो का व्याकरण

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इस वीडियो से अभ्यास कैसे करें

शैडोइंग तकनीक क्या है?