Does the arithmetic derivative have a geometric interpretation
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The standard derivative, for example $frac{d}{dx}x^2 = 2x, $ gives the slope of a function at a particular point. Does the arithmetic derivative have a similarly simple geometric interpretation?
As a specific example, how might I show graphically that $1'=0$, that $2'=1$ or that $8'=12$?
EDIT: The arithmetic derivative can be defined as follows:
$p'=1$ for any prime, $p$
$(pq)'= p'q + pq'$ for any $p,q in mathbb{N}$
(Source: Wikipedia)
From the above, rules analogous to the power and quotient rules can be derived.
For a frenetic introduction, please see this YouTube video.
elementary-number-theory prime-numbers
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
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add a comment |
$begingroup$
The standard derivative, for example $frac{d}{dx}x^2 = 2x, $ gives the slope of a function at a particular point. Does the arithmetic derivative have a similarly simple geometric interpretation?
As a specific example, how might I show graphically that $1'=0$, that $2'=1$ or that $8'=12$?
EDIT: The arithmetic derivative can be defined as follows:
$p'=1$ for any prime, $p$
$(pq)'= p'q + pq'$ for any $p,q in mathbb{N}$
(Source: Wikipedia)
From the above, rules analogous to the power and quotient rules can be derived.
For a frenetic introduction, please see this YouTube video.
elementary-number-theory prime-numbers
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
$endgroup$
4
$begingroup$
What is this "arithmetic derivative", exactly?
$endgroup$
– Arthur
Dec 22 '18 at 18:52
$begingroup$
I will add more detail. I thought it might be an obscure topic.
$endgroup$
– Adam Hrankowski
Dec 22 '18 at 18:59
2
$begingroup$
@AdamHrankowski what actually we have to do in this question?
$endgroup$
– Dynamo
Dec 23 '18 at 12:11
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You know how you can graph a function and draw the tangent line at a point. You can say, "The slope of this line is the same as the derivative at this point." Is there a picture you can draw and say, "This picture helps you see the meaning of the arithmetic derivative"?
$endgroup$
– Adam Hrankowski
Dec 23 '18 at 20:20
$begingroup$
Further discussion here: reddit.com/r/3Blue1Brown/comments/a90drf/…
$endgroup$
– Adam Hrankowski
Dec 26 '18 at 16:55
add a comment |
$begingroup$
The standard derivative, for example $frac{d}{dx}x^2 = 2x, $ gives the slope of a function at a particular point. Does the arithmetic derivative have a similarly simple geometric interpretation?
As a specific example, how might I show graphically that $1'=0$, that $2'=1$ or that $8'=12$?
EDIT: The arithmetic derivative can be defined as follows:
$p'=1$ for any prime, $p$
$(pq)'= p'q + pq'$ for any $p,q in mathbb{N}$
(Source: Wikipedia)
From the above, rules analogous to the power and quotient rules can be derived.
For a frenetic introduction, please see this YouTube video.
elementary-number-theory prime-numbers
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
$endgroup$
The standard derivative, for example $frac{d}{dx}x^2 = 2x, $ gives the slope of a function at a particular point. Does the arithmetic derivative have a similarly simple geometric interpretation?
As a specific example, how might I show graphically that $1'=0$, that $2'=1$ or that $8'=12$?
EDIT: The arithmetic derivative can be defined as follows:
$p'=1$ for any prime, $p$
$(pq)'= p'q + pq'$ for any $p,q in mathbb{N}$
(Source: Wikipedia)
From the above, rules analogous to the power and quotient rules can be derived.
For a frenetic introduction, please see this YouTube video.
elementary-number-theory prime-numbers
elementary-number-theory prime-numbers
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
edited Dec 22 '18 at 21:48
Adam Hrankowski
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
asked Dec 22 '18 at 18:44
Adam HrankowskiAdam Hrankowski
2,094930
2,094930
4
$begingroup$
What is this "arithmetic derivative", exactly?
$endgroup$
– Arthur
Dec 22 '18 at 18:52
$begingroup$
I will add more detail. I thought it might be an obscure topic.
$endgroup$
– Adam Hrankowski
Dec 22 '18 at 18:59
2
$begingroup$
@AdamHrankowski what actually we have to do in this question?
$endgroup$
– Dynamo
Dec 23 '18 at 12:11
$begingroup$
You know how you can graph a function and draw the tangent line at a point. You can say, "The slope of this line is the same as the derivative at this point." Is there a picture you can draw and say, "This picture helps you see the meaning of the arithmetic derivative"?
$endgroup$
– Adam Hrankowski
Dec 23 '18 at 20:20
$begingroup$
Further discussion here: reddit.com/r/3Blue1Brown/comments/a90drf/…
$endgroup$
– Adam Hrankowski
Dec 26 '18 at 16:55
add a comment |
4
$begingroup$
What is this "arithmetic derivative", exactly?
$endgroup$
– Arthur
Dec 22 '18 at 18:52
$begingroup$
I will add more detail. I thought it might be an obscure topic.
$endgroup$
– Adam Hrankowski
Dec 22 '18 at 18:59
2
$begingroup$
@AdamHrankowski what actually we have to do in this question?
$endgroup$
– Dynamo
Dec 23 '18 at 12:11
$begingroup$
You know how you can graph a function and draw the tangent line at a point. You can say, "The slope of this line is the same as the derivative at this point." Is there a picture you can draw and say, "This picture helps you see the meaning of the arithmetic derivative"?
$endgroup$
– Adam Hrankowski
Dec 23 '18 at 20:20
$begingroup$
Further discussion here: reddit.com/r/3Blue1Brown/comments/a90drf/…
$endgroup$
– Adam Hrankowski
Dec 26 '18 at 16:55
4
4
$begingroup$
What is this "arithmetic derivative", exactly?
$endgroup$
– Arthur
Dec 22 '18 at 18:52
$begingroup$
What is this "arithmetic derivative", exactly?
$endgroup$
– Arthur
Dec 22 '18 at 18:52
$begingroup$
I will add more detail. I thought it might be an obscure topic.
$endgroup$
– Adam Hrankowski
Dec 22 '18 at 18:59
$begingroup$
I will add more detail. I thought it might be an obscure topic.
$endgroup$
– Adam Hrankowski
Dec 22 '18 at 18:59
2
2
$begingroup$
@AdamHrankowski what actually we have to do in this question?
$endgroup$
– Dynamo
Dec 23 '18 at 12:11
$begingroup$
@AdamHrankowski what actually we have to do in this question?
$endgroup$
– Dynamo
Dec 23 '18 at 12:11
$begingroup$
You know how you can graph a function and draw the tangent line at a point. You can say, "The slope of this line is the same as the derivative at this point." Is there a picture you can draw and say, "This picture helps you see the meaning of the arithmetic derivative"?
$endgroup$
– Adam Hrankowski
Dec 23 '18 at 20:20
$begingroup$
You know how you can graph a function and draw the tangent line at a point. You can say, "The slope of this line is the same as the derivative at this point." Is there a picture you can draw and say, "This picture helps you see the meaning of the arithmetic derivative"?
$endgroup$
– Adam Hrankowski
Dec 23 '18 at 20:20
$begingroup$
Further discussion here: reddit.com/r/3Blue1Brown/comments/a90drf/…
$endgroup$
– Adam Hrankowski
Dec 26 '18 at 16:55
$begingroup$
Further discussion here: reddit.com/r/3Blue1Brown/comments/a90drf/…
$endgroup$
– Adam Hrankowski
Dec 26 '18 at 16:55
add a comment |
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4
$begingroup$
What is this "arithmetic derivative", exactly?
$endgroup$
– Arthur
Dec 22 '18 at 18:52
$begingroup$
I will add more detail. I thought it might be an obscure topic.
$endgroup$
– Adam Hrankowski
Dec 22 '18 at 18:59
2
$begingroup$
@AdamHrankowski what actually we have to do in this question?
$endgroup$
– Dynamo
Dec 23 '18 at 12:11
$begingroup$
You know how you can graph a function and draw the tangent line at a point. You can say, "The slope of this line is the same as the derivative at this point." Is there a picture you can draw and say, "This picture helps you see the meaning of the arithmetic derivative"?
$endgroup$
– Adam Hrankowski
Dec 23 '18 at 20:20
$begingroup$
Further discussion here: reddit.com/r/3Blue1Brown/comments/a90drf/…
$endgroup$
– Adam Hrankowski
Dec 26 '18 at 16:55