How much bigger is 3↑↑↑↑3 compared to 3↑↑↑3?
$begingroup$
3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms of 3↑↑↑3?
big-numbers
$endgroup$
add a comment |
$begingroup$
3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms of 3↑↑↑3?
big-numbers
$endgroup$
$begingroup$
No, it is much much much much much much much much much much much much much much much much much much much much much much much much much much much larger.
$endgroup$
– Yves Daoust
Oct 22 '18 at 21:29
add a comment |
$begingroup$
3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms of 3↑↑↑3?
big-numbers
$endgroup$
3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms of 3↑↑↑3?
big-numbers
big-numbers
edited Oct 22 '18 at 21:48
anomaly
17.4k42664
17.4k42664
asked Oct 22 '18 at 21:22
Shaan PaymasterShaan Paymaster
111
111
$begingroup$
No, it is much much much much much much much much much much much much much much much much much much much much much much much much much much much larger.
$endgroup$
– Yves Daoust
Oct 22 '18 at 21:29
add a comment |
$begingroup$
No, it is much much much much much much much much much much much much much much much much much much much much much much much much much much much larger.
$endgroup$
– Yves Daoust
Oct 22 '18 at 21:29
$begingroup$
No, it is much much much much much much much much much much much much much much much much much much much much much much much much much much much larger.
$endgroup$
– Yves Daoust
Oct 22 '18 at 21:29
$begingroup$
No, it is much much much much much much much much much much much much much much much much much much much much much much much much much much much larger.
$endgroup$
– Yves Daoust
Oct 22 '18 at 21:29
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
By definition, $3uparrowuparrowuparrowuparrow 3 = 3uparrowuparrowuparrow(3uparrowuparrowuparrow(3uparrowuparrowuparrow 3))$.
begin{matrix}
3uparrowuparrowuparrow 3= & underbrace{3^{3^{3^{3^{cdot^{cdot^{cdot^{cdot^{3}}}}}}}}} \
& mbox{7,625,597,484,987 copies of 3}
end{matrix}
which should make it a little clearer how much bigger $3uparrowuparrowuparrowuparrow 3$ is than $3uparrowuparrowuparrow 3$. It's hard to even give a more concrete answer because the numbers of powers of $3$ get unreasonably large to compute or type.
$endgroup$
$begingroup$
What's the name of the operator?
$endgroup$
– 0x90
Oct 22 '18 at 22:16
$begingroup$
I believe it's hyperexponentiation. The symbol is from Knuth's up-arrow notation.
$endgroup$
– kcborys
Oct 23 '18 at 16:25
1
$begingroup$
Correction: $3uparrow^k3=3uparrow^{k-1}3uparrow^{k-1}3$, and generally $auparrow^kb=auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a$ with $b$ many $a$'s.
$endgroup$
– Simply Beautiful Art
Jan 9 at 21:25
add a comment |
$begingroup$
Knuth's up-arrow notation is defined so that we have
$$auparrow^kb=underbrace{auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a}_{btext{ many }atext{'s}}$$
where $uparrow^k=underbrace{uparrowuparrowdotsuparrowuparrow}_k~$ and we evaluate from right to left (e.g. $auparrow buparrow c=auparrow(buparrow c)ne(auparrow b)uparrow c$ in general).
This means we have:
$$3uparrowuparrowuparrowuparrow3=3uparrowuparrowuparrow3uparrowuparrowuparrow3$$
and we also have
$$3uparrowuparrowuparrow3=3uparrowuparrow3uparrowuparrow3uparrowuparrow3$$
and
$$3uparrowuparrow3=3uparrow3uparrow3=3^{3^3}=7,625,597,484,987$$
Using this result, we may further calculate $3uparrowuparrowuparrow3$ as
$$underbrace{3^{3^{3^{.^{.^.}}}}}_{7,625,597,484,987}$$
which is pretty large. Keep in mind that this is $3uparrowuparrowuparrow3$, or $3uparrowuparrow3uparrowuparrow3$. If we took this result and did $uparrowuparrow$ that many times, we'd basically arrive at $3uparrowuparrowuparrowuparrow3$, which is equivalent to:
$$3uparrowuparrowuparrow3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrowdotsuparrowuparrow3}_{3uparrowuparrowuparrow3}$$
which is much much much larger compared to the relatively puny
$$3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrow3}_3$$
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2966679%2fhow-much-bigger-is-3%25e2%2586%2591%25e2%2586%2591%25e2%2586%2591%25e2%2586%25913-compared-to-3%25e2%2586%2591%25e2%2586%2591%25e2%2586%25913%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
By definition, $3uparrowuparrowuparrowuparrow 3 = 3uparrowuparrowuparrow(3uparrowuparrowuparrow(3uparrowuparrowuparrow 3))$.
begin{matrix}
3uparrowuparrowuparrow 3= & underbrace{3^{3^{3^{3^{cdot^{cdot^{cdot^{cdot^{3}}}}}}}}} \
& mbox{7,625,597,484,987 copies of 3}
end{matrix}
which should make it a little clearer how much bigger $3uparrowuparrowuparrowuparrow 3$ is than $3uparrowuparrowuparrow 3$. It's hard to even give a more concrete answer because the numbers of powers of $3$ get unreasonably large to compute or type.
$endgroup$
$begingroup$
What's the name of the operator?
$endgroup$
– 0x90
Oct 22 '18 at 22:16
$begingroup$
I believe it's hyperexponentiation. The symbol is from Knuth's up-arrow notation.
$endgroup$
– kcborys
Oct 23 '18 at 16:25
1
$begingroup$
Correction: $3uparrow^k3=3uparrow^{k-1}3uparrow^{k-1}3$, and generally $auparrow^kb=auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a$ with $b$ many $a$'s.
$endgroup$
– Simply Beautiful Art
Jan 9 at 21:25
add a comment |
$begingroup$
By definition, $3uparrowuparrowuparrowuparrow 3 = 3uparrowuparrowuparrow(3uparrowuparrowuparrow(3uparrowuparrowuparrow 3))$.
begin{matrix}
3uparrowuparrowuparrow 3= & underbrace{3^{3^{3^{3^{cdot^{cdot^{cdot^{cdot^{3}}}}}}}}} \
& mbox{7,625,597,484,987 copies of 3}
end{matrix}
which should make it a little clearer how much bigger $3uparrowuparrowuparrowuparrow 3$ is than $3uparrowuparrowuparrow 3$. It's hard to even give a more concrete answer because the numbers of powers of $3$ get unreasonably large to compute or type.
$endgroup$
$begingroup$
What's the name of the operator?
$endgroup$
– 0x90
Oct 22 '18 at 22:16
$begingroup$
I believe it's hyperexponentiation. The symbol is from Knuth's up-arrow notation.
$endgroup$
– kcborys
Oct 23 '18 at 16:25
1
$begingroup$
Correction: $3uparrow^k3=3uparrow^{k-1}3uparrow^{k-1}3$, and generally $auparrow^kb=auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a$ with $b$ many $a$'s.
$endgroup$
– Simply Beautiful Art
Jan 9 at 21:25
add a comment |
$begingroup$
By definition, $3uparrowuparrowuparrowuparrow 3 = 3uparrowuparrowuparrow(3uparrowuparrowuparrow(3uparrowuparrowuparrow 3))$.
begin{matrix}
3uparrowuparrowuparrow 3= & underbrace{3^{3^{3^{3^{cdot^{cdot^{cdot^{cdot^{3}}}}}}}}} \
& mbox{7,625,597,484,987 copies of 3}
end{matrix}
which should make it a little clearer how much bigger $3uparrowuparrowuparrowuparrow 3$ is than $3uparrowuparrowuparrow 3$. It's hard to even give a more concrete answer because the numbers of powers of $3$ get unreasonably large to compute or type.
$endgroup$
By definition, $3uparrowuparrowuparrowuparrow 3 = 3uparrowuparrowuparrow(3uparrowuparrowuparrow(3uparrowuparrowuparrow 3))$.
begin{matrix}
3uparrowuparrowuparrow 3= & underbrace{3^{3^{3^{3^{cdot^{cdot^{cdot^{cdot^{3}}}}}}}}} \
& mbox{7,625,597,484,987 copies of 3}
end{matrix}
which should make it a little clearer how much bigger $3uparrowuparrowuparrowuparrow 3$ is than $3uparrowuparrowuparrow 3$. It's hard to even give a more concrete answer because the numbers of powers of $3$ get unreasonably large to compute or type.
answered Oct 22 '18 at 22:14
kcboryskcborys
45728
45728
$begingroup$
What's the name of the operator?
$endgroup$
– 0x90
Oct 22 '18 at 22:16
$begingroup$
I believe it's hyperexponentiation. The symbol is from Knuth's up-arrow notation.
$endgroup$
– kcborys
Oct 23 '18 at 16:25
1
$begingroup$
Correction: $3uparrow^k3=3uparrow^{k-1}3uparrow^{k-1}3$, and generally $auparrow^kb=auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a$ with $b$ many $a$'s.
$endgroup$
– Simply Beautiful Art
Jan 9 at 21:25
add a comment |
$begingroup$
What's the name of the operator?
$endgroup$
– 0x90
Oct 22 '18 at 22:16
$begingroup$
I believe it's hyperexponentiation. The symbol is from Knuth's up-arrow notation.
$endgroup$
– kcborys
Oct 23 '18 at 16:25
1
$begingroup$
Correction: $3uparrow^k3=3uparrow^{k-1}3uparrow^{k-1}3$, and generally $auparrow^kb=auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a$ with $b$ many $a$'s.
$endgroup$
– Simply Beautiful Art
Jan 9 at 21:25
$begingroup$
What's the name of the operator?
$endgroup$
– 0x90
Oct 22 '18 at 22:16
$begingroup$
What's the name of the operator?
$endgroup$
– 0x90
Oct 22 '18 at 22:16
$begingroup$
I believe it's hyperexponentiation. The symbol is from Knuth's up-arrow notation.
$endgroup$
– kcborys
Oct 23 '18 at 16:25
$begingroup$
I believe it's hyperexponentiation. The symbol is from Knuth's up-arrow notation.
$endgroup$
– kcborys
Oct 23 '18 at 16:25
1
1
$begingroup$
Correction: $3uparrow^k3=3uparrow^{k-1}3uparrow^{k-1}3$, and generally $auparrow^kb=auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a$ with $b$ many $a$'s.
$endgroup$
– Simply Beautiful Art
Jan 9 at 21:25
$begingroup$
Correction: $3uparrow^k3=3uparrow^{k-1}3uparrow^{k-1}3$, and generally $auparrow^kb=auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a$ with $b$ many $a$'s.
$endgroup$
– Simply Beautiful Art
Jan 9 at 21:25
add a comment |
$begingroup$
Knuth's up-arrow notation is defined so that we have
$$auparrow^kb=underbrace{auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a}_{btext{ many }atext{'s}}$$
where $uparrow^k=underbrace{uparrowuparrowdotsuparrowuparrow}_k~$ and we evaluate from right to left (e.g. $auparrow buparrow c=auparrow(buparrow c)ne(auparrow b)uparrow c$ in general).
This means we have:
$$3uparrowuparrowuparrowuparrow3=3uparrowuparrowuparrow3uparrowuparrowuparrow3$$
and we also have
$$3uparrowuparrowuparrow3=3uparrowuparrow3uparrowuparrow3uparrowuparrow3$$
and
$$3uparrowuparrow3=3uparrow3uparrow3=3^{3^3}=7,625,597,484,987$$
Using this result, we may further calculate $3uparrowuparrowuparrow3$ as
$$underbrace{3^{3^{3^{.^{.^.}}}}}_{7,625,597,484,987}$$
which is pretty large. Keep in mind that this is $3uparrowuparrowuparrow3$, or $3uparrowuparrow3uparrowuparrow3$. If we took this result and did $uparrowuparrow$ that many times, we'd basically arrive at $3uparrowuparrowuparrowuparrow3$, which is equivalent to:
$$3uparrowuparrowuparrow3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrowdotsuparrowuparrow3}_{3uparrowuparrowuparrow3}$$
which is much much much larger compared to the relatively puny
$$3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrow3}_3$$
$endgroup$
add a comment |
$begingroup$
Knuth's up-arrow notation is defined so that we have
$$auparrow^kb=underbrace{auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a}_{btext{ many }atext{'s}}$$
where $uparrow^k=underbrace{uparrowuparrowdotsuparrowuparrow}_k~$ and we evaluate from right to left (e.g. $auparrow buparrow c=auparrow(buparrow c)ne(auparrow b)uparrow c$ in general).
This means we have:
$$3uparrowuparrowuparrowuparrow3=3uparrowuparrowuparrow3uparrowuparrowuparrow3$$
and we also have
$$3uparrowuparrowuparrow3=3uparrowuparrow3uparrowuparrow3uparrowuparrow3$$
and
$$3uparrowuparrow3=3uparrow3uparrow3=3^{3^3}=7,625,597,484,987$$
Using this result, we may further calculate $3uparrowuparrowuparrow3$ as
$$underbrace{3^{3^{3^{.^{.^.}}}}}_{7,625,597,484,987}$$
which is pretty large. Keep in mind that this is $3uparrowuparrowuparrow3$, or $3uparrowuparrow3uparrowuparrow3$. If we took this result and did $uparrowuparrow$ that many times, we'd basically arrive at $3uparrowuparrowuparrowuparrow3$, which is equivalent to:
$$3uparrowuparrowuparrow3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrowdotsuparrowuparrow3}_{3uparrowuparrowuparrow3}$$
which is much much much larger compared to the relatively puny
$$3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrow3}_3$$
$endgroup$
add a comment |
$begingroup$
Knuth's up-arrow notation is defined so that we have
$$auparrow^kb=underbrace{auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a}_{btext{ many }atext{'s}}$$
where $uparrow^k=underbrace{uparrowuparrowdotsuparrowuparrow}_k~$ and we evaluate from right to left (e.g. $auparrow buparrow c=auparrow(buparrow c)ne(auparrow b)uparrow c$ in general).
This means we have:
$$3uparrowuparrowuparrowuparrow3=3uparrowuparrowuparrow3uparrowuparrowuparrow3$$
and we also have
$$3uparrowuparrowuparrow3=3uparrowuparrow3uparrowuparrow3uparrowuparrow3$$
and
$$3uparrowuparrow3=3uparrow3uparrow3=3^{3^3}=7,625,597,484,987$$
Using this result, we may further calculate $3uparrowuparrowuparrow3$ as
$$underbrace{3^{3^{3^{.^{.^.}}}}}_{7,625,597,484,987}$$
which is pretty large. Keep in mind that this is $3uparrowuparrowuparrow3$, or $3uparrowuparrow3uparrowuparrow3$. If we took this result and did $uparrowuparrow$ that many times, we'd basically arrive at $3uparrowuparrowuparrowuparrow3$, which is equivalent to:
$$3uparrowuparrowuparrow3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrowdotsuparrowuparrow3}_{3uparrowuparrowuparrow3}$$
which is much much much larger compared to the relatively puny
$$3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrow3}_3$$
$endgroup$
Knuth's up-arrow notation is defined so that we have
$$auparrow^kb=underbrace{auparrow^{k-1}auparrow^{k-1}dotsuparrow^{k-1}a}_{btext{ many }atext{'s}}$$
where $uparrow^k=underbrace{uparrowuparrowdotsuparrowuparrow}_k~$ and we evaluate from right to left (e.g. $auparrow buparrow c=auparrow(buparrow c)ne(auparrow b)uparrow c$ in general).
This means we have:
$$3uparrowuparrowuparrowuparrow3=3uparrowuparrowuparrow3uparrowuparrowuparrow3$$
and we also have
$$3uparrowuparrowuparrow3=3uparrowuparrow3uparrowuparrow3uparrowuparrow3$$
and
$$3uparrowuparrow3=3uparrow3uparrow3=3^{3^3}=7,625,597,484,987$$
Using this result, we may further calculate $3uparrowuparrowuparrow3$ as
$$underbrace{3^{3^{3^{.^{.^.}}}}}_{7,625,597,484,987}$$
which is pretty large. Keep in mind that this is $3uparrowuparrowuparrow3$, or $3uparrowuparrow3uparrowuparrow3$. If we took this result and did $uparrowuparrow$ that many times, we'd basically arrive at $3uparrowuparrowuparrowuparrow3$, which is equivalent to:
$$3uparrowuparrowuparrow3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrowdotsuparrowuparrow3}_{3uparrowuparrowuparrow3}$$
which is much much much larger compared to the relatively puny
$$3uparrowuparrowuparrow3=underbrace{3uparrowuparrow3uparrowuparrow3}_3$$
answered Jan 9 at 21:35
Simply Beautiful ArtSimply Beautiful Art
50.5k578181
50.5k578181
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2966679%2fhow-much-bigger-is-3%25e2%2586%2591%25e2%2586%2591%25e2%2586%2591%25e2%2586%25913-compared-to-3%25e2%2586%2591%25e2%2586%2591%25e2%2586%25913%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
No, it is much much much much much much much much much much much much much much much much much much much much much much much much much much much larger.
$endgroup$
– Yves Daoust
Oct 22 '18 at 21:29