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How To Draw Mushroom Paint Mushroom Coloring Pages for Kids Funny Coloring Book
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HAPPY LIFE 💖 BEST OF VOCAL DEEP HOUSE, TROPICAL & CHILL OUT MIX | PERFECT SUMMER MIX BY XDEEP HOUSE - Duration: 1:03:19.
HAPPY LIFE 💖 BEST OF VOCAL DEEP HOUSE, TROPICAL & CHILL OUT MIX | PERFECT SUMMER MIX BY XDEEP HOUSE
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Storm ! Working special FORCES operational shooting - Duration: 2:48.
prepared in the assault of a brothel in an apartment
ALL LYING .... on the floor ...
LYING
Muzzle THE FLOOR
mom mom
Hands behind your back!
Get up!
You get up!
You get up get up!
FAST FAST HERE GO HERE ...
Hands behind his back holding ...
Hands behind your back!
You get up!
Get down!
You get up!
GO-GO
Accept it!
SEREGA GIVE WOMEN HERE ...
A man whose ...
Get up!
ONE stood up!
RURAL VILLAGE!
SO THAT IN RED ..
We could not be here to go ...
Handsome man takes off his shirt ..
Arise HERE ..
Arise HERE ..
HERE HERE Stand up ....
Turn HERE ..
Turn from the other side ..
THE OTHER SIDE
Stop! BACK ..
LET turns
ALL lie ...
All dresses ..
ARM Whew ... Wait VYSUNI
ARMS..
All dresses!
Well, you surprise that the police have gone?
A ?
SURPRISED that the police have gone?
- I'm not guilty ...
THANKS FOR WATCHING .PODPISHISYA
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Fiat Doblò Cargo 1.3 MULTIJET ACTUAL,AIRCO,ELEK RAMEN, CENTR VERGR. MEERDERE FOTO"S VOLGEN - Duration: 0:54.
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J / I know - Duration: 4:19.
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Liouville's number, the easiest transcendental and its clones - Duration: 18:24.
Welcome to another Mathologer video.
Liouville's number the monster up there
consists of infinitely many isolated
islands of 1s at the 1! th, 2! th,
3! th, etc. digits with
exploding gaps of zeros between them. As
I promised you at the end of the last
video, today's mission is to show you a
nice visual way of seeing that this
number is transcendental. I'm pretty
convinced that as far as transcendence proofs
for specific numbers goes, what I've put
together here is as simple and as
accessible as it will ever get. Still, if
you make it to the end and understand
all my arguments I think you can be
pretty proud of yourself since not even
that many professional mathematicians
know any transcendence proofs. And, if you
make it to the end, I've got a special
treat for you. I'll show you how you can
use Liouville's number as a template to make
a clone of the real numbers within the
real numbers. This clone is made up of
transcendental numbers like this strange
cousin of Pi, but perhaps the craziest
thing about this clone is that although
it is as large at the set of real
numbers itself, it is of measure 0. This
means that in a sense it takes up no
space within the real numbers. For those
of you in the know this set is closely
related to Cantor sets.
What's a transcendental number again?
Well for a detailed introduction just
watch the last video. So here just real
quick: the transcendental numbers are
defined in terms of what they are NOT.
They are NOT among the real numbers that
you encounter when you're doing
classical algebra, like the integers, the
rational numbers, any of the real numbers
that can be written as rooty expressions
like these, or any of the real numbers
that pop up as solutions to polynomial
equations with integer coefficients like
these. This last set of numbers is called the
algebraic numbers and includes all the
other types of numbers that I mentioned
before: integers, rationals and rooty
numbers. Anyway, we call a real number
transcendental if it is not algebraic,
if it is not a solution of any
polynomial equation with integer
coefficients. Liouville's number or Liouville's
constant was shown to be transcendental
in 1851 by the great French
mathematician Joseph Liouville. It was one of
the first numbers shown to be
transcendental and Liouville's proof of this fact
is probably the easiest transcendence
proof for any specific number. Liouville's
proof is accessible to anybody who has
had some exposure to proofs at
university level real analysis and I have
concluded links to the original French
paper and a modern version of the proof
in English in the description. But since
many of you would struggle with any of
the proofs in these papers or in
textbooks, I've tried to come up with an
alternative way of seeing the
transcendence of Liouville's number that can be
understood and hopefully also enjoyed by
anybody who watches these sorts of maths
videos, including all Mathologer fans who
are still in high school. I'm still
working on the primary school version
but and I am probably never going to get there.
Okay, Liouville's number, lots and lots of
0s and 1s and the 1s are the 1! th,
2! th, 3! th, etc.
digits. Just a quick reminder, 1!
is equal to 1, 2! is 1 times 2 is
equal to 2, 3! is 1 times 2
times 3 is equal to 6, 4! is 24.
Then 120, 720, 5040, etc. a sequence of
numbers that grows rapidly which means
that the stretches of zeros between
consecutive 1s get longer and longer
extremely quickly. Okay, here then is my
proof that Liouville's number is
transcendental. Let me know in the
comments whether this proof works for
you. When dealing with a complicated
number like pi
approximate values often suffice or are at
least a good place to start. To get such
an approximation we often chop off the
decimal expansion at some point.
Now let's approximate the number pi
squared with the squares of these
truncations.
Then these approximations for pi squared
will be spot-on
to a certain digit and maybe good enough
for some application we have in mind.
However from some point on these
approximations or go off target. For
example, the last approximation down
there coincides with pi squared in only
the first four digits and the remaining
digits that you can see here are wrong.
Liouville's number L behaves quite
different in this respect but to start
with the only distinguishable
truncations are those that cut off
after the 1s and these are the first
four of these truncations of L. Now
squaring gives these numbers here and in
contrast to what we had before all the
digits of the squared truncations seem to
be correct. And looks are not deceiving,
this is actually always true and the
reason for this is not too hard to pin
down. It's because all the nonzero digits
of Liouville's
number are spaced further and further
apart and those further down the line
simply don't play any role in producing
the earlier digits of the square of our
number. For the moment I'll gloss over the
technical details but I'll return to
them after I've finished
outlining the proof. Anyway, what happens
if you raise L to a different power, say
5? Well then, as you can see, while the
approximations are still definitely damn
good, things are no longer spot on
for all digits. Here, for example, one digit
is wrong and here three but if you look
closely you'll find that the next
approximation is spot on again, all
digits are correct. And actually things
will be spot on from this truncation on.
So, all digits of truncation L_5^5, L_6^5,
and so on will be correct. In fact,
something similar will be true for all
powers. So as we've seen, all squared
truncations are spot on. The same turns
out to be the case for the cubed ones,
fourth powers too, fifth powers we've
already seen that all digits are spot-on
from a certain truncation on. Same thing
for sixth powers, except this happens a
little bit later, and so on, for all
powers. Now we find that
the same is true if, instead of just
powers, we look at any polynomial with
integer coefficients. So if, instead of
just powers of the number L, we're
approximating a polynomial like this
evaluated at the number L, then we find
that all the digits of our
approximations will be correct from some
truncation onward. In particular, in the
case of this polynomial all digits would be
correct from L_6 onward. Okay, now we've
almost arrived at the punch line.
Remember, what we have to show is that L
does not solve any equation like this.
Well, let's assume L does solve this
equation. Then in decimal form the left
part would evaluate to 0.000...
all 0s, right? But
then we already know that the digits we
get when we substitute truncation L_6
instead are all correct. Well all the digits
are 0s and so all digits of our
approximation have to be 0s, too. But
that means that L_6 is also a solution
of our polynomial equation. The same is
true for L_7 and all the following
infinitely many truncations. But
infinitely many solutions is impossible
because a polynomial equation such as
the one above can only have as many
solutions as its degree. For example, our
degree six polynomial can have at most
six solutions and so the assumption that
our number solves a polynomial
equation with integer coefficients leads
the absolutely impossible conclusion
that this polynomial has infinitely many
solutions. This means that our assumption
that L solves an equation like this was
wrong in the first place and we conclude
that L is a transcendental number. Very
neat proof by contradiction, right? Of
course, what I did not show you is that
all the approximations will eventually be
spot-on no matter what polynomial
equation we're dealing with. Okay
so let's have a closer look. The
different 1s in our number L just stand
for different powers of 10. For example,
the first one stands for 10 to the minus
1, the second for 10 to the minus 2,
the third for ten to the minus six, and
so on. The number L is just a sum of
these powers of 10. Now we started by
pondering L squared. So let's have a look
at how we would calculate this number
here. Actually before we do that, let's
calculate the square of the second
truncation.
So here we have to multiply every term
at the top with every one at the bottom
and then add up. So 10 to the minus 1
times 10 to the minus 1 is 10 to the
minus 2. 10 to the minus 1 times 10 to
the minus 2 is 10 to the minus 3. This
times that 10 to the minus 3 again. This
times that 10 to the minus 4. Add up
everything to get the square we're after.
For L squared we have to multiply again
every term at the top with every one at
the bottom and then add up (might
take a while, right?) So 10 to the minus 1
times 10 to the minus 1 is 10 to the
minus 2, minus 1 minus 2 minus 3, minus 1
minus 6 minus 7, and so on. Minus 2 minus
1 minus 3, minus 2 minus 2 minus 4, and so
on, and so on. Now why are the digits of
the squared truncation correct? Well, its
rightmost digit 1 comes from multiplying
the rightmost term of the truncation
with itself. For the other terms to
interfere with the digit of the squared
truncation one of the terms outside the
box at the top multiplied by one of the
terms at the bottom should be at least
as big as 10 to the minus 4. But the
largest number we can make this way is
10 to the minus 6 times 10 to the minus
1 which is 10 to the minus 7, a lot
smaller than 10 to minus 4. Okay, so there's
no danger here. What about these critical
numbers for the next truncation? Also no
danger and the gap between the exponents
minus 12 and minus 25 is larger than
before, so even less danger. One more. The
gap has increased even further. Now it's
a one-liner to show
that this trend continues and I actually
leave it to you to sort out the details.
And, well, actually you only get partial
credit if it takes you more than one line.
Let's have a look at all this for
the 4th truncation of the 5th power
of L, that's the next thing we looked at,
right? As I said before all digits are
correct for this 4th truncation. Let's
double check. Yes, the green 124 is larger
than the yellow 116, tick. Again, I'll
leave it to you to fill in the details
to show that this will also be the case
for the higher truncations. Remember
there were problems with the lower
truncations. Let's see where these show up
here. Okay the order has flipped, the 30
is now larger than the 28, and so there
will be wrong digits in the approximation.
One step down there's also trouble, with
the green terms interfering with the
digits of the power of the second
truncation. Okay so here are the first
digits of L squared.
I just like to highlight again the gap
between the yellow lowest terms of the
squared truncations
together with the corresponding
worst-case-scenario green terms. As we've
already observed this gap is increasing,
right? In fact, they'll get arbitrarily
large and the same is true for all the
other powers want to get starts
appearing from some truncation onward.
Now those of you who did those one-line
calculations earlier on will already
have convinced yourself of this fact. Okay
now what about polynomials with integer
coefficients? Well they are built from
powers. Take, for example, L squared plus L
highlight the second truncation. First
note that the corresponding green digits
are basically aligned. That's because the
top one essentially comes about from
multiplying the bottom one by ten to the
minus one. That's a shift of one position
anyway this relative alignment will
always be the same as we move to the
right. Also as we move towards the right
the gap of zeros in front of the green
digits gets as large as you wish. Given
this gap of zeros, it is clear that when we
add up L_3 and L_3 squared
none of the green digits or beyond will
add to the sum within L squared and so
the digits of the approximation would
all be correct okay but what happens is
the individual powers in the polynomial
are multiplied by non-trivial constants.
For example, let's multiply the linear
term by something huge say 10. Okay so
the overall effect of this changed term
are its nonzero islands possibly
growing and the greens shifting to the
left. This may lead to the overall gaps
in front of the greens shrinking or even
vanishing for really large coefficients
like googleplex. However since the gaps
grow unlimited in size they will
eventually absorb the shrinkage and gaps
will always be prevalent from a certain
point on and this means that no matter
what polynomial we're dealing with all
our approximations will be spot on from a
certain truncation onward. Anyway that's
basically it. Of course I could still
fill in all the nitty-gritty
calculations, for example, about the ever
growing gaps. But if you've made it up to
here I'm sure you'll be able to fill in
those details yourself. In any case if
there's anything you don't understand
just ask. Here are a couple of super cool
facts that can be shown in the same way.
First, even when interpreted as base 2, 3,
4 etc. numbers Liouville's monster will
always be a transcendental number. Second,
even when we replace the ones by other
digits we get a transcendental number as
long as infinitely many of these digits
are nonzero. Now, adding an integer to a
transcendental number gives a number
that is still transcendental. Maybe
someone prove this in the comments. So
adding 5 to this transcendental
number up there gives a new
transcendental number: five point one two
zero zero zero and so on. Here's a fun
idea then. Let's take a real number like
pi and use its digits to create a new
number like this. Well, it's
super-tough to prove that pi is
transcendental whereas proving the
transcendence of this weird clone is not
any harder and what I've showed you
earlier. Okay
next trick. Let's do the same for every
real number. What this does is create a
clone of the set of real numbers within
the set of real numbers that consists
entirely of transcendental numbers. Even
the clones of the algebraic numbers are
transcendental. We just have to be a
little bit careful here when we
translate a number that features a
terminating decimal expansion like for
example the number 1.23.
Using that terminating decimal expansion
would also result in a clone with a
terminating decimal expansion and any
number like this is rational and
therefore not transcendental. Of course
that's easily fixed because every number
with a terminating decimal expansion has
a second decimal expansion with an
infinite tail of nines. For example,
1.23 is equal to
1.2999... and so on.
Using this alternative decimal
expansion which features infinitely many
nonzero digits will then give a
transcendental clone. Well one mini
problem remains. The number 0 cannot
be massaged in this way to give a
transcendental number. But okay as it
happens quite often 0 is a little
exception here, big deal. Anyway we now
have got a clone of the real numbers
within the real numbers consisting of
transcendental numbers. Of course since
it is a clone there's a one-to-one
correspondence between it and the
original set. In other words, just like
the real numbers the clone is an
uncountably infinite set. But of course
in addition all its elements are easy
transcendentals, super cool, right?
However this uncountably infinite set
also has the paradoxical property of
having measure 0 so although it is as
large as the whole set of real numbers
it is so well hidden within that in a
sense it's not even there. To really be
able to appreciate this paradoxical set
you should watch the previous video
where I also give a little intro to how
you can assign a length or measure to a
subset of the real numbers.
Or you could head over to Infinite
series the Kelsey is also discussing
these sorts of sets at the moment. I've
also prepared a proof of the fact that
the clone has measure 0 but I think
this video is already getting quite long
and so I'll put this proof on Mathologer 1
sometime in the next couple of days.
Okay let's finish here. I hope you all
understood and enjoyed this video but as
usual please let me know what worked for
you and what didn't and if you are
struggling with anything just ask.
And that's it for today.
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Размножение глицинии отводками, обрезка глицинии - Duration: 2:25.
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Does Jesus Care - Duration: 4:18.
Does Jesus care when my heart is pained
Too deeply for mirth and song
As the burdens press, and the cares distress
And the way grows weary and long
Oh, yes, He cares, I know He cares
His heart is touched with my grief
When the days are weary, the long nights dreary
I know my Savior cares
Does Jesus care when my way is dark
With a nameless dread and fear
As the daylight fades into deep night shades
Does He care enough to be near
Oh, yes, He cares, I know He cares
His heart is touched with my grief
When the days are weary, the long nights dreary
I know my Savior cares
Does Jesus care when I've tried and failed
To resist some temptation strong
When for my deep grief I find no relief
Though my tears flow all the night long
Oh, yes, He cares, I know He cares
His heart is touched with my grief
When the days are weary, the long nights dreary
I know my Savior cares
Does Jesus care when I've said "goodbye"
To the dearest on earth to me
And my sad heart aches till it nearly breaks
Is it aught to Him - does He see
Oh, yes, He cares, I know He cares
His heart is touched with my grief
When the days are weary, the long nights dreary
I know my Savior cares
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Fiat Panda 1.2 EDIZIONE COOL - Duration: 0:59.
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Jamais pour le mal ... (2/3) - Duration: 3:33.
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Storm ! Working special FORCES operational shooting - Duration: 2:48.prepared in the assault of a brothel in an apartment
ALL LYING .... on the floor ...
LYING
Muzzle THE FLOOR
mom mom
Hands behind your back!
Get up!
You get up!
You get up get up!
FAST FAST HERE GO HERE ...
Hands behind his back holding ...
Hands behind your back!
You get up!
Get down!
You get up!
GO-GO
Accept it!
SEREGA GIVE WOMEN HERE ...
A man whose ...
Get up!
ONE stood up!
RURAL VILLAGE!
SO THAT IN RED ..
We could not be here to go ...
Handsome man takes off his shirt ..
Arise HERE ..
Arise HERE ..
HERE HERE Stand up ....
Turn HERE ..
Turn from the other side ..
THE OTHER SIDE
Stop! BACK ..
LET turns
ALL lie ...
All dresses ..
ARM Whew ... Wait VYSUNI
ARMS..
All dresses!
Well, you surprise that the police have gone?
A ?
SURPRISED that the police have gone?
- I'm not guilty ...
THANKS FOR WATCHING .PODPISHISYA
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sirf 21 baar ya muqtadir o parhian | ya muqtadiro ka wazifa in urdu |y a muqtadiro ki faziflat - Duration: 3:05.
sirf 21 baar ya muqtadir o parhian | ya muqtadiro ka wazifa in urdu |y a muqtadiro ki faziflat
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Facebook Beğeni Hilesi %100 (KANITLI) - Duration: 2:20.
Hepinize Merhaba
Bugün Sizlerle Facebook Beğeni Hilesi Yapçaz
:D
Şimdi Nasıl Yapılır İzle Ve Gör :D
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How To Draw Mushroom Paint Mushroom Coloring Pages for Kids Funny Coloring Book - Duration: 11:24.
How To Draw Mushroom Paint Mushroom Coloring Pages for Kids Funny Coloring Book
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