On Tue, 20 Jul 2021 12:03:19 +0100, Roderick Stewart
<rjfs@escapetime.myzen.co.uk> wrote:

>On Tue, 20 Jul 2021 11:22:19 +0100, Max Demian
><max_demian@bigfoot.com> wrote:
>
>>On 20/07/2021 08:43, MrSpud_z8iO1@3v77bw.tv wrote:
>>> On Mon, 19 Jul 2021 17:36:50 +0100
>>> gareth evans <headstone255@yahoo.com> wrote:
>>>> On 19/07/2021 16:17, MrSpud_k9@j5ei2on7o.org wrote:
>>>>>
>>>>> No they're not. Either the square root of -1 doesn't exist in which case
>>>>> complex numbers are simply a hack to get over the mathematical equivalent of
>>>>> an energy barrier, or sqr(-1) does exist and the whole basis of our
>>>>> mathematics is wrong. You can't have it both ways.
>>>>
>>>> The difficulty is in trying to understand something as an integer
>>>> when it is not an integer, it is an operator that when applied
>>>> twice changes +1 into -1
>>>>
>>>> YMMV
>>>>
>>>> It again is part of our limited sensory perception, for the
>>>> real number system that we experience is a subset of
>>>> the complete number system of complex numbers, a
>>>> 2-dimensional thing, that we cannot experience.
>>>
>>> Because they don't exist, they're an invention, a hack to get other bits of
>>> maths to work.
>>0! = 1 is another "hack".
>
>The most annoying one for me is 1+2+3+4+5...(to infinity) = -1/12.

Better than I think you are wrong.
See
http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html

and
http://physicsbuzz.physicscentral.com/2014/02/so-what-does-1234-equal-we-give-you_11.html

{snip}

--
brightside s9

On 20/07/2021 11:18, Roderick Stewart wrote:
> If a Roman had II sheep and somebody gave him another III, then
> he'd have V

If he then gave all V to someone else, the Roman would have no way of
describing the outcome, because there was no concept of zero in Roman
numerals.

Jim

On 20/07/2021 13:30, Indy Jess John wrote:
> On 20/07/2021 11:18, Roderick Stewart wrote:

>> If a Roman had II sheep and somebody gave him another III, then
>> he'd have V
>
> If he then gave all V to someone else, the Roman would have no way of
> describing the outcome, because there was no concept of zero in Roman
> numerals.

He would say he had none; no sheep (or whatever the Latin is). It's only
zero as a *number* that could be manipulated as any other number that
wasn't realised. They knew what nothing was.

--
Max Demian

On 20/07/2021 13:02, BrightsideS9 wrote:
> On Tue, 20 Jul 2021 12:03:19 +0100, Roderick Stewart
> <rjfs@escapetime.myzen.co.uk> wrote:
>> On Tue, 20 Jul 2021 11:22:19 +0100, Max Demian
>> <max_demian@bigfoot.com> wrote:
>>
>>> On 20/07/2021 08:43, MrSpud_z8iO1@3v77bw.tv wrote:

>>>> Because they don't exist, they're an invention, a hack to get other bits of
>>>> maths to work.

>>> 0! = 1 is another "hack".
>>
>> The most annoying one for me is 1+2+3+4+5...(to infinity) = -1/12.
>
> Better than I think you are wrong.
> See
> http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html
>
> and
> http://physicsbuzz.physicscentral.com/2014/02/so-what-does-1234-equal-we-give-you_11.html
>
> {snip}
>

Well that's a relief!

Of course, on some computers 32767 + 1= *minus* 32768.

There is a silly story about an airliner travelling at 32767 feet that
flips over if it tries to go any higher.

--
Max Demian

On Tue, 20 Jul 2021 10:43:29 +0100
Roderick Stewart <rjfs@escapetime.myzen.co.uk> wrote:
>would have to use. Even if we regard the square root of minus one as a
>notational bodge that we use to perform certain tasks, if those aliens
>want to perform the same tasks they'll have to invent the same bodge,

Not necessarily. Has anyone proven that there is no other method to solve
certain problems - eg FFTs - without complex numbers? There may well be an
entire class of mathematical bodges and hacks that we're currently completely
ignorant of.

On Tue, 20 Jul 2021 10:45:10 +0100
gareth evans <headstone255@yahoo.com> wrote:
>On 20/07/2021 08:43, MrSpud_z8iO1@3v77bw.tv wrote:
>>
>> Because they don't exist, they're an invention, a hack to get other bits of
>> maths to work.
>
>But all of mathematics is an abstract invention.
>
>Let's take something that you seem to be happy about, the
>natural numbers used for counting.
>
>You can have two fingers, two apples, two of anything
>but two, by itself, on its own, does not exist!

Obviously not, but the quantity it represents does.

>OK, you might say, what about "two", "2" or "II" (as in Roman)?
>.... and I will say to you that they are just different ways of
>describing something, but are not the thing itself.

The notation and the concept are not the same thing.

On Tue, 20 Jul 2021 11:22:19 +0100
Max Demian <max_demian@bigfoot.com> wrote:
>On 20/07/2021 08:43, MrSpud_z8iO1@3v77bw.tv wrote:
>>> It again is part of our limited sensory perception, for the
>>> real number system that we experience is a subset of
>>> the complete number system of complex numbers, a
>>> 2-dimensional thing, that we cannot experience.
>>
>> Because they don't exist, they're an invention, a hack to get other bits of
>> maths to work.
>0! = 1 is another "hack".

Yup, and clearly nonsense given anything multiplied by zero is zero but
presumably it works better in some equations.

On Tue, 20 Jul 2021 12:03:19 +0100
Roderick Stewart <rjfs@escapetime.myzen.co.uk> wrote:
>On Tue, 20 Jul 2021 11:22:19 +0100, Max Demian
><max_demian@bigfoot.com> wrote:
>>> Because they don't exist, they're an invention, a hack to get other bits of
>>> maths to work.
>>0! = 1 is another "hack".
>
>The most annoying one for me is 1+2+3+4+5...(to infinity) = -1/12.

Bollocks, it'll be infinity.

<MrSpud_yml6Fmogu@pt_rbnaifsjd7.net> wrote in message
news:sd6o78$1fee$1@gioia.aioe.org...
> On Tue, 20 Jul 2021 12:03:19 +0100
> Roderick Stewart <rjfs@escapetime.myzen.co.uk> wrote:
>>On Tue, 20 Jul 2021 11:22:19 +0100, Max Demian
>><max_demian@bigfoot.com> wrote:
>>>> Because they don't exist, they're an invention, a hack to get other
>>>> bits of
>>>> maths to work.
>>>0! = 1 is another "hack".
>>
>>The most annoying one for me is 1+2+3+4+5...(to infinity) = -1/12.
>
> Bollocks, it'll be infinity.

Exactly. It turns out that it is a half-remembered half-truth.

http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html

the -1/12 is the area "under" (actually above) the curve between the two
zero-crossings of the curve which generates factorials.

"The area of the little region below the horizontal axis equals -1/12, and
the infinite area under the curve on the right gives you 1+2+3+4+. . . ,
which goes to infinity as you add terms, not to -1/12."

On 20/07/2021 15:50, MrSpud_8hz3fxx2_m@7f30vb0p.com wrote:
> Not necessarily. Has anyone proven that there is no other method to solve
> certain problems - eg FFTs - without complex numbers? There may well be an
> entire class of mathematical bodges and hacks that we're currently completely
> ignorant of.
>
>

The Fourier Transform, whether Fast, Slow, or otherwise dilatory, is
derived using complex numbers, so attemptng a treatment without the
benefit of complex numbers puts you onto a hiding to nothing.

However, you might be amused by something I produced 24 years ago ...

Alimentary Fourier Anal-ysis
----------------------------
(C) Copyright 1997 Gareth Alun Evans G4SDW

The Fourier Transform is rather like the Human Alimentary canal....

The Alimentary Canal takes food, and then transforms it into
something that is a function of the rear end, and hence may
be referred to as, "The Food-Rear Transform".

The Fourier Transform causes integration over a one second
period, and results in an infinite series of sines.
The Food-Rear Transform causes disintegration over a 12-hour
period, and results in an infinite series of signs...."Gents",
"Ladies", "Rest Room", etc. etc.

The Fourier Transform (Tukey's reasoning?) takes Time and
transforms it to a frequency spectrum.
The Food-Rear Transform takes Parsley, Sage and Thyme (Turkey's
seasoning?) and transforms it to a frequently active rectum.

"Laplace", literally, "The Place", is how you deal with
the transform.
"Walsh" is something you do to your hands afterwards.
"JPeg" is what you hang your jacket on to. It is a form of
compression such that you take up less space and spend less time
there (although the output may be more granular).

The Fast Fourier Transform must be done through an appropriate
window, otherwise an effect known as the "Picket Fence" problem,
also known as "Spectral Leakage", can occur.
Diarrhoea is known as the FAST Food-Rear Transform, and if done
through the wrong window can result in a problem (also known
as Rectal Leakage) all over the Picket Fence.

The Picket Fence problem is also known as, "Scalloping",
something which is normally done to pie crusts. In the old
days, pies used to be made with "Four And Twenty Blackbirds";
but nowadays, members of the Jay family are used. When you go to
someone's house for tea, you politely eat everything that is
put in front of you, but your reaction might be,
"Eat the Jay (Oh my God!) Tea???".
Similarly, the Fourier Transform is a great consumer
of e^(jwT)... "E to the j, omega, t".

The Fast Fourier Transform is famous for its Butterflies.
The Fast Food-Rear Transform is famous for its
Butterflies-in-the-stomach.

Dirichlet's are the Fourier Transform's conditions, without which
the result is divergent.
Diarrhoea Shits are the Food-Rear's conditions, with which the
result is VERY divergent.

Having experienced the Fast Fourier Transform, you are then faced
with the Inverse, which can be very daunting. It is a paper
exercise but it can be quite messy.
Having experienced the Fast Food-Rear Transform, you are then
faced with the "In Verde", literally "In The Grass", which can be
very daunting indeed. (The paper exercise can be quite messy.)

On 20/07/2021 15:55, MrSpud_j4ql50B@qtlyefd8scid4za.tv wrote:
> On Tue, 20 Jul 2021 11:22:19 +0100
> Max Demian <max_demian@bigfoot.com> wrote:
>> On 20/07/2021 08:43, MrSpud_z8iO1@3v77bw.tv wrote:
>>>> It again is part of our limited sensory perception, for the
>>>> real number system that we experience is a subset of
>>>> the complete number system of complex numbers, a
>>>> 2-dimensional thing, that we cannot experience.
>>>
>>> Because they don't exist, they're an invention, a hack to get other bits of
>>> maths to work.
>> 0! = 1 is another "hack".
>
> Yup, and clearly nonsense given anything multiplied by zero is zero but
> presumably it works better in some equations.
>

Not at all nonsense because ...

(n-1)! = n! / n

So, 0! is derived from 1! by 1!/1

QED

On Tue, 20 Jul 2021 16:38:16 +0100
gareth evans <headstone255@yahoo.com> wrote:
>On 20/07/2021 15:50, MrSpud_8hz3fxx2_m@7f30vb0p.com wrote:
>> Not necessarily. Has anyone proven that there is no other method to solve
>> certain problems - eg FFTs - without complex numbers? There may well be an
>> entire class of mathematical bodges and hacks that we're currently completely
>
>> ignorant of.
>>
>>
>
>The Fourier Transform, whether Fast, Slow, or otherwise dilatory, is
>derived using complex numbers, so attemptng a treatment without the
>benefit of complex numbers puts you onto a hiding to nothing.

Journeys from town to town are made using the horse, so attempting to do so
without the benefit of one puts you on a hiding to nothing.

Said probably someone in 1890.

On Tue, 20 Jul 2021 16:42:01 +0100
gareth evans <headstone255@yahoo.com> wrote:
>On 20/07/2021 15:55, MrSpud_j4ql50B@qtlyefd8scid4za.tv wrote:
>> On Tue, 20 Jul 2021 11:22:19 +0100
>> Max Demian <max_demian@bigfoot.com> wrote:
>>> On 20/07/2021 08:43, MrSpud_z8iO1@3v77bw.tv wrote:
>>>>> It again is part of our limited sensory perception, for the
>>>>> real number system that we experience is a subset of
>>>>> the complete number system of complex numbers, a
>>>>> 2-dimensional thing, that we cannot experience.
>>>>
>>>> Because they don't exist, they're an invention, a hack to get other bits of
>
>>>> maths to work.
>>> 0! = 1 is another "hack".
>>
>> Yup, and clearly nonsense given anything multiplied by zero is zero but
>> presumably it works better in some equations.
>>
>
>Not at all nonsense because ...
>
>(n-1)! = n! / n
>
>So, 0! is derived from 1! by 1!/1

All that proves is that equation only works for N > 0.

In message <sd6rob$1dcq$1@gioia.aioe.org>, MrSpud_phi@wwen2c.org writes
>On Tue, 20 Jul 2021 16:38:16 +0100
>gareth evans <headstone255@yahoo.com> wrote:
>>On 20/07/2021 15:50, MrSpud_8hz3fxx2_m@7f30vb0p.com wrote:
>>> Not necessarily. Has anyone proven that there is no other method to solve
>>> certain problems - eg FFTs - without complex numbers? There may well be an
>>> entire class of mathematical bodges and hacks that we're currently
>>>completely
>>
>>> ignorant of.
>>>
>>>
>>
>>The Fourier Transform, whether Fast, Slow, or otherwise dilatory, is
>>derived using complex numbers, so attemptng a treatment without the
>>benefit of complex numbers puts you onto a hiding to nothing.
>
>Journeys from town to town are made using the horse, so attempting to do so
>without the benefit of one puts you on a hiding to nothing.
>
>Said probably someone in 1890.
>
>

1790, perhaps? There would have been no one in 1890's Britain who hadn't
heard of the railways. Unless your "horse" includes the iron horse. :)
--
John Hall "Hard work often pays off after time, but laziness always
pays off now." Anon

On 20/07/2021 18:15, John Hall wrote:
> 1790, perhaps? There would have been no one in 1890's Britain who hadn't
> heard of the railways. Unless your "horse" includes the iron horse.

I am sure there would be plenty of people in rural villages and islands
in 1890 where people had not heard of the railway.

On Tue, 20 Jul 2021 14:56:40 -0000 (UTC),
MrSpud_yml6Fmogu@pt_rbnaifsjd7.net wrote:

>On Tue, 20 Jul 2021 12:03:19 +0100
>Roderick Stewart <rjfs@escapetime.myzen.co.uk> wrote:
>>On Tue, 20 Jul 2021 11:22:19 +0100, Max Demian
>><max_demian@bigfoot.com> wrote:
>>>> Because they don't exist, they're an invention, a hack to get other bits of
>>>> maths to work.
>>>0! = 1 is another "hack".
>>
>>The most annoying one for me is 1+2+3+4+5...(to infinity) = -1/12.
>
>Bollocks, it'll be infinity.
>
I agree. Normal verbal reasoning based on commonsense leads me fairly
quickly to the conclusion that it's bollocks.

But then, I'm not a mathematician. Apparently there is a mathematical
argument using valid notation that leads to the conclusion that all
the natural numbers add up to minus one twelfth. It makes no sense to
me, but some mathematicians appear to accept that it must be correct
in some sense because the notation is valid. I don't understand it.

There are no limits or restrictions in the English language that make
it impossible to create statements that are either manifestly not true
or simply make no coherent sense, so it isn't particularly surprising
to me that mathematical language should have the same weakness, but to
some the notation is sacrosanct even if it leads to bollocks.

Yesterday upon the stair
I met a man who wasn't there
He wasn't there again today
Oh how I wish he'd go away.

Oddly, sometimes it's OK to reject a mathematical conclusion on this
basis. A quadratic equation, for instance, gives two answers, one of
which is usually nonsense. The mathematical notation is equally valid
for both of them, but you have to step out of the world of numbers and
use ordinary everyday commonsense to decide which one to reject.

Rod.

On Tue, 20 Jul 2021 18:15:40 +0100
John Hall <john_nospam@jhall.co.uk> wrote:
>In message <sd6rob$1dcq$1@gioia.aioe.org>, MrSpud_phi@wwen2c.org writes
>>On Tue, 20 Jul 2021 16:38:16 +0100
>>gareth evans <headstone255@yahoo.com> wrote:
>>>On 20/07/2021 15:50, MrSpud_8hz3fxx2_m@7f30vb0p.com wrote:
>>>> Not necessarily. Has anyone proven that there is no other method to solve
>>>> certain problems - eg FFTs - without complex numbers? There may well be an
>>>> entire class of mathematical bodges and hacks that we're currently
>>>>completely
>>>
>>>> ignorant of.
>>>>
>>>>
>>>
>>>The Fourier Transform, whether Fast, Slow, or otherwise dilatory, is
>>>derived using complex numbers, so attemptng a treatment without the
>>>benefit of complex numbers puts you onto a hiding to nothing.
>>
>>Journeys from town to town are made using the horse, so attempting to do so
>>without the benefit of one puts you on a hiding to nothing.
>>
>>Said probably someone in 1890.
>>
>>
>
>1790, perhaps? There would have been no one in 1890's Britain who hadn't
>heard of the railways. Unless your "horse" includes the iron horse. :)

Yeah ok, but you know what I mean :)

On Tue, 20 Jul 2021 20:44:27 +0100
Roderick Stewart <rjfs@escapetime.myzen.co.uk> wrote:
>On Tue, 20 Jul 2021 14:56:40 -0000 (UTC),
>MrSpud_yml6Fmogu@pt_rbnaifsjd7.net wrote:
>
>>On Tue, 20 Jul 2021 12:03:19 +0100
>>Roderick Stewart <rjfs@escapetime.myzen.co.uk> wrote:
>>>On Tue, 20 Jul 2021 11:22:19 +0100, Max Demian
>>><max_demian@bigfoot.com> wrote:
>>>>> Because they don't exist, they're an invention, a hack to get other bits
>of
>>>>> maths to work.
>>>>0! = 1 is another "hack".
>>>
>>>The most annoying one for me is 1+2+3+4+5...(to infinity) = -1/12.
>>
>>Bollocks, it'll be infinity.
>>
>I agree. Normal verbal reasoning based on commonsense leads me fairly
>quickly to the conclusion that it's bollocks.

And using logical reasoning:
For all natural positive integers N(n+1) > Nn therefor Nn + N(n+1) > Nn
so there is no 'n' for which adding the next integer will produce a lower value.

>There are no limits or restrictions in the English language that make
>it impossible to create statements that are either manifestly not true
>or simply make no coherent sense, so it isn't particularly surprising
>to me that mathematical language should have the same weakness, but to
>some the notation is sacrosanct even if it leads to bollocks.

English is a poor language to be precise in even in normal speech. eg:

Tom saw Harry and he waved.

Who waved, Tom or Harry?

Other languages have case endings on the nouns that would clarify, English
doesn't.

>Oddly, sometimes it's OK to reject a mathematical conclusion on this
>basis. A quadratic equation, for instance, gives two answers, one of
>which is usually nonsense. The mathematical notation is equally valid
>for both of them, but you have to step out of the world of numbers and
>use ordinary everyday commonsense to decide which one to reject.

True.

In message <sd8oec$1esi$1@gioia.aioe.org>, MrSpud_5nkd0w@62zd.gov writes

>
>English is a poor language to be precise in even in normal speech. eg:
>
>Tom saw Harry and he waved.
>
>Who waved, Tom or Harry?
>
>Other languages have case endings on the nouns that would clarify, English
>doesn't.

I'm not sure which languages would clarify the statement. Any examples?
>
>
>

--
Ian

On 21/07/2021 01:25 pm, Ian Jackson wrote:

> In message <sd8oec$1esi$1@gioia.aioe.org>, MrSpud_5nkd0w@62zd.gov writes
>
>> English is a poor language to be precise in even in normal speech. eg:
>> Tom saw Harry and he waved.
>> Who waved, Tom or Harry?
>
>> Other languages have case endings on the nouns that would clarify,
>> English doesn't.
>
> I'm not sure which languages would clarify the statement. Any examples?

Latin had that to an extent, but not (IIRC) for proper nouns.

But it was a very long time ago that I studied that language and while
verbs were declined, nouns were conjugated.

"Ian Jackson" <ianREMOVETHISjackson@g3ohx.co.uk> wrote in message
news:ztjr+dBwIB+gFwfn@brattleho.plus.com...
> In message <sd8oec$1esi$1@gioia.aioe.org>, MrSpud_5nkd0w@62zd.gov writes
>
>
>>
>>English is a poor language to be precise in even in normal speech. eg:
>>
>>Tom saw Harry and he waved.
>>
>>Who waved, Tom or Harry?
>>
>>Other languages have case endings on the nouns that would clarify, English
>>doesn't.
>
> I'm not sure which languages would clarify the statement. Any examples?

Yes I can't think of a language that I know (French, German, a piss-poor
amount of Latin) that could distinguish. Tom is the subject and Harry is the
object of "saw", but either one of them could be the subject of "waved".

"Tom saw Harry and waved at him": Tom waved.

"Tom saw Harry who waved at him": Harry waved.

I think I'd probably rewrite the sentence in one of those two ways, to avoid
ambiguity, or else replace "he" with the person's name, even that's
repetitious.

It's like the marriage details in IMDB (Internet Movie Database) for an
actor/actree. There is a field for defining why the marriage ended:
divorced, his death, her death. That is unambiguous for male-female
relationships, but ambiguous for male-male or female-female ones.

On Wed, 21 Jul 2021 13:25:20 +0100
Ian Jackson <ianREMOVETHISjackson@g3ohx.co.uk> wrote:
>In message <sd8oec$1esi$1@gioia.aioe.org>, MrSpud_5nkd0w@62zd.gov writes
>
>
>>
>>English is a poor language to be precise in even in normal speech. eg:
>>
>>Tom saw Harry and he waved.
>>
>>Who waved, Tom or Harry?
>>
>>Other languages have case endings on the nouns that would clarify, English
>>doesn't.
>
>I'm not sure which languages would clarify the statement. Any examples?

German would be one if I remember from school correctly.

In message <ilqjanFkrcfU3@mid.individual.net>, JNugent
<jennings&co@fastmail.fm> writes
>On 21/07/2021 01:25 pm, Ian Jackson wrote:
>
>> In message <sd8oec$1esi$1@gioia.aioe.org>, MrSpud_5nkd0w@62zd.gov writes
>>
>>> English is a poor language to be precise in even in normal speech.
>>>eg:
>>> Tom saw Harry and he waved.
>>> Who waved, Tom or Harry?
>>
>>> Other languages have case endings on the nouns that would clarify,
>>>English doesn't.
>> I'm not sure which languages would clarify the statement. Any
>>examples?
>
>Latin had that to an extent, but not (IIRC) for proper nouns.
>
>But it was a very long time ago that I studied that language and while
>verbs were declined, nouns were conjugated.

The other way round.

In "Tom saw Harry and he waved", "Tom and "he" would be in the
nominative case (=subject) and Harry would be in the accusative case
(=object), but that still wouldn't tell you whether "he" referred to Tom
or to Harry.
--
John Hall "Hard work often pays off after time, but laziness always
pays off now." Anon

On 21/07/2021 13:49, NY wrote:

> Yes I can't think of a language that I know (French, German, a piss-poor
> amount of Latin) that could distinguish. Tom is the subject and Harry is
> the
> object of "saw", but either one of them could be the subject of "waved".
>
> "Tom saw Harry and waved at him": Tom waved.
>
> "Tom saw Harry who waved at him": Harry waved.
>
> I think I'd probably rewrite the sentence in one of those two ways, to
> avoid
> ambiguity, or else replace "he" with the person's name, even that's
> repetitious.
>
I'd rewrite it as "Tom saw Harry, and waved. Fewer words, clearer meaning.

On 21/07/2021 02:47 pm, John Hall wrote:
> In message <ilqjanFkrcfU3@mid.individual.net>, JNugent
> <jennings&co@fastmail.fm> writes
>> On 21/07/2021 01:25 pm, Ian Jackson wrote:
>>
>>> In message <sd8oec$1esi$1@gioia.aioe.org>, MrSpud_5nkd0w@62zd.gov writes
>>>
>>>> English is a poor language to be precise in even in normal speech. eg:
>>>> Tom saw Harry and he waved.
>>>> Who waved, Tom or Harry?
>>>
>>>> Other languages have case endings on the nouns that would clarify,
>>>> English doesn't.
>>>  I'm not sure which languages would clarify the statement. Any examples?
>>
>> Latin had that to an extent, but not (IIRC) for proper nouns.
>>
>> But it was a very long time ago that I studied that language and while
>> verbs were declined, nouns were conjugated.
>
> The other way round.
>
> In "Tom saw Harry and he waved", "Tom and "he" would be in the
> nominative case (=subject) and Harry would be in the accusative case
> (=object), but that still wouldn't tell you whether "he" referred to Tom
> or to Harry.

As I said, it was a long time ago.

But I'm fairly sure that Julius Caesar (who seemed to feature in every
section of Hillard & Botting) never had his name ended differently for case.