Calculator test

[Commie]

Hi Ladies and Gents,

Euler, in the 18th century wrote down a very important equation, this being in one form can be written:

e^(pi.i)=cos(pi)+i.sin(pi) which equates to e^(pi.i)=-1

Which can be written as e^(pi.i)+1=0

Know we can also deduce that pi.i=ln(-1)

Key in ln(-1) into your best calculator and hit return, your calculator should give pi.i, if it throws up a domain error try putting into complex rectangular mode and try again.

As expected, my hp15ce and ti30x pro mp throws up error 0, which didn't surprise me.

Cheers
Darren

[Steve Simpkin]

I must admit that I am surprised that the HP-15C shows an error for this. I thought it would provide a complex answer like the HP-42S. The oldest HP I have that can answer this is the HP-28C, as do all of the RPL machines. The oldest TI I have that provides an answer is the TI-92 (symbolic πi). The TI-84 Plus returns an answer if it is set for complex answers. The HP 35s returns a "LOG(NEG)" error.

Edit: Other have confirmed the the HP-15C will produce a complex answer if it is set to complex mode first (SF 8).

Edit 2: The HP 35s does return an answer to LN(-1) if the argument is complex. For example LN(-1i0) = 0i3.14159265359.

[Johnh]

That's very interesting, a new insight for me. I hadn't realised there was such a relationship between Pi, natural logs and complex numbers.

Hp15c-ce actually does do it! Need to set to complex mode first (SF 8), then do LN of -1 and it displays 0 as the real part, and the imaginary part is displayed as Pi by revealing it with (i).

On a few emulators for Hp15c such as TouchRPN, JRPN 15 and Pockemul, it works the same. And Free42 and Plus42 display it directly.

I'm not sure about Hp35s. I have this but never spent time with its complex mode. But it doesn't seem to do it in its default mode.and puts up a LOG(NEG) message.

[naddy]

I'm not sure what the point of the exercise is. Obviously, there are three possibilities:

1. Error because complex numbers are not supported. (E.g., HP-11C.)
2. Error unless a complex operation is explicitly requested. (E.g., HP-15C, HP-32S.)
3. The calculator automatically switches into the complex domain. (E.g., HP-42S, HP-28C/S.)
   

The prototypical test for complex support would be \(\sqrt{-1}\).

[Thomas Klemm]

(12-06-2024 11:19 PM)Steve Simpkin Wrote:  I thought it would provide a complex answer like the HP-42S.

Complex results for real-number operations can be disabled with REALRES.
This function sets flag 74.

Example

-1
LN

Invalid Data

But it still works for a complex number:

-1 i0
LN

0 i3.14159265359

It can be enabled with CPXRES.
This function clears flag 74 (default).

[rprosperi]

(12-06-2024 10:27 PM)Commie Wrote:  Hi Ladies and Gents,

Euler, in the 18th century wrote down a very important equation, this being in one form can be written:

e^(pi.i)=cos(pi)+i.sin(pi) which equates to e^(pi.i)=-1

Which can be written as e^(pi.i)+1=0

Know we can also deduce that pi.i=ln(-1)

Key in ln(-1) into your best calculator and hit return, your calculator should give pi.i, if it throws up a domain error try putting into complex rectangular mode and try again.

As expected, my hp15ce and ti30x pro mp throws up error 0, which didn't surprise me.

Cheers
Darren

Is the notation used here "xyz.i" some kind of standard shorthand for "xyz*i" ? Not saying it's bad or wrong, I'm just not familiar with it. Where is it from?

[StephenG1CMZ]

I have seen that notation used before, though the "." is meant to be level with the characters rather than lower down.
https://en.m.wikipedia.org/wiki/Multipli...erminology
Apparently it was introduced by Leibniz in 1689:
https://jeff560.tripod.com/operation.html
Ànd it was occasionally seen in India before that.

[Idnarn]

(12-06-2024 10:27 PM)Commie Wrote:  Key in ln(-1) into your best calculator and hit return, your calculator should give pi.i, if it throws up a domain error try putting into complex rectangular mode and try again.

As expected, my hp15ce and ti30x pro mp throws up error 0, which didn't surprise me.

Cheers
Darren

• HP 15C CE and Nonpareil with the HP 15C both provide (0,3.141592654i) as the answer (the calculator has to be put in complex mode first by g SF 8).
• HP 12C "Gold" has the ln function and displays "Error 0". It has no advertised complex number support.
  

In the case of Casios:

• The wonderful fx-260 SOLAR II has no support for complex numbers and understandably displays "-E-". (However it's a wonderful little $10 pocket scientific calculator, and perhaps one of the very few that have an unshifted pi key - if you look at a photo, it's shifted, but you can press EXP without shifting and get pi when no other digits have been entered. Its solar cell generates sufficient power in warm white lit dimmer rooms at night and it has a cap that holds state for more than a minute without light. Immediate eval like yesteryear calculators - it's not RPN but close enough; none of that SVPAM entry.)
• fx-100MS II has support for complex numbers, but displays "Math ERROR" in complex mode.
• fx-CG50 has support for complex numbers and displays \pi i.
  

A more interesting test of complex numbers support is trying trigonometry on complex numbers.

The HP 15C CE performs sin(), cos(), tan() of complex numbers.

All of the fore-mentioned Casio calculators don't have any support for these functions on complex numbers. It's unexpected that Casio in this age still doesn't support it esp. on the latter two calculators that support complex numbers as it'd be trivial to implement considering everything else these calculators do. The fx-CG50 has the KhiCAS (xcas/giac) addon though which kinda makes up for it.

[Commie]

(12-06-2024 11:20 PM)Johnh Wrote:  Hp15c-ce actually does do it! Need to set to complex mode first (SF 8), then do LN of -1 and it displays 0 as the real part, and the imaginary part is displayed as Pi

I just want to confirm Johnh and others are correct, the hp15ce does indeed work and gives the correct answer in complex mode. This is quite amazing given that the hp15c debut in 1982.

Cheers
Darren

[rprosperi]

(12-07-2024 08:49 AM)StephenG1CMZ Wrote:  I have seen that notation used before, though the "." is meant to be level with the characters rather than lower down.
https://en.m.wikipedia.org/wiki/Multipli...erminology
Apparently it was introduced by Leibniz in 1689:
https://jeff560.tripod.com/operation.html
Ànd it was occasionally seen in India before that.

Sure, I've seen dotted notation with a dot symbol for multiplication but as you say that dot is vertically centered on the characters, just as "*", "+", "-", etc. are. I've never seen the period used in this way??

[AnnoyedOne]

In my PC Hewlett Packard HP-15C emulator (with the 192 register patch)

https://www.hpmuseum.org/forum/thread-22453.html

I activated complex mode (g SF 8), entered -1, then g Ln and got

0
f Re<>Im gave 3.1416

which I believe is 0 + pi.i = pi.i

In real (non-complex) mode I got "Error 0" which is correct.

Of course you have to know that the result of Ln(-1) is complex just as the designers did. Read

"Section 3: Calculating in Complex Mode" of the Advanced Functions Handbook (AFH), Ln(z), [page 70 in the original book].

A1

[Johnh]

I grew up in the UK, with high-school maths in the 1970's.

I often use the "." in the lower position to denote multiplication, but only for algebraic descriptions such as a = b.c .

As with other english-speaking countries, we also use "." as the decimal point, so with numbers it has to be an explicit × or x eg 2 × 3 = 6 at least for me.

And if it's code or excel, a "*' is used.

But who taught me that use of the period "." ? Did I learn maths wrongly? or should I start a class action in my old maths class?

[AnnoyedOne]

(12-07-2024 07:17 PM)Johnh Wrote:  I often use the "." in the lower position to denote multiplication...

I learnt it as "dot product" for vector multiplication (bigger dot).

https://en.wikipedia.org/wiki/Dot_product

That said, on the internet etc, I've seen *, x, and . used. Some have browser extensions etc to enter mathematical symbols but I (and others) usually don't bother.

A1

[Geoff Quickfall]

Interesting thread!

What I remember from school:

[naddy]

(12-07-2024 03:34 AM)rprosperi Wrote:  

(12-06-2024 10:27 PM)Commie Wrote:  Which can be written as e^(pi.i)+1=0

Is the notation used here "xyz.i" some kind of standard shorthand for "xyz*i" ? Not saying it's bad or wrong, I'm just not familiar with it. Where is it from?

As the Wikipedia article explains:

Quote:Historically, in the United Kingdom and Ireland, the middle dot was sometimes used for the decimal to prevent it from disappearing in the ruled line, and the period/full stop was used for multiplication.

[JurgenRo]

(12-07-2024 07:50 PM)Geoff Quickfall Wrote:  Interesting thread!

What I remember from school:

I am a mathematician, but I have never seen A(B) as a product. A(B) is rather a common abbreviation for “A is a function of B”. On the other hand you missed AB, which is very common and called "implicit notation" for A*B. In general, the meaning of a product is derived from the context. Ambiguities (scalar product, vector product) therefore generally have special notations (A x B: vector product, cross product; <A,B>: scalar product etc.).

Edit: typo

[Geoff Quickfall]

Your correct, forgot about the AB form.

The A(B) should really have been described as:

A(B +C) to imply priority and the A(B) was incorrect as the parentheses are redundant in a strictly multiplicational (word?) aspect.

Did say, from memory :-)

Cheers

[Steve Simpkin]

(12-07-2024 08:12 PM)JurgenRo Wrote:  ...
I am a mathematician, but I have never seen A(B) as a product. A(B) is rather a common abbreviation for “A is a function of B”. On the other hand you missed AB, which is very common and called "implicit notation" for A*B. In general, the meaning of a product is derived from the context. Ambiguities (scalar product, vector product) therefore generally have special notations (A x B: vector product, cross product; <A,B>: scalar product etc.).

Edit: typo

Oh no, the dreaded Implicit Multiplication Calculator Conundrum!
Order of operations - what is 6÷2(1+2) ?

See also:
https://sites.google.com/view/hp-plus-ca...rehj3fwg80

[Commie]

Hi Guys,

Just wanna point out, for those calculators that do work, it doesn't have to be fixed at ln(-1), you can use any negative number which can deliver a real and a complex number. For example, ln(-0.5)=-0.69+pi.i

I've checked my hp15ce and yes it works, I'm growing to like my 15ce!

Cheers
Darren

[C.Ret]

(12-06-2024 10:27 PM)Commie Wrote:  Key in \(ln(^-1)\) into your best calculator and hit return, your calculator should give \(\pi\cdot i\)

Hi Commie, your best calculators have a RETURN key ?

(12-07-2024 03:33 AM)Thomas Klemm Wrote:  

(12-06-2024 11:19 PM)Steve Simpkin Wrote:  I thought it would provide a complex answer like the HP-42S.

Complex results for real-number operations can be disabled with REALRES. This function sets flag 74.
Example
-1 LN
Invalid Data
But it still works for a complex number:
-1 i0 LN
0 i3.14159265359
It can be enabled with CPXRES. This function clears flag 74 (default).

This observation exactly match how the HP-15C works, except there is no specific command apart setting Flg 8 which is the complex-mode toggle or indicator.

(12-07-2024 01:22 PM)AnnoyedOne Wrote:  I activated complex mode (g SF 8), entered -1, then g Ln and got
0.0000
f Re<>Im gave 3.1416
which I believe is \(0+\pi\cdot i=\pi\cdot i/)

Right.
There is an alternative way of promoting complex mode on the HP-15C; it is simply to enter arguments as complex number.
So the same observation can be made, from 'standard' calculator state (the state we get after a reset [ON]+[ - ] ).

In 'real entry mode', results are strictly real:
[ 1 ][CHS][g][ LN ] display Error  0.
In 'extended complex stack mode', using one of the many entry methods:
[ 1 ][CHS][ENTER][ 0 ][f][ I ][g][ LN ] set the flag 8 and display real part 0.0000 and [f][(i)] or [f][Re⇋Im] display imaginary part 3.1416.

Unfortunately, there is no other way to quit the complex extend mode than clearing flag 8 by [f][ CF ][ 8 ] or by resetting the calculator.

Note, that trying the equation \(e^{\pi\cdot i}=^-1\) the opposite direction, show an interesting fact (or flow) about HP-15C precision in complex computations:
[g][ ∏ ][f][Re⇋Im][ ◀- ][ e^x ] exactly display real part -1.0000 but the imaginary part is not exactly what we expect; [f][(i)] or [f][Re⇋Im] display 4.1000  -10.

(12-07-2024 08:23 PM)Commie Wrote:  I've checked my hp15ce and yes it works, I'm growing to like my 15ce!

Really Commie? Even is there is no RETURN key on our beloved device?