i^x = 2

[dm319]

I was watching with fascination this video which shows how to solve:

i^x = 2

Having never done complex maths at school, this is quite a brain warp.

But I was very impressed that if you let

x = -2·ln2·i/π

The HP-15c, DM42, TI-83+ and RPN83P got the correct answer. My fx-115W and fx-992s didn't.

[Ren]

I really should "waste" more of my time watching YT math videos, to make up for all the times I was doodling during math courses in in Elementary and High School instead of learning valuable skills for engineering and science.

[dm319]

(11-03-2024 10:13 PM)Ren Wrote:  I really should "waste" more of my time watching YT math videos

I've been tricked into learning something! Darn it!

[Albert Chan]

i^x = 2

x * ln(i) = x * (0 + pi/2*i) = ln(2) + 2n*pi*i

x = 4n - ln(2)/(pi/2)*i

[RPNerd]

(11-03-2024 10:13 PM)Ren Wrote:  I really should "waste" more of my time watching YT math videos

In particular this guy, blackpenredpen. He has several youtube channels that explore different levels of maths and calculus, some of which deal with topics you could probably work on blindfolded in your sleep with both arms tied behind your back, but some of which really make you think.

[dm319]

I don't know what is involved in having functions work with complex numbers, but it's very impressive that HP's first attempt at this is more capable (fractional powers, trig) than any scientific calculator on the market today.

[naddy]

(11-05-2024 09:49 PM)dm319 Wrote:  I don't know what is involved in having functions work with complex numbers, but it's very impressive that HP's first attempt at this is more capable (fractional powers, trig) than any scientific calculator on the market today.

It's perspiration rather than inspiration. If you're new to complex numbers, the key insight is that every number \(z = a + j b\) has an alternative representation \(z = r e^{j \theta}\), which is a lot more convenient for many operations if you remember the rules for exponents and logarithms, and the rectangular<>polar conversion functions convert between the two representations.

I've had an HP-15C for decades, so when I picked up an HP-11C two months ago, one of the first things I did was just write an ad-hoc program that implemented complex multiplication, division, exponentiation, logarithm, ... and then I started losing interest and stopped, because it's basically rote work. Only later did I discover that the HP-11C Solutions Handbook contains an example program that does much the same thing as my throwaway program did (and some more), and in pretty much the same way. It's really a straightforward exercise. And since then, people have pointed to the HP-41C Math Pac as the origin. I bet a lot of people wrote similar programs for themselves already on older programmables.

The HP-15C Advanced Functions Handbook has a section (p. 68ff) listing the definitions of the various functions—you may notice much builds on \(e^z\) and \(\ln z\)—and explains the choices of some finer details. I'm sure it was a lot of work.

[RPNerd]

(11-06-2024 01:04 AM)naddy Wrote:  It's perspiration rather than inspiration. If you're new to complex numbers, the key insight is that every number \(z = a + j b\) has an alternative representation \(z = r e^{j \theta}\), which is a lot more convenient for many operations if you remember the rules for exponents and logarithms, and the rectangular<>polar conversion functions convert between the two representations.

Just for reference, "j" is how electrical engineers write what mathematicians call the imaginary number \(i\). In the HP manuals you're more likely to see \(z = a + ib\) and \(z = re^{i\theta}\).

[Thomas Klemm]

(11-05-2024 09:49 PM)dm319 Wrote:  I don't know what is involved in having functions work with complex numbers (…)

This article might give you some ideas: [HP-48] calculating complex arccos and arcsin functions

[dm319]

(11-06-2024 01:04 AM)naddy Wrote:  It's perspiration rather than inspiration.

That's a great line. I'm still surprised that after all these years, Casio and TI didn't implement trig, powers and other transcendental functions on their scientifics. It won't be for lack of memory or for lack of code given their graphic calculators do.

Bit off topic, and for no good reason, I'd love to see a scientific with arbitrary base, but for which every function works in whichever base - powers, trig, fractions, decimals, scientific notation. It's a silly idea but there is an appeal for having every 'type' of number work with every function, as much as is possible.

[John Keith]

(11-10-2024 12:01 PM)dm319 Wrote:  

(11-06-2024 01:04 AM)naddy Wrote:  It's perspiration rather than inspiration.

That's a great line. I'm still surprised that after all these years, Casio and TI didn't implement trig, powers and other transcendental functions on their scientifics. It won't be for lack of memory or for lack of code given their graphic calculators do.

Bit off topic, and for no good reason, I'd love to see a scientific with arbitrary base, but for which every function works in whichever base - powers, trig, fractions, decimals, scientific notation. It's a silly idea but there is an appeal for having every 'type' of number work with every function, as much as is possible.

That wouldn't require any change in the algorithms. All computers perform calculations in either binary or BCD. The base is only for display- every computer that does calculations in binary converts the result to base 10 for the user. The only thing that need change is the numeric display code.

[dm319]

(11-10-2024 12:57 PM)John Keith Wrote:  That wouldn't require any change in the algorithms. All computers perform calculations in either binary or BCD. The base is only for display- every computer that does calculations in binary converts the result to base 10 for the user. The only thing that need change is the numeric display code.

I guess the implementation could be a particular base, though that may result in rounding errors that shouldn't happen in the native base. But whatever the implementation, I haven't seen a scientific that can do 1÷3 in base 12!

[C.Ret]

(11-10-2024 05:25 PM)dm319 Wrote:  

(11-10-2024 12:57 PM)John Keith Wrote:  [...]The only thing that need change is the numeric display code.

I guess the implementation could be a particular base, though that may result in rounding errors that shouldn't happen in the native base. But whatever the implementation, I haven't seen a scientific that can do 1÷3 in base 12!

Bonsoir,

It's interesting that you're worried about rounding errors, as this example actually gives an exact result: \( \frac{1}{3}=0.4_{(12)} \) exactly.

Base 12 is practical in contexts like measuring, counting and currency:
— one-third of a foot is 4 inches,
— one-third of a shilling is 4 pence,
— one-third of a year is 4 months, …

[AnnoyedOne]

(11-10-2024 06:52 PM)C.Ret Wrote:  Base 12 is practical in contexts like measuring, counting and currency:

Wasn't it the French who invented the metric measurement system where everything is a power of 10 (base 10)? And most of the world, except North America, uses it?

Ten fingers = base 10 = "natural". Cats can be Polydactyl but humans have 10 fingers and toes.

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

A1

PS: OT but interesting.

https://worldpopulationreview.com/countr...ric-system

[dm319]

(11-10-2024 06:52 PM)C.Ret Wrote:  It's interesting that you're worried about rounding errors, as this example actually gives an exact result: \( \frac{1}{3}=0.4_{(12)} \) exactly.

That's why I gave that example! If your internal calculations are in binary or decimal it isn't exactly representable. The DC calculator returns .3BBBBBBBBB...

(11-10-2024 06:52 PM)C.Ret Wrote:  Base 12 is practical in contexts like measuring, counting and currency:
— one-third of a foot is 4 inches,
— one-third of a shilling is 4 pence,
— one-third of a year is 4 months, …

Base 12 is my favourite. If I had my way, we would all switch to base 12. But we do need an extra 2 digits to do it properly (not A or B!).

Base 12 has the highest factors to digits ratio of any base*.

*I just made this up but it might be true.

[dm319]

(11-10-2024 07:06 PM)AnnoyedOne Wrote:  Wasn't it the French who invented the metric measurement system where everything is a power of 10 (base 10)? And most of the world, except North America, uses it?

Ten fingers = base 10 = "natural". Cats can be Polydactyl but humans have 10 fingers and toes.

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

I'm all for using base 10 when we're already using a base 10 number system!

Also TIL about polydactyl cats.

D

[AnnoyedOne]

(11-10-2024 07:34 PM)dm319 Wrote:  Base 12 is my favourite.

You should be a baker (Dozen = 12) or live in North America (1 foot = 12 inches). I'm not sure if there's a "standard" inch but it is 2.54cm.

And, unless you work for NASA, converting between systems is easy.

Quote:In September of 1999, after almost 10 months of travel to Mars, the Mars Climate Orbiter burned and broke into pieces. On a day when NASA engineers were expecting to celebrate, the ground reality turned out to be completely different, all because someone failed to use the right units, i.e., the metric units!

https://www.simscale.com/blog/nasa-mars-...ter-metric

A1

[AnnoyedOne]

(11-10-2024 05:25 PM)dm319 Wrote:  ...I haven't seen a scientific that can do 1÷3 in base 12!

https://github.com/Jefik37/base-12-calculator

A1

[dm319]

(11-10-2024 07:49 PM)AnnoyedOne Wrote:  You should be a baker (Dozen = 12) or live in North America (1 foot = 12 inches). I'm not sure if there's a "standard" inch but it is 2.54cm.

Or use a 12 hour clock, or a 12 month calendar, or 12 zodiacs, or 12 pennies in a shilling! While I think base 12 is superior I prefer to use multiples of 10 when our number system is base 10. I do wish the UK had just gone all-in on metric. I buy fuel in litres but my car reports efficiency in miles per gallon. Road signs give distances in miles but shorter distances are often in metres. I could go on.

That's a great find on that calculator, I love the figures for 11 and 12.

I quite like this cartoon, which I'm reminded of whenever I forget my input base in DC, and suddenly I have to work backwards to figure out what on earth I've mistakenly set the current base to.

   

[naddy]

(11-10-2024 07:49 PM)AnnoyedOne Wrote:  I'm not sure if there's a "standard" inch but it is 2.54cm.

All those American units are defined in terms of SI units, so metrologically speaking, Americans are metric, they just use very odd multiples of metric units.

But yes, it is presumably no coincidence that a circle is traditionally partitioned into 360 degrees. That number has a lot of even divisors (32SII):

Code:

A01 LBL A
A02 0
A03 STO I
# CK=256C 004.5
L01 LBL L
L02 ISG I
L03 +/-
L04 360
L05 RCL I
L06 x=y?
L07 RTN
L08 ÷
L09 FP
L10 x≠0?
L11 GTO L
L12 RCL I
L13 PSE
L14 GTO L
# CK=E73F 029.0

XEQ A: 1, 2, 3, 4, 5, 6, 8, ...