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### Fundamental Mathematics

Posted: Tue Apr 02, 2019 8:19 am
Claim: Pocket calculators as we know them can be seen as number crunchers calculating with whatever numbers are thrown at them.

Proof: Take the famous processor check arcsin(arccos(arctan(tan(cos(sin(9°))))) and calculate inside out as we all were taught:
1. sin(9°) returns a real number A, certainly not an angle; but what shall be cos(A) then? And tan(cos(A))?
2. With the latter returning a real number B, however, arctan(B) will definitely return an angle C; but what shall arccos(C) return then? And arcsin(arccos(C))?
3. Q.e.d.
Though what is this? Rubbish? Sloppiness? Freedom?!?

### Re: Fundamental Mathematics

Posted: Tue Apr 02, 2019 10:36 am
Interesting question, here is my take on this:

arcsin(arccos(arctan(tan(cos(sin(9°))))) works as is only with non-typed arguments and results.

Else I would say that it depends if the typing is strong or weak. So it’s a designer’s choice

With strong typing, then arcsin(arccos(arctan(tan(cos(sin(9°))))) is not valid, but I would expect it would be easy to add or drop a type to/from a value through built-in commands.

With weak typing, I would expect that if the argument of a trigonometric function expecting an angle (such as SIN, COS, TAN) is a real number not typed as an angle (such as 9°), the trigonometric function will consider it as an angle expressed in current angle mode. And a trigonometric function expecting a real argument would convert a typed real (such as an angle) to a real by dropping the type, with maybe a warning message. In such case it would be nice to have a flag to enable/disable such warning messages.

### Re: Fundamental Mathematics

Posted: Tue Apr 02, 2019 11:21 am
Walter wrote:
Tue Apr 02, 2019 8:19 am
Claim: Pocket calculators as we know them can be seen as number crunchers calculating with whatever numbers are thrown at them.
That is obviously true. No proof is necessary, and I'm sure the people who design calculators would agree. The same thing also goes for all the programming languages I'm familiar with.

It sounds like you're not happy with this state of affairs. Do you have an alternative in mind? Like numbers with attached units, and making the calculator refuse to perform operations when the units make no sense, like taking the sine of something that isn't an angle, or the arc sine of something that isn't dimensionless, or adding kilometers to liters...

### Re: Fundamental Mathematics

Posted: Tue Apr 02, 2019 12:28 pm
Walter wrote:
Tue Apr 02, 2019 8:19 am
Claim: Pocket calculators as we know them can be seen as number crunchers calculating with whatever numbers are thrown at them.

Though what is this? Rubbish? Sloppiness? Freedom?!?
So what's your point? Calculators and computers have always crunched numbers this way. It's up to the human to make sure the numbers make sense.
I grew up using a slide rule and we even had to have some idea of the magnitude of our answer before solving a problem. Anyone who uses any calculating device blindly deserves what (s)he gets.

### Re: Fundamental Mathematics

Posted: Tue Apr 02, 2019 12:29 pm
Let's not forget that the argument of a trigonometric function is not an angle but an arc length representing an angle. When we write sin (9 °) we do not speak of an angle of 9 ° but of the projection of an arc corresponding to 9 °.

### Re: Fundamental Mathematics

Posted: Tue Apr 02, 2019 1:15 pm
Perhaps one way to think of arcsin(arccos(arctan(tan(cos(sin(9°))))) is that sin, cos, and tan decrement the power of the degrees units by 1, and that arcsin, arccos, and arctan increment the power of the degrees units by 1. I'm not sure if that's a good way to think about it though.

Angle units are often considered to be dimensionless, but I guess it depends on the context they are used, i.e. whether the trig functions are operating on pure numbers (as the Taylor series expansion would suggest) or on actual angles.

### Re: Fundamental Mathematics

Posted: Tue Apr 02, 2019 1:50 pm
Quite interesting replies so far.
toml_12953 wrote:
Tue Apr 02, 2019 12:28 pm
Walter wrote:
Tue Apr 02, 2019 8:19 am
Claim: Pocket calculators as we know them can be seen as number crunchers calculating with whatever numbers are thrown at them.

Though what is this? Rubbish? Sloppiness? Freedom?!?
So what's your point? Calculators and computers have always crunched numbers this way. It's up to the human to make sure the numbers make sense.
I grew up using a slide rule and we even had to have some idea of the magnitude of our answer before solving a problem. Anyone who uses any calculating device blindly deserves what (s)he gets.
Tom, I interpret your vote as for "Freedom".

### Re: Fundamental Mathematics

Posted: Wed Apr 03, 2019 2:46 am
Walter wrote:
Tue Apr 02, 2019 1:50 pm
Quite interesting replies so far.
toml_12953 wrote:
Tue Apr 02, 2019 12:28 pm
Walter wrote:
Tue Apr 02, 2019 8:19 am
Claim: Pocket calculators as we know them can be seen as number crunchers calculating with whatever numbers are thrown at them.

Though what is this? Rubbish? Sloppiness? Freedom?!?
So what's your point? Calculators and computers have always crunched numbers this way. It's up to the human to make sure the numbers make sense.
I grew up using a slide rule and we even had to have some idea of the magnitude of our answer before solving a problem. Anyone who uses any calculating device blindly deserves what (s)he gets.
Tom, I interpret your vote as for "Freedom".
Right on! (60's talk)

### Re: Fundamental Mathematics

Posted: Wed Apr 03, 2019 6:13 am
toml_12953 wrote:
Wed Apr 03, 2019 2:46 am
Walter wrote:
Tue Apr 02, 2019 1:50 pm
toml_12953 wrote:
Tue Apr 02, 2019 12:28 pm
So what's your point? Calculators and computers have always crunched numbers this way. It's up to the human to make sure the numbers make sense.
I grew up using a slide rule and we even had to have some idea of the magnitude of our answer before solving a problem. Anyone who uses any calculating device blindly deserves what (s)he gets.
Tom, I interpret your vote as for "Freedom".
Right on! (60's talk)
Got it.

Btw, also I grew up using a slide rule. Even in university lab class. Error propagation without a calculator isn't real fun. But that's another story.

### Re: Fundamental Mathematics

Posted: Sun Apr 07, 2019 9:30 pm
Thomas Okken wrote:
Tue Apr 02, 2019 11:21 am
Walter wrote:
Tue Apr 02, 2019 8:19 am
Claim: Pocket calculators as we know them can be seen as number crunchers calculating with whatever numbers are thrown at them.
That is obviously true. No proof is necessary, and I'm sure the people who design calculators would agree. The same thing also goes for all the programming languages I'm familiar with.

It sounds like you're not happy with this state of affairs. Do you have an alternative in mind? Like numbers with attached units, and making the calculator refuse to perform operations when the units make no sense, like taking the sine of something that isn't an angle, or the arc sine of something that isn't dimensionless, or adding kilometers to liters...
Background: We were thinking of angular data. I.e. arcsin would operate on reals and return tagged angles, for instance. Primary reason for these considerations was getting rid of D.MS+ and similar special operations on such "special real" numbers. Actually, D.MS data are the only nondecimal data I'm aware of having survived in our decimal world of today (please don't come with feet, quarts, and other obsolete stuff). And D.MS is coexisting with DEG and RAD and other true decimal angular units.

Why bother? On one hand, calculatory freedom's nice. OTOH, it allows for quite some meaningless or even false operations. All a matter of user responsibility, sure ... but I'm telling this also to people who need to be advised not to dry pets in microwave ovens.