negative number raised to even power

[Gerald H]

What a waste of time! Just use negadecimal:

http://www.hpmuseum.org/forum/thread-3710.html

[Joe Horn]

(05-04-2015 06:10 AM)Dirk. Wrote:  So how do you explain your students that 2(+/-)² is -4 at the prime when entering it using the keyboard?

It's quite simple, really. Notice that you typed it as "2(+/-)²", which shows that you are assuming that keystroke order should always matter, which is true in RPN... but it does not (and SHOULD not) always matter in command-line calculators (the reason follows shortly). Prime is a command line calculator, and the command line is not evaluated until Enter is pressed. At that time, the order in which the keys were pressed is totally immaterial; the ONLY thing that matters are the rules of parsing the algebraic syntax on the command line. In those rules, "-2²" means "start with 2, square it, then negate the result".

In ANY command-line algebraic-notation calculator which follows standard algebraic notation rules of operator precedence, [2][+/-][x²] yields the same result as [+/-][2][x²], because both key sequences create identical command lines, namely, "-2²", which of course IN STANDARD ALGEBRAIC NOTATION equals -4.

(05-04-2015 06:10 AM)Dirk. Wrote:  In my opinion this is simply a bug which needs to get fixed as soon as possible.
(Sorry for my harsh tone, since I'm German, I can't do better :-) )

I'm half German, so I half understand. But my Irish half wants to write a sad song about how we algebraic users are so often misunderstood. Let's compromise by enjoying some Weihenstephaner Hefeweissbier and Guinness, and let the others fight it out.

[Dirk.]

Well, due to my home, I have to recommend Licher.
But anyway: One of the main points for me is the inconsistency. The calculator behaves differenty when processing a previously calculated result and when entering something leading to the same display. I think the way the algebraic SHARP-Calculators (SHARP EL-512S and SHARP EL-520V) behave is much more consistent.

... and now I'll go out for an Edinger Weißbier :-)

[John Colvin]

To add to the confusion, even in C/C++, it gets weird:

int a = (-2)^2;
printf("%d", a);

gives -4 as an answer??
But
int a = pow(-2, 2);
printf("%d", a);

gives 4 as an answer
So it seems that precedence rules aren't always followed.

[debrouxl]

In C/C++, ^ is exclusive or (xor). In C (but not in C++), from C99 onwards, when using the alternative spellings iso646.h header, xor is expanded to ^.

With 2's complement representation of negative numbers:
* the binary representation of -1 is the pattern of 1 (0b11...111 in GCC and Clang for years, C++14 made the 0b syntax standard);
* the binary representation of -2 is 0b11...110;
* the binary representation of 2 is 0b00...010;
* the binary representation of (-2)^2 is therefore 0b11...100, i.e. -4.

[John Colvin]

(05-05-2015 05:17 AM)debrouxl Wrote:  In C/C++, ^ is exclusive or (xor). In C (but not in C++), from C99 onwards, when using the alternative spellings iso646.h header, xor is expanded to ^.

With 2's complement representation of negative numbers:
* the binary representation of -1 is the pattern of 1 (0b11...111 in GCC and Clang for years, C++14 made the 0b syntax standard);
* the binary representation of -2 is 0b11...110;
* the binary representation of 2 is 0b00...010;
* the binary representation of (-2)^2 is therefore 0b11...100, i.e. -4.

That explains it. I knew ^ meant xor in the context of boolean operations, but that symbol is used so frequently to represent exponentiation, I just forgot about that quirk C. My bad.

[Thomas Klemm]

(05-05-2015 06:12 PM)John Colvin Wrote:  That explains it. I knew ^ meant xor in the context of boolean operations, but that symbol is used so frequently to represent exponentiation, I just forgot about that quirk C. My bad.

And I assumed you made a clever joke.

Cheers
Thomas

[Wes Loewer]

(05-05-2015 05:17 AM)debrouxl Wrote:  In C/C++, ^ is exclusive or (xor). In C (but not in C++), from C99 onwards, when using the alternative spellings iso646.h header, xor is expanded to ^.

(just for clarification)
But in Standard C++, there's no need to include the header as xor is already a C++ keyword.

[Wes Loewer]

I'm really baffled by all the discussion on this issue. Everyone agrees with the following two statements:

1) The square of negative two is four: (-2)^2 = 4 (In 41C rpn: 2 CHS ENTER 2 y^x)
2) The negative of the square of two is negative four: -(2^2) = -4 (In 41C rpn: 2 ENTER 2 y^x CHS)

What people are disagreeing about is the meaning of -2^2. Even though some calculators and computer programs may not follow it, the standard algebra order of operations indicates that exponentiation proceeds negation, so -2^2 means -(2^2) = -4.

[Claudio L.]

(05-05-2015 07:11 PM)Thomas Klemm Wrote:  

(05-05-2015 06:12 PM)John Colvin Wrote:  That explains it. I knew ^ meant xor in the context of boolean operations, but that symbol is used so frequently to represent exponentiation, I just forgot about that quirk C. My bad.

And I assumed you made a clever joke.

Cheers
Thomas

I thought the same thing: I wish he hadn't clarified that it was a mistake.
(-2)^2 = -4 was the smartest thing I read in the whole thread! The way it applies to the discussion makes it an excellent joke, could've caught a lot of people off-guard.

[PANAMATIK]

(04-26-2015 06:57 PM)Marcus von Cube Wrote:  

(04-26-2015 06:22 PM)TASP Wrote:  2
chs
^
2
chs
y\x


yields 'error'

You are computing (-2)^(-2) (not the original expression!) which is most probably computed as exp(ln(-2)*(-2)). A calculator that only handles real numbers cannot compute ln(-2). Try the x² key instead.

2
chs
^
2
y\x

The original expression (-2)^2 yields 'error' too on HP-21 and HP-25, however HP-29C computes 4.

2 chs ^ 3 y\x shows -8 and works only for integer exponents (-2)*(-2)*(-2)=-8

Bernhard

[Thomas Radtke]

(05-05-2015 07:32 PM)Wes Loewer Wrote:  What people are disagreeing about is the meaning of -2^2.

But then 0-2^2 is clear again. It's the missing zero that confuses some people.

[Wes Loewer]

(05-06-2015 07:03 AM)Thomas Radtke Wrote:  

(05-05-2015 07:32 PM)Wes Loewer Wrote:  What people are disagreeing about is the meaning of -2^2.

But then 0-2^2 is clear again. It's the missing zero that confuses some people.

I agree that -2^2 = 0-2^2, but someone could argue that these are different as the left side involves negation while the right side involves subtraction, two operators that use similar looking symbols.

I think I've got something that will convince any doubters...
Consider the function f(x)=exp(-x^2)
   
I think everyone will agree that the graph of this function is a bell curve, which means that the function is interpreted as exp(-(x^2)) and not exp((-x)^2).

[Thomas Radtke]

(05-06-2015 02:37 PM)Wes Loewer Wrote:  I think I've got something that will convince any doubters...
Consider the function f(x)=exp(-x^2)

Interesting, but here we have a variable instead of a number. This leads me to this thought:

I find it indeed difficult to realize we have no other representation of a negative number than a positive one subtracted from zero. An implicitly atomic expression -2 looks quite attrative to consider, but is unfortunately no convention. Hence a new symbol (raised minus) has been introduced on some calculators like the 32SII.

[Thomas Radtke]

(05-04-2015 02:51 PM)Joe Horn Wrote:  In ANY command-line algebraic-notation calculator which follows standard algebraic notation rules of operator precedence, [2][+/-][x²] yields the same result as [+/-][2][x²], because both key sequences create identical command lines, namely, "-2²", which of course IN STANDARD ALGEBRAIC NOTATION equals -4.

Maybe it has been mentioned before: Just found out the 32SII does it wrong.
[EQN][-][2][y^x][2][ENTER] gives 4 as well as [EQN][2][+/-][y^x][2][ENTER]. It was a hard day and it has been some time since I last get into the details of this calculator, so what am I missing here?

The 17BII equation solver does it right with [A][=][-][2][y^x][2][calc] resulting in -4.

[Wes Loewer]

(04-26-2015 07:20 AM)Thomas Radtke Wrote:  

(04-26-2015 06:38 AM)parisse Wrote:  Could you give a list of softwares/calcs that evaluate -2^2 (algebraic notation) to 4?

MS Excel.

Check out https://support.microsoft.com/en-us/kb/kbview/132686 where Microsoft says "This order of precedence may mean that a formula returns a positive value when you expect it to return a negative value."

Here's a table showing how various calculators/software handle the following order of operations issues:
-2^2 = (a) -4 (b) 4
2^3^2 = (a) 512 (b) 64
1/2pi = (a) 1.57 (b) 0.159

Calculators
aaa: HP Prime Algebraic Entry
aab: HP Prime Textbook Entry/50g EQW (but 1/2pi is shown as stacked fraction and is not ambiguous)
aaa: HP 50g Alg mode
abb: Casio fx-9960G
abb: TI 81/82/85/86
aba: TI 73/83/83+/84+(Classic Mode)
aaa: TI 84+(MathPrint Mode)/89/Nspire

Software
bb-: Excel
bb-: OpenOffice
ba-: Google Docs Sheet
ba-: Gnumeric (formula bar shows "(-2)^2" and "2^(3^2)")
ab-: Corel Quattro Pro
ba-: Corel WordPerfect Table
aa-: Maxima
aaa: Wolfram Alpha
aaa: GeoGebra

Legacy Software
bb-: VisiCalc
??-: Lotus 123 (can someone test this)
ab-: Lotus Improv (anybody remember this spreadsheet?)

This is certainly not a complete list, just what I had handy to test. All the calculators I tested agreed that -2^2=-4, but the other issues were mixed. The more recent HP and TI calculators agree with -4, 512, and 1.57.

[Wes Loewer]

(05-06-2015 03:40 PM)Thomas Radtke Wrote:  

(05-06-2015 02:37 PM)Wes Loewer Wrote:  Consider the function f(x)=exp(-x^2)

Interesting, but here we have a variable instead of a number.

True, but I figured that either negation precedes exponentiation or it doesn't.
I'm curious, for those that say -2^2 = 4, what about -(2)^2 ?

[Dirk.]

Well Wes,
Rule of thumb: negation as always higher priority when the calculator does it (Variable^2, Ans^2, ...) but not when you type it.

[timetraveller]

A fascinating discussion.

From the HP 32S II Owner's Manual:

Page 1-11.

Making Numbers Negative

The +/- key changes the sign of a number.

To key in a negative number, type the number, then press +/-.

To change the sign of a number that was entered previously, just press +/-. (If the number has an exponent, +/- affects only the mantissa - the non-exponent part of the number.)

===================================

Note that last part. There is no mention of negation of the entire calculation, only negation of the mantissa.

The Microsoft support article link on MS EXCEL in a previous post basically says the same thing.

I come from a software environment and have written code since 1968, so I'm well aware of orders of precedence. In the software environment, -4 is the correct answer by virtue of the rules of precedence. But back to the original question of this thread: intuitively on a calculator, when I press the +/- key I want to change the sign of a single entity, not the entire subsequent calculation. For me at least, that was always the purpose of the +/- key.

What I find interesting is how the calculator's role has changed over the last 40 years or so. Today it has assumed the role of a software device (read: computer), thus assuming the computer's methodology of doing things. User beware! Today's calculator user must have at least a rudimentary knowledge of computer-speak!

Bill

[Gerald H]

(05-06-2015 07:56 PM)Wes Loewer Wrote:  

(04-26-2015 07:20 AM)Thomas Radtke Wrote:  MS Excel.

Check out https://support.microsoft.com/en-us/kb/kbview/132686 where Microsoft says "This order of precedence may mean that a formula returns a positive value when you expect it to return a negative value."

Here's a table showing how various calculators/software handle the following order of operations issues:
-2^2 = (a) -4 (b) 4
2^3^2 = (a) 512 (b) 64
1/2pi = (a) 1.57 (b) 0.159

Calculators
aaa: HP Prime Algebraic Entry
aab: HP Prime Textbook Entry/50g EQW (but 1/2pi is shown as stacked fraction and is not ambiguous)
aaa: HP 50g Alg mode
abb: Casio fx-9960G
abb: TI 81/82/85/86
aba: TI 73/83/83+/84+(Classic Mode)
aaa: TI 84+(MathPrint Mode)/89/Nspire

Software
bb-: Excel
bb-: OpenOffice
ba-: Google Docs Sheet
ba-: Gnumeric (formula bar shows "(-2)^2" and "2^(3^2)")
ab-: Corel Quattro Pro
ba-: Corel WordPerfect Table
aa-: Maxima
aaa: Wolfram Alpha
aaa: GeoGebra

Legacy Software
bb-: VisiCalc
??-: Lotus 123 (can someone test this)
ab-: Lotus Improv (anybody remember this spreadsheet?)

This is certainly not a complete list, just what I had handy to test. All the calculators I tested agreed that -2^2=-4, but the other issues were mixed. The more recent HP and TI calculators agree with -4, 512, and 1.57.

Lotus 123, 2.4: aba