BIMDAS - Order of Calculation

[Guest]

I've recently noticed that Excel flies in the face of standard scientific,
mathematical and engineering convention in the calculation of powers for
numbers that are then multiplied by a negative.

The convention of mathematics, "BIMDAS" (or similar acronyms), states that
_I_ndices (or powers, or exponents), should be calculated before
_M_ultiplication. Because of this, the following is accepted as correct:

-3^2 = -9.

This is because it is the equivalent of (-1) * 3^2 = -1 * 9 = -9

However, Excel chooses to recognise this as (-3)^2 = 9.

This error is particularly problematic when doing algebraic computations in
such a tool as Mathematica and then copying the result into Excel in input
form. That is,

-x^2 -> -A1^2
-> -A1^2
(Mathematica) (Mathematica with reference substituted) (Excel)

To correct the error, one must manually change it to: -(A1^2)

QUESTIONS
1: Why does Excel have this convention!
2: Is there any way to change it/make is more convenient?

I've only recently noticed this (which is quite scary to think how many
errors I may have made in the past!)

Thanks for your time,

Cheers,
Peter

 

[Myrna Larson]

Why? Because that's the way they wrote it. And no, there's no way to change
this. You must write your Excel formulas to a suit the way Excel does the
calculations.

 

[Guest]

Why? Because that's the way they wrote it.

Why is the sky blue? Because.

I was hoping for something more informative than this. e.g.
1. This convention was considered more intuitive to the majority of expected
users.
2. Computer programmers live in their own world and have their own
conventions.
3. Other...

And no, there's no way to change this.

Perhaps this should be included in the next version of Excel. Some other
options can be changed with regards to the calculations (by going to "Tools",
"Options", "Calculation"). This would be quite a useful feature.

Thanks
Peter

 

[Alan]

I've recently noticed that Excel flies in the face of standard
scientific, mathematical and engineering convention in the
calculation of powers for numbers that are then multiplied by a
negative.

The convention of mathematics, "BIMDAS" (or similar acronyms), states
that _I_ndices (or powers, or exponents), should be calculated before
_M_ultiplication. Because of this, the following is accepted as
correct:

-3^2 = -9.

This is because it is the equivalent of (-1) * 3^2 = -1 * 9 = -9

However, Excel chooses to recognise this as (-3)^2 = 9.

This error is particularly problematic when doing algebraic
computations in such a tool as Mathematica and then copying the
result into Excel in input form. That is,

-x^2 -> -A1^2
-> -A1^2
(Mathematica) (Mathematica with reference substituted)
(Excel)

To correct the error, one must manually change it to: -(A1^2)

QUESTIONS
1: Why does Excel have this convention!
2: Is there any way to change it/make is more convenient?

I've only recently noticed this (which is quite scary to think how
many errors I may have made in the past!)

Thanks for your time,

Cheers,
Peter

Hi Peter,

See this thread for a *very* comprehensive discussion on the topic:

http://groups.google.co.nz/groups?h...r=&selm=VJROc.4%24975.1%40fe2.columbus.rr.com

Or this (same thing but shorter URL):

http://tinyurl.com/5t69s


Enjoy!

Alan.

 

[Royman101]

Maybe is skewed. I was taught that -3^2=9, and not -9. The integer used
is -3, not -1*3. You may think I was taught wrong. I submit to you I was not
taught in this country, rather, I grew up in Europe. Could it be that Excel
had some European programmers....

Roy

 

[Harlan Grove]

I was hoping for something more informative than this. e.g.
1. This convention was considered more intuitive to the majority of
expected users.

....

Perhaps now, but not necessarily back when spreadsheets made their big debut
in the mid 1980s. FWLIW, this is COBOL's sign convention, and maybe it
wasn't unreasonable for Microsoft's original Excel programmers to decide
that it'd be a good idea to follow COBOL operator precedence.

Then again, Lotus 123 follows standard mathematical conventions and gives
exponentiation higher precedence than unary minus (sign change), so
Microsoft's original programmers gave Excel a different operator precedence
convention than the leading spreadsheet on the market back when they were
developing the original version of Excel. That alone makes it VERY LIKELY
this was a design screw-up, but once made it can't be unmade because it'd
break existing formulas relying on current operator precedence.

If you really believe you want to read about this, follow these archived
threads.

http://groups-beta.google.com/group/sci.math/browse_frm/thread/81214f129ec55664

http://groups-beta.google.com/group...rogramming/browse_frm/thread/74fc5978cbbd95a8

Perhaps this should be included in the next version of Excel. Some other
options can be changed with regards to the calculations (by going to
"Tools", "Options", "Calculation"). This would be quite a useful feature.

Don't count on this happening. Excel's formula parser, in which its operator
precedence is implemented, seems to be one of the oldest, most myopically
designed bits of code in all of Excel. If Microsoft hasn't made any
fundamental changes since Excel 4 (3D referencing within .XLW workbooks),
over a decade ago, why would you believe they have any inclination to fix
this any time soon?

Also, flipping operator precedence on the fly would require a different
formula parser of every possible operator precedence combination. That'd add
considerable bulk to the Excel .EXE - not a good thing.

 

[Guest]

Maybe is skewed. I was taught that -3^2=9, and not -9. The integer used

is -3, not -1*3. You may think I was taught wrong. I submit to you I was not
taught in this country, rather, I grew up in Europe. Could it be that Excel
had some European programmers....

This could have something to do with it.

However, the convention -x^2 == -(x^2) is not negotiable when you are
working with algebra. If you are dealing with real numbers, then writing -x^2
would be pointless if -x^2 really means (-x)^2 == ( -1)^2 * (x)^2 = x^2

Therefore, if the "European convention" you mention holds, then -x^2 == x^2.
(!)

So then, -x^2 should be taken as -(x^2), to mean the opposite of the square
of x, and to differentiate it from simplifying to x^2.

Cheers,
Peter

**PS**

The post that Alan mentioned (http://tinyurl.com/5t69s) is good reading on
this topic. Apparently the issue cost someone $2000 in time and effort to
resolve!

The convention as I've learnt it is that the power operator is only applied
to the argument immediately under the indice. So for instance:
2 * 3^2 = 2 * 9 = 18 AND
(2 * 3)^2 = 6^2 = 36
(2 * 3)^2 = 2^2 * 3 ^ 2 = 4 * 9 = 36

Therefore, if the negative is meant to be included in the power, it should
be included in brackets. So then:
-3^2 = -1 * 3^2 AND
(-3)^2 = 9
(-3)^2 = (-1 * 3)^2 = (-1)^2 * (3)^2 = 1 * 9 = 9

 

[Ken Russell]

-3^2 is the same as -3*-3 which equals 9 not -9

At least that's what I was taught over 60 years ago.

--
Ken Russell

(e-mail address removed)
Remove yourhat to reply by e-mail
..

 

[Guest]

But then Ken, what reason would one have for writing:

-x^2

if it really simplifies down to -x * -x = x^2?

How would one write the opposite of the square of x?

Cheers,
Peter

 

[Ken Russell]

Doesn't -(x^2) satisfy -9?

--
Ken Russell

(e-mail address removed)
Remove yourhat to reply by e-mail
..

 

[Guest]

Doesn't -(x^2) satisfy -9?

Yes, according to your convention, you would have to write -(x^2). So
consider the two systems we could have:

1. x^2 the opposite of which is -x^2
2. x^2 the opposite of which is -(x^2)

System number one (which, in my experience, is standard in maths, science
and engineering textbooks) makes more sense to me. Indices or powers have
precedence over multiplication.

Consider graphs of parabolas. According to your definition, the graphs:

y = x^2

AND

y = -x^2

would be the same graph. However, in my experience, eveyone knows that y =
x^2 is a "U" shape while y = -x^2 is an "upsidedown-U" shape.

Would you agree?

Cheers,
Peter

 

[Ken Russell]

I am a simple man who bows to your superior knowledge. At least all my
maths teachers are dead now

Cheers,
Ken Russell

(e-mail address removed)
Remove yourhat to reply by e-mail
..

 

[Guest]

Ken, I am in immense respect of you! I've just read the entire topic on this
bug on the google newsgroup... no-one really listening, just arguing, ego's
flying etc. Your humilty and gentleness is very pleasantly refreshing!

Cheers to you!

 

[Dana DeLouis]

I think the consensus is that Excel is wrong in this calculation, and most
likely will never be fixed. I think the problem is that Excel can not look
ahead in its interpretation of the equation to see that ^ would come first,
and then take the negative of this number. I agree with Harlan as it
appears Excel can only read Left to Right, and that's it. A program like
Mathematica can read the whole expression correctly.

For a small demo, Excel does the following left to right only...
=4^3^2
4096

But Mathematica will do this correctly as 4^(3^2)
4^3^2
262144

As you can see, Excel just can't look ahead to do it properly.

I've never liked the help file explanation on Operator precedence.
"Negation (as in -1)" Negation I think usually means True / False. I think
we have to guess that what they mean is that it will flip the sign bit of
the number if this is what's seen first (and disregard anything later as in
^). My thoughts are this is not a very good explanation.

http://mathworld.wolfram.com/Negation.html

Just some other thoughts. A nice feature of Excel though is its ability to
interpret text as numbers where appropriate. This is a "nice" feature for
Excel, but not for a math program. For example, if A1 had the text '5

You could use = - - A1 to get the number 5.
Same with =A1+0

Of course, in Mathematica you could not add the text "5" and zero to get 5.
And the - - A1 is the PreDecrement operator, so this would not make sense in
Mma.

If A1 held the number 5, in Excel, you could have a formula like:
=A1--------2
7

This would not make sense in Mathematica.

You may want to look at the "InputForm" of Mma equation and use the same
"Power" function with Excel. Using Excel's Power function is a good way to
make it clear what you are doing. I have experimented with putting a "Hold"
around the equation (via "HoldForm") and work with the Power pattern, but
never had much luck with this approach myself. Once you put a Release on
the hold, the equation will simplify again. I do have a custom //Vba
function that transforms the output into the format for Excel's vba, but it
doesn't cover everything.

For a simple demo. If you were not sure, and want to enter =4^3^2 in Excel,
you may want to take a look at how this is done, and use the power function
in Excel.

FullForm[HoldForm[4^3^2]]

Power[4,Power[3,2]

I would take the hint and do it like this in Excel
=POWER(4,POWER(3,2))

And for those interested ...
FullForm[HoldForm[-5^2]]
Times[-1, Power[5, 2]]

 

[Harlan Grove]

Royman101 wrote...

Maybe is skewed. I was taught that -3^2=9, and not -9. The integer used
is -3, not -1*3. You may think I was taught wrong. I submit to you I was not
taught in this country, rather, I grew up in Europe. Could it be that Excel
had some European programmers....

I have a few German language math texts that show equations that
clearly show they follow exponentiation before negation operator
precedence. Perhaps it may be unreasonable to extrapolate from that
that Germany follows the same operator precedence convention as the US,
but I'd be willing to bet that was so.

 

[Harlan Grove]

Dana DeLouis wrote...

I think the consensus is that Excel is wrong in this calculation, and most
likely will never be fixed. I think the problem is that Excel can not look
ahead in its interpretation of the equation to see that ^ would come first,
and then take the negative of this number. I agree with Harlan as it
appears Excel can only read Left to Right, and that's it. A program like
Mathematica can read the whole expression correctly.

For a small demo, Excel does the following left to right only...
=4^3^2
4096

But Mathematica will do this correctly as 4^(3^2)
4^3^2
262144

As you can see, Excel just can't look ahead to do it properly.

You're misunderstanding what I wrote *AND* attributing to me things I
didn't state.

VisiCalc uses simple L-to-R evaluation: 2+3*4 returns 20.

123 uses an operator precedence hierarchy: 2+3*4 returns 14. It gives ^
higher precedence than unary -: -3^2 returns -9.

Excel uses an operator precedence hierarchy: 2+3*4 returns 14. However,
it gives ^ *lower* precedence than unary -: -3^2 returns 9.

With regard to 4^3^2, it has nothing to do with operator *precedence* -
it's operator *associativity*. FWIW, most programming languages (other
than VB[?] and oddballs like APL and its descendants) provide R-to-L
associativity for exponentiation and L-to-R associativity for +-*/ (+
and * are associative for integers, rational, algebraic, real and
complex numbers, but not always so for binary floating point
'numbers').

Precedence determines evaluation order of *DIFFERENT* operators.
Associativity determines evaluation order of the *SAME* operator
applied multiple times in sequence.

I've never liked the help file explanation on Operator precedence.
"Negation (as in -1)" Negation I think usually means True / False. I think
we have to guess that what they mean is that it will flip the sign bit of
the number if this is what's seen first (and disregard anything later as in
^). My thoughts are this is not a very good explanation.

....

Yeah, yeah. Agreed, but in colloquial discussions it's too much of a
PITA to say additive inverse. Sign change may be an alternative.

Of course, in Mathematica you could not add the text "5" and zero to get 5.
And the - - A1 is the PreDecrement operator, so this would not make sense in
Mma.

....

Nor in C or all the other languages it's spawned. So use -(-A1).

And for those interested ...
FullForm[HoldForm[-5^2]]
Times[-1, Power[5, 2]]

And there are some like me who consider this to be a really stupid
approach due to its circular nature. Sign and exponentiation together
only make sense in rings, rings are necessarily additive groups,
additive groups have well-defined additive inverses, and

AdditiveInverse(MultiplicativeIdentity) * x = AdditiveInverse(x)

is a derived truth that necessarily relies upon additive inverse to
provide -1. So why not express -5^2 as -(5^2) or perhaps

ChangeSign[Power[5, 2]]

?

This entire problem is due to the laziness of mathematicians in
previous centuries who used the same character/token/sign to express
numeric sign, sign change and subtraction. An alternative convention
might have been to interpret a dash *not* immediately after a complete
expression but immediately before a literal number [e.g., x*-3^2 == x *
((-3) ^ 2)] as part of the number (so, technically, not subject to
operator precedence because it wouldn't be an operator), but between
incomplete expressions with the expression to the right *not* a literal
number treat it as a sign change operator with standard precedence
[e.g., x*-y^2 == x * (-(y^2))]. One argument against this convention is
that you'd need to remember that literal numbers and variables would be
treated differently, e.g., y = 3, -3^2 wouldn't equal -y^2.

Computers wouldn't have a problem with this now. Lexical analysis
precedes syntactic parsing, so just need to include leading - and + as
part of literal number tokens when they clearly couldn't be dyadic
operators. However that caveat implies the presence of noncapturing
assertions in the lexical analyzer, and those weren't part of most
regular expression packages until the mid 1980s.

 

[Dana DeLouis]

Thanks Harlan. I wish there were more of these discussions. I was trying
to come up with another example for the op to show that Excel is unable to
do a R-to-L "associative" operation. For the op, I guess the best one can
offer is just be aware of Excel's limitations, and try to anticipate the
differences between the two programs.

(A little off topic I know) For the op, a slightly different point from
what was mentioned is that in Mma, the Power function does not have the
"Associative property", as it does not have the Flat attribute. Flat
corresponds to the mathematical property of associativity.
Functions like Times & Plus are "Flat", and associative. Therefore...

Power[4, 3, 2]
262144

Attributes[Power]
{Listable, NumericFunction, OneIdentity, Protected}

Attributes[Times]
{Flat, ..., OneIdentity, Orderless, ....}

Again, just be aware of the differences...
--
Dana DeLouis
Win XP & Office 2003

Dana DeLouis wrote...

I think the consensus is that Excel is wrong in this calculation, and most
likely will never be fixed. I think the problem is that Excel can not look
ahead in its interpretation of the equation to see that ^ would come first,
and then take the negative of this number. I agree with Harlan as it
appears Excel can only read Left to Right, and that's it. A program like
Mathematica can read the whole expression correctly.

For a small demo, Excel does the following left to right only...
=4^3^2
4096

But Mathematica will do this correctly as 4^(3^2)
4^3^2
262144

As you can see, Excel just can't look ahead to do it properly.

You're misunderstanding what I wrote *AND* attributing to me things I
didn't state.

VisiCalc uses simple L-to-R evaluation: 2+3*4 returns 20.

123 uses an operator precedence hierarchy: 2+3*4 returns 14. It gives ^
higher precedence than unary -: -3^2 returns -9.

Excel uses an operator precedence hierarchy: 2+3*4 returns 14. However,
it gives ^ *lower* precedence than unary -: -3^2 returns 9.

With regard to 4^3^2, it has nothing to do with operator *precedence* -
it's operator *associativity*. FWIW, most programming languages (other
than VB[?] and oddballs like APL and its descendants) provide R-to-L
associativity for exponentiation and L-to-R associativity for +-*/ (+
and * are associative for integers, rational, algebraic, real and
complex numbers, but not always so for binary floating point
'numbers').

Precedence determines evaluation order of *DIFFERENT* operators.
Associativity determines evaluation order of the *SAME* operator
applied multiple times in sequence.

I've never liked the help file explanation on Operator precedence.
"Negation (as in -1)" Negation I think usually means True / False. I think
we have to guess that what they mean is that it will flip the sign bit of
the number if this is what's seen first (and disregard anything later as in
^). My thoughts are this is not a very good explanation.

...

Yeah, yeah. Agreed, but in colloquial discussions it's too much of a
PITA to say additive inverse. Sign change may be an alternative.

Of course, in Mathematica you could not add the text "5" and zero to get 5.
And the - - A1 is the PreDecrement operator, so this would not make sense in
Mma.

...

Nor in C or all the other languages it's spawned. So use -(-A1).

And for those interested ...
FullForm[HoldForm[-5^2]]
Times[-1, Power[5, 2]]

And there are some like me who consider this to be a really stupid
approach due to its circular nature. Sign and exponentiation together
only make sense in rings, rings are necessarily additive groups,
additive groups have well-defined additive inverses, and

AdditiveInverse(MultiplicativeIdentity) * x = AdditiveInverse(x)

is a derived truth that necessarily relies upon additive inverse to
provide -1. So why not express -5^2 as -(5^2) or perhaps

ChangeSign[Power[5, 2]]

?

This entire problem is due to the laziness of mathematicians in
previous centuries who used the same character/token/sign to express
numeric sign, sign change and subtraction. An alternative convention
might have been to interpret a dash *not* immediately after a complete
expression but immediately before a literal number [e.g., x*-3^2 == x *
((-3) ^ 2)] as part of the number (so, technically, not subject to
operator precedence because it wouldn't be an operator), but between
incomplete expressions with the expression to the right *not* a literal
number treat it as a sign change operator with standard precedence
[e.g., x*-y^2 == x * (-(y^2))]. One argument against this convention is
that you'd need to remember that literal numbers and variables would be
treated differently, e.g., y = 3, -3^2 wouldn't equal -y^2.

Computers wouldn't have a problem with this now. Lexical analysis
precedes syntactic parsing, so just need to include leading - and + as
part of literal number tokens when they clearly couldn't be dyadic
operators. However that caveat implies the presence of noncapturing
assertions in the lexical analyzer, and those weren't part of most
regular expression packages until the mid 1980s.

 

[Royman101]

This could have something to do with it.

However, the convention -x^2 == -(x^2) is not negotiable when you are
working with algebra. If you are dealing with real numbers, then writing -x^2
would be pointless if -x^2 really means (-x)^2 == ( -1)^2 * (x)^2 = x^2

Therefore, if the "European convention" you mention holds, then -x^2 == x^2.
(!)

I'll add my comment - Yes, this is correct, and this is what I was taught
(many moons ago). However, I see you point as well and find myself more
agreeable with it than my own 'paradigm', if you will. Thanks!

 

[Guest]

Yes, obviously I should have been more careful transferring Mathematica
formuals into Excel. I though that changing the Mma output into "Input Form"
would have been safe enough - obviously not. Otherwise one must enter the
formulas by hand - not a fun task.

I nice trick I've discovered is to make the cell-reference substitutions in
Mathematica (e.g. a -> G3, b -> G4...). This avoids doing it by hand as well.

 

[Myrna Larson]

Who would would possibly know why they did it this way? What's the point of
wasting time speculating about WHY?

You have a problem to solve. I would think your time would be better spent
on that aspect. Fundamentally, you need to add parentheses to an expression
like -X^2 so it becomes -(X^2).

It is VERY unlikely this will be changed. It could "break" existing
spreadsheet formulas that have been written to accommodate Excel's
calculation order.