How should I evaluate 2*-2^3?

Which one of these two is the correct one?

2*((-2)^3)

2*(-(2^3))

I was wondering what was the correct operator precedence. Everyone was taught at school to evaluate expressions this way:

1. First exponentiation and roots
   


2. Then multiplications and divisions
   


3. Finally additions and subtractions
   

However, for more complex operations, this table is incomplete. For instance, which precedence do unary operators have regarding to the others?

On the one hand, we can see that unary operators have lower precedence than the exponentiation in many cases.

-2^2 = -(2^2) = -4 (exponentiation first)

On the other hand, we sometimes use it differently (unary operators take precedence over exponentiation):

2^-2 = 2^(-2) = 1/4 = 0.25 (unary minus first)

I am a bit confused about this, so these are my questions:

1. Which would be the general correct rule for this kind of operations? I know this ambiguous problem can easily be solved putting brackets around the unary operators (but many programming languages perform those operations without needing them). Because of this, I think there should be a rule or an international standard.
   



2. 
   

   How should I evaluate the following expressions?

   
   
   

   
   

   2*-2^3 <-- the most important
   
   2^-3*4
   
   2*-3*4
   
   -2^-3

   
   

   
   

In order to research how modern calculators behave, I have tested 2^-3*4 in many different calculators and each one gives me a different result:

 - Google                    2^-3*4 = (2^(-3))*4 = 0.5



 - http://web2.0calc.com/    2^-3*4 = 2^(-(3*4)) ~ 0.000244140625

Not everyone was taught what you say. I was not, for example. I was never taught how to write expressions with exponents in-line, so I never found out what the canonic meaning of x^a+b actually is.

What I was taught is that whenever there is some confusion and there may exist two ways of interpreting an expression, I should use parentheses. And that is exactly what you need to start doing.

The thing is that by now, the notation has become so widely used with no central rule telling us what the only proper way of evaluation is, that it no longer makes much sense to try to impose a world-wide standard.

Taking this into consideration, the answers are:

1. There is no general correct rule for this kind of operation. Brackets are the way to go. There is no international standart.


2. The following expressions should be evaluated as "input unclear". If you get an expression like that to evaluate, ask the author of the expression to further explain what they meant.

2^-2 can only be interpreted one way, because the minus sign is next to the second argument, and the exponentiation sign isn't. It's not a matter of operator precedence.

It's only when a parameter has an operator on each side that we have to use precedence to decide. For a concrete example, FORTRAN has an exponential operator built into the language, and exponentiation has the highest precedence. Unary minus has the same precedence as binary plus and minus (with left-to-right evaluation to break ties, as e.g. -4 + 3 = (-4) + 3, not -(4+3).

The numerical results I give below are consistent with Julia language, the latest and most sophisticated language for scientific computing, created by a team of MIT team involving mathematicians. All computations are also consistent with Google. Good for Google! When there are more than one equal signs, i.e, more than three members, in an expression the computations in Julia and Google are made with the first member, which states the problem, so I am not cheating.

In regular mathematical writing,

a^b^c = a^(b^c).                [a^b^c in not equal to (a^b)^c = a^(b*c)].



Numerical example: 2^1^2 = 2^(1^2) = 2^(1) = 2.

Note that the order of computation of exponentiation operators is from right to left. [Otherwise, the answer would have been 4].

Now, let us introduce the negation operator, i.e., the unary - operator that changes the sign of the expression that follows. It can be distinguished from the binary subtraction operator because the negation operator does not have a minuend just to the left.

Rule 1: The negation operator has the same level of precedence as exponentiation.

Rule 2: At this level, the operators of Rule 1 will be solved from right to left. Examples: a^-b = a^(-b) and -c^d = -(c^d). Numerical examples: 2^-3 = 2^(-3) = 1/8 and -2^3 = -(2^3) = -8.

Following the previous two rules, you can solve all the required computations:

2 * -2^3 = 2 * (-(2^3)) = 2 * (-8) = -16



2^-3 * 4 = (2^(-3)) * 4 = 1/(2^3) * 4 = 1/8 * 4 = 1/2



2 * -3 * 4 = 2 * (-3) * 4 = -24



-2^-3 = -(2^(-3)) = -(1 / 2^3) = -1/8



2^-3^4 = 2^(-(3^4)) = 2^(-81) = 1 / 2^81 = a positive number very close to 0



4^-1^2 = 4^(-(1^2)) = 4^(-1) = 1/4

The second last example was posed in a comment. The last example is an easy-to-compute variation of the previous. [If it were correct to compute the first member of the last expression from left to right the answer would be 4^-1^2 = (4^-1)^2 = (1/4)^2 = 1/16. However this goes against the right-to-left mathematical convention to compute a^b^c reviewed above.]

As an alternative with equal results to Rules 1 and 2, one can state the following rules:

• Exponentiation has precedence over negation, as the documentation of mathematically reasonable languages typically state.


• Exponentiation operators are computed from right to left.


• Negation operators are computed from right to left.


• (following @TonyK's answer) The consecutive operators in a^-b can only be interpreted one way, because the minus sign is next to b, and the exponentiation sign isn't. Therefore, operators precedence need not be considered.

Thanks to @BrainOverflow for telling us about Google. It uses the right mathematical conventions unlike Matlab and Excel for a^b^c, and unlike Excel for -a^(2*n), where n is an integer.

I know that you have accepted an answer but I am tempted to add this anyway.

The notation in your question is more typical of computing rather than mathematics. Multiplication is rarely represented by * in mathematics and exponentiation is rarely represented by ^. Your 2*-2^3 would be more likely to be written as $2 (-2)^3$. Parentheses are required less often with traditional mathematical notation. E.g. 2 ^ x + y could be written as $2^x + y$ or $2^{x + y}$ both of which are fairly unambiguous without parentheses. Similarly, $a + b / c + d$ could be written as $a + frac{b}{c} + d$ or $frac{a + b}{c + d}$.

However, traditional mathematical notation is not as perfect and precise as many might expect. Here are some oddities:

It is quite well agreed that $mathbb{N}$ represents the natural numbers but less well agreed whether they include $0$.

You occasional see formulae in which $e$ is an arbitrary value (probably along with $a$, $b$ ,$c$, and $d$) but also used to represent the exponential function. Similarly $pi$ might be the usual famous number in one place but the prime counting function in another.

$sin^2(x)$ almost always means $(sin(x))^2$ rather than $sin(sin(x))$ yet $sin^{-1}(x)$ almost always means $arcsin(x)$ rather than $(sin(x))^{-1}$.

Mathematics questions will be more welcome here in traditional mathematical notation using MathJax. Computing questions are probably best in a computing group.