Wikipedia:Reference desk/Archives/Mathematics/2015 January 28

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January 28

typical function

In entire function, it says "According to J. E. Littlewood, the Weierstrass sigma function is a 'typical' entire function" Can anyone provide a formal citable reference for this? Thanks, Robinh (talk) 02:25, 28 January 2015 (UTC)

Precedence of unary negation (unary -)

I know unary negation has a lower precedence than exponentation, but where exactly should it be in the order of operations?
Using this (http://mathworld.wolfram.com/Precedence.html) as a guide, where should negation be on the list exactly?
Also, is there a name given to "unary +"? Is there an antonym for "negation"?

Note that I'm not interested in applied mathematics (such as programming / computer science, etc). I'm interested in pure mathematics only and what the precedence rules are in mathematics.

Thanks.
The Transcendent One 08:25, 28 January 2015 (UTC)

It definitely must be before addition and subtraction, so that $-1+2$ reads $(-1)+2=1$, otherwise it would be $-(1+2)=-3$. It can be either before or after multiplication and division, as $(-a)b=-(ab)$ so the precedence change does not change calculation results. --CiaPan (talk) 08:47, 28 January 2015 (UTC)

According to Order_of_operations#The_standard_order_of_operations, there is no universal standard. But it's commonly considered equivalent to multiplication, since it is in fact multiplication by -1 (and thus, as CiaPan notes, its place relative to multiplication doesn't matter). Mentally, I at least think of it as a lower priority than multiplication - so I read $-ab$ as $-(ab)$ rather than $(-a)b$. -- Meni Rosenfeld (talk) 09:55, 28 January 2015 (UTC)

That's right. However the 'unary minus' may be defined as 'multiplying by negative unit' in introduction to algebra for beginners only. Formally it is a symbol of an additive inverse, which is defined based on addition ('−x' is such element of a group, which added to x results in 0), then serves as a base to define subtraction: x−y = x+(−y). Then −ab is an inverse element of ab, just as −(ab) notation suggests. --CiaPan (talk) 11:37, 28 January 2015 (UTC)

So, entry number 4 on the list at MathWorld should be "4. Multiplication and division and negation"?

Also, is the expression ${-n}!$ valid without brackets for $n\geq 0$, but must be written with brackets as ${(-n)}!$ for $n<0$?

Also, how do I resolve the expression ${-a}\times {-b}$? Are brackets needed anywhere? ${(-a)}\times {(-b)}$ can be used to force negation to be done first, but I don't think multiplication can be done before the negation of the b term in ${-(a}\times {-b)}$.
Furthermore, does the "left to right" rule in precedence apply when negation is considered with other operations? For example, ${a}\div {b}\times {c}\quad =\quad {(a}\div {b)}\times {c}\quad \neq \quad {a}\div {(b}\times {c)}$.
The Transcendent One 11:07, 28 January 2015 (UTC)

It might be useful to rephrase CiaPan: there are three distinct ways to use the − sign, and these are frequently confounded in everyday use. First, we use the sign to indicate the additive inverse of a positive member of the field of real numbers; in this case, the sign is not being used as a function, but merely as part of a sign denoting a particular real number. Second, we use the sign to indicate the unary function that maps a real number to its additive inverse; this is how (for example) the "CHS" ("change sign") key works on my calculator, since I can apply it at any time in exactly the same way as I might apply any other unary function (such as "Sin", "Cos" or "Exp"). These first two uses are almost invariably confounded in normal mathematical notation. Third, we use the sign to indicate a binary function which, as CiaPan points out, is x−y = x+(−y). Interestingly, not all notational systems permit this confusion. One well-known system that does not is Iverson Notation and the related APL_(programming_language), which uses "¯" to indicate a negative number, and "−" to indicate unary negation. All of this suggests one thing fairly clearly: where the − sign is used specifically in denoting a member of the field of real numbers (that is, forms part of the sign itself) it must take the highest precedence, because it is actually part of the sign indicating the number under consideration, and it would be meaningless to do otherwise. RomanSpa (talk) 13:22, 28 January 2015 (UTC)

Yes, I'd put it at priority zero on the MathWorld list, definitely not with multiplication and division except as explained by Meni above for the most convenient interpretation of -ab, and before exponentation where there is an unwritten bracket. Dbfirs 08:51, 30 January 2015 (UTC)

Aside from the interpretation being a little flexible (e.g. −a × b and −ab might be interpreted as (−a)b and −(ab) respectively), the precedence of unary minus and exponentiation depends on the order. Thus, −a^b = −(a^b), but a^−b = a^(−b). In general, mathematical notation does not have universal rules, making it unambiguous. —Quondum 17:19, 30 January 2015 (UTC)

a^−b means a^(−b) because there's nothing else it could mean, not because of any precedence rule. a^b+c = a^(b+c) too, but no one would say that + has higher precedence than exponentiation. In programming languages that have an infix exponentiation operator (often **), a**b+c means (a**b)+c. -- BenRG (talk) 05:11, 31 January 2015 (UTC)

There's no left-to-right rule in algebraic notation. x + y + z just means x + y + z. It doesn't matter how you parenthesize it because addition is associative. x − y + z is a shorthand for x + (−y) + z, and again the addition is associative. The ÷ symbol is not used in mathematics. Division may be represented by a / symbol, but that is a typographical variant of the horizontal division line, not a proper infix operator, and its interpretation depends on context. 1/xy definitely means ${\frac {1}{xy}}$, not ${\frac {1}{x}}y$, but 1/x·y could mean either one.

Also, don't confuse operator precedence with the order of operations. Operator precedence is a real thing, whereas the order of operations exists only in the minds of grade school teachers. When evaluating 1×2+3+4, it's fine to start by evaluating 3+4. You can't start with 2+3, but that's not because "multiplication comes first"; it's because the expression really means (1×2)+3+4, and 2)+3 is not a subexpression. -- BenRG (talk) 05:11, 31 January 2015 (UTC)