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I want to express the sign of $a$ multiplied by $b$ in a formula but do not know how to mathematically express the sign of $a$?

I am not sure but using the plus-minus sign isn't correct, is it?

In latex style I would write:

$\pm a \times b$

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Sign function –  Henning Makholm Dec 16 '12 at 14:53

2 Answers 2

up vote 1 down vote accepted

Supposing $a,b\neq 0$, the sign of $ab$ is $$\text{sgn}(ab)=\frac{\left|ab\right|}{ab}$$ The sign of $a$ is $$\text{sgn}(a)=\frac{\left|a\right|}{a}$$ and so the [sign of $a$] times $b$ is nothing but $$\frac{\left|a\right|}{a}b$$

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What about $a= 0$? Is there convention for $\mathrm{sgn}(0)$? –  user50554 Dec 16 '12 at 14:56
    
We define $\text{sgn}(0)=0$ –  Nameless Dec 16 '12 at 14:57
    
So $\mathrm{sgn}(a) :=\left\{\begin{array}{cc} \frac{|a|}{a} & \text{if } a \neq 0 \\ 0 & \text{if } a = 0\end{array}\right. $ –  user50554 Dec 16 '12 at 14:59
    
@user50554 Yes it is –  Nameless Dec 16 '12 at 14:59
    
Many thanks for this! Would sgn(a) = a / |a| also be correct or is it really defined as |a| / a ? –  Johnny Dec 16 '12 at 15:16

The correct way to formalize the sign of $a$ multiplied by $b$ is $$ \mathrm{sgn} (a) \cdot b.$$

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