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In college, I had been taught the famous formula $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ Here $\pm$ means that I choose either $+$ or $-$. But I have seen sometimes in physics that $\pm$ can mean some interval. If $x=5 \text{ m}\pm 0.05\text{ m}$ then $4.95\text{ m}\leq x\leq 5.05\text{ m}$. It looks like there are two different definitions for a symbol $\pm$. On the other hand, I have seen that physicist are not always as rigorous in maths as mathematicians. So, my question is that is there just one definition for the symbol $\pm$ and physicist uses non-standard definition or two (maybe more) different definitions?

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I think you're right. It has two different meanings depending on the context. It can either mean "either plus or minus" in mathematics (which is a concise way to write that there are two possibilities), or "give or take" in physics (which yields a range of possibilities). –  Joel Cohen Oct 10 '12 at 20:37
I also skimmed document given in the link en.wikipedia.org/wiki/ISO/IEC_80000 but I didn't find the definition. –  Student Oct 10 '12 at 20:41
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5 Answers

up vote 7 down vote accepted

There are simply two different meanings for this symbol.

In mathematics

It is generally used as shorthand, either to indicate the presence of two possible values:

$$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

or to condense two equations into one:

$$\sin(a\pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)$$

Note that it is sometimes used in tandem with the flipped plus/minus sign, as in

$$\cos(a\pm b) = \cos (a) \cos (b) \mp \sin (a) \sin (b)$$

in which $\mp$ and $\pm$ should be interpreted as having different signs.

In experimental sciences

It is used to indicate precision of a measurement. The interpretation can be a bit fuzzy. For example, the expression $X = 10\pm 1.5$ might mean

$$8.5 \leq X \leq 11.5$$

or it might mean that a particular confidence interval (for example, the 95% confidence interval) for $X$ is $[8.5, 11.5]$.

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There is no mathematical definition of the $\pm$ symbol. However, it is a possibility of writing two or more equations into one easily memorizable form.

1) In the formula above it is an abbreviation for the two solutions of a quadratic equation.

2) As you stated in physics it may also be used to define a symmetric interval.

3) In statistics it is also used to describe the distribution of a random variable:

$x = 4 \pm 2.5$

would tell the reader that $x$ is a random variable with mean 4 and standard deviation 2.5 . However, there is no implicit assumption that the variable is Gaussian distributed.

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(assuming $b\neq0$)

In Mathematics $x=a\pm b$ indicates a spectrum of two point-solutions to an equation, $$\text{either}\quad x=a+b,\\\text{or}\quad x=a-b$$ that is, $$x\in\{a+b,a-b\}$$ spectra

In Physics $x=a\pm b$ refers to a value somewhere in the interval between the two boundary points $$x\geq a-b\quad\text{and}\quad x\leq a+b$$ or $$x\in[a-b,a+b]$$ enter image description here

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(-1) "either $x = a + b$ or $x = a - b$" is not the same as "$x = \{a + b, a - b\}$", and the same for the second example. –  TMM Jan 5 '13 at 16:26
@UnkleRhaukus You should change the $=$ (equals) to a $\in$ (is contained in). I've just done this for you (also, +1 for the pictures). –  Chris Taylor Jan 5 '13 at 20:02
thanks @Chris Taylor. –  Elements in Space Jan 6 '13 at 7:40
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I would say that in math, we use $\pm$ in a way that means "plus or minus", hence the quadratic formula with this notation.

In Physics, it is more interpreted as "more or less", which brings the interval signification.

As long as the context is clear it should not be misleading to understand what one means

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Another way $\pm$ is used is when there is an expression with multiple terms that can be either positive of negative. If both "$\pm$" and "$\mp$" are used, it often means that all the terms with "$\pm$" have the same sign and all the with "$\mp$" have the opposite sign.

For example, the expression "$a\pm b \mp c \pm d \mp e$" would mean that $b$ and $d$ would have the same sign (either "+" or "-"), and $c$ and $e$ would have the opposite sign.

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