What is the meaning of ± in mathematical equations?

[Davewilliams]

I was on the bus home from a hockey match and my maths teacher asked what is the the square root of 9.
he said it was ±3 but i thought it was only +3
he put forth
-3 x-3 = 9 and 3x3 =9 so it was both so 9^1/2 = ±3

i said that 3^2=9 then 3=9^1/2 so 3=3 and not 3=-3 so 3 ≠±3

doing it algebraically
a^2=b
a=b^1/2 but instead i would write a=±(b^1/2)

so if the square root is already plus and minus whey have the second plus and minus?

 

[pasmith]

I was on the bus home from a hockey match and my maths teacher asked what is the the square root of 9.
he said it was ±3 but i thought it was only +3
he put forth
-3 x-3 = 9 and 3x3 =9 so it was both so 9^1/2 = ±3

i said that 3^2=9 then 3=9^1/2 so 3=3 and not 3=-3 so 3 ≠±3

doing it algebraically
a^2=b
a=b^1/2 but instead i would write a=±(b^1/2)

so if the square root is already plus and minus whey have the second plus and minus?

Every strictly positive number has two square roots, so the phrase "the square root" is ambiguous. By convention it is taken to mean only the positive root. Thus $\sqrt 9 = 9^{1/2} = 3$, and the square roots of 9 are 3 and -3.

 

[SteveL27]

Your teacher is sadly wrong. The square root of a positive real number x is the larger of the two numbers a with a^2 = x. That's a convention that allows us to make sqrt into a function.

 

[tiny-tim]

Hi Davewiliams! Welcome to PF!

Your teacher is sadly wrong. The square root of a positive real number x is the larger of the two numbers a with a^2 = x. That's a convention that allows us to make sqrt into a function.

Absolutely.

 

[Mark44]

I was on the bus home from a hockey match and my maths teacher asked what is the the square root of 9.

If the question was "what are the square roots of 9", the answer would be ±3.

he said it was ±3 but i thought it was only +3
he put forth
-3 x-3 = 9 and 3x3 =9 so it was both so 9^1/2 = ±3

i said that 3^2=9 then 3=9^1/2 so 3=3 and not 3=-3 so 3 ≠±3

doing it algebraically
a^2=b
a=b^1/2 but instead i would write a=±(b^1/2)

so if the square root is already plus and minus whey have the second plus and minus?

Based solely on the title of this thread, "what is the 9^1/2 equal to?", no one here has given the answer to that question.
As you wrote it, 9^1/2 = 9/2 = 4.5.
Since you undoubtedly meant the 1/2 power of 9, it should have been written as 9^(1/2). The exponent operation has a higher precedence than the division, so 9^1/2 is the same as (9^1)/2.

 

[1MileCrash]

Mark beat me to it. Remember to be careful with formatting. 9^1/2 can only be reasonably interpreted as 4.5.

As for the intended question, √9 is 3, and only 3. Yes, 9 has two square roots, but √9 is a notation and that notation refers to the positive root of 9.

 

[Mark44]

doing it algebraically
a^2=b
a=b^1/2 but instead i would write a=±(b^1/2)

I should add that although you used parentheses (which is commendable) in the expression ±(b^1/2), they're not in the right place. The solutions to the equation a² = b should be written as a = ±b^(1/2). A few sticklers might even balk at this, and insist it should be ±(b^(1/2)).

 

[stevmg]

Ok, 9^(1/2) = 3

 

[olivermsun]

If the question was "what are the square roots of 9", the answer would be ±3.
Based solely on the title of this thread, "what is the 9^1/2 equal to?", no one here has given the answer to that question.
As you wrote it, 9^1/2 = 9/2 = 4.5.

Since you undoubtedly meant the 1/2 power of 9, it should have been written as 9^(1/2). The exponent operation has a higher precedence than the division, so 9^1/2 is the same as (9^1)/2.

LOL, another one of these notation-convention arguments... I've always wondered: if it's clear what a writer means by some vaguely ambiguous expression, then why do some people insist on parsing it as something different but responding to what everyone knows is the intended meaning anyway?

P.S.: Wolfram Alpha interprets that string of characters, as written, as 9^1/2 = √9 = 3.

 

[Mark44]

LOL, another one of these notation-convention arguments... I've always wondered: if it's clear what a writer means by some vaguely ambiguous expression, then why do some people insist on parsing it as something different but responding to what everyone knows is the intended meaning anyway?

Part of my bias comes from my professional experience at writing computer code. Although computers execute code blindingly fast, they are also extremely narrow in their interpretation of that code, and will do exactly what you tell them to do, not necessarily what you mean for them to do.

Writing mathematical expressions in an online forum is more difficult than communicating them on a piece of paper or blackboard/whiteboard. You can write something like this on paper:
$$\frac{x + 2}{y + 3}$$
The meaning is clear and no parentheses are needed, due to the two-dimensional nature of what we're writing on.
OTOH, when a new member here writes x + 2/y + 3, thinking this is the same as the above, we as helpers have to interpret what is written. Do we gear our help to what he wrote or to what we think he meant? I have seen countless posts by naive new members where the people here weren't sure what was meant.

Writing 9^1/2 falls in the same category as the fractional expression above. With an online help system, it is harder to discern what was meant from what was written. If the goal of a person posting here is to get help with his or her problem, an important consideration is to communicate the problem as clearly as possible. In part, that means to write unambiguously, using the universally accepted conventions for the order of operations.

P.S.: Wolfram Alpha interprets that string of characters, as written, as 9^1/2 = √9 = 3.

 

[Tobias Funke]

I agree with the others who say he's wrong, but there are different levels of wrong and I don't think this is really a big deal. He's saying "the" square root and then not following the convention of choosing the non-negative one, choosing instead to use a multivalued function. I believe I have a complex analysis book that defines a^(m/n) as the set of all z such that z^(n/m)=a, or something like that. In some cases I think multivalued functions are more elegant. Anyway, I wouldn't go around telling other students that the teacher is wrong and making him out to be an idiot (not that you would) over something relatively unimportant like this. Now, if he divides by 0 and "proves" that 0=1 while claiming it's valid, as I've unfortunately seen teachers do, then he might be fair game...

As for the notation, if anyone sees 9^1/2=3 and thinks "you fool! 9/2=4.5", they've got bigger issues than notation

But I still think it's a good idea to point out the technically incorrect notation for the reasons Mark44 mentioned. Not all computers will be able to figure it out.

 

[glappkaeft]

P.S.: Wolfram Alpha interprets that string of characters, as written, as 9^1/2 = √9 = 3.

The weird part is that that violates not only the standard precedence rules for that notation but also Wolfram Alphas own precedence rules which clearly states that exponentiation is done prior to division.

 

[olivermsun]

I agree with all the statements above that 9^1/2 is an ambiguous way to write what seems likely to be 9^1/2, and this problem should be pointed out to the poster. I just don't agree that this expression unambiguously means (9^1) / 2, even to a computer.

I cited Wolfram Alpha because you can usually find a counterexample for this sort of thing if you try 2-3 online calculators! I don't think this is even necessarily a violation of order of operations for WA — instead, the parser seems to assume that "glued" divisions (without spaces) denote fractions like those written as in Mark44's post above. By contrast if you type in 9 ^ 1 / 2 with spaces throughout, it's read as $\dfrac{9^1}{2}$, which I think is what most of us would expect.

So yes, the "better part of valor," either for asking questions (or for turning in homework!) is to make things unambiguous, and here that means typing 9^(1/2) or using the handy typesetting functions of this forum: $9^{1/2}$.

 

[pwsnafu]

P.S.: Wolfram Alpha interprets that string of characters, as written, as 9^1/2 = √9 = 3.

Alpha is natural language interpreter, which is why it does this.
But using CAS as a citation is useless. Mathematica treats "xy" as a variable xy. If you want to multiply x and y you need to input "x y" with the space. They use spaces as context markers. It's a different environment than standard mathematics, which doesn't use spacing.

He's saying "the" square root and then not following the convention of choosing the non-negative one, choosing instead to use a multivalued function. I believe I have a complex analysis book that defines a^(m/n) as the set of all z such that z^(n/m)=a, or something like that. In some cases I think multivalued functions are more elegant.

Yes, in complex analysis $z^{1/n}$ is multivalued. But this is dealing with non-negative reals where it isn't multivalued.

 

[Tobias Funke]

Yes, in complex analysis $z^{1/n}$ is multivalued. But this is dealing with non-negative reals where it isn't multivalued.

Non-negative reals are still complex numbers though, and according to that book's (2nd. ed of Saff and Snyder's long-titled book) definition, 9^(1/2)=+-3. They make the distinction between sqrt(9) and 9^(1/2), the latter being multivalued. I kind of like that approach.

 

[Victor II]

The following is the case:

sqrt(9) = +3, -3,

because if 3^2 = 9 and (-3)^2 = 9.

 

[pwsnafu]

The following is the case:

sqrt(9) = +3, -3,

because if 3^2 = 9 and (-3)^2 = 9.

No. As has been said earlier in the thread, $\sqrt{9} = 3$. That is not the same thing as a finding all x such that $x^2 = 9$. Why do you think the quadratic formula has the $\pm$ sign?

 

[1MileCrash]

LOL, another one of these notation-convention arguments... I've always wondered: if it's clear what a writer means by some vaguely ambiguous expression, then why do some people insist on parsing it as something different but responding to what everyone knows is the intended meaning anyway?

P.S.: Wolfram Alpha interprets that string of characters, as written, as 9^1/2 = √9 = 3.

Please explain to me how I am to interpret 9^1/2 as √9 in terms of operation precedence.

I will send a bug report to Wolfram.

 

[1MileCrash]

Why do you think the quadratic formula has the $\pm$ sign?

Clearly because the quadratic formula must account for adding the positive root, subtracting the negative root, adding the negative root, and subtracting the positive root.

 

[lurflurf]

Please explain to me how I am to interpret 9^1/2 as √9 in terms of operation precedence.

I will send a bug report to Wolfram.

That is a feature not a bug. Clearly a^1/n should be interpreted as a^(1/n) because (a^1)/n would be better written as a/n. On the other hand a^b/n is interpreted as (a^b)/n when b!=1.

 

[pwsnafu]

That is a feature not a bug. Clearly a^1/n should be interpreted as a^(1/n) because (a^1)/n would be better written as a/n. On the other hand a^b/n is interpreted as (a^b)/n.

Pretty much. One of devs gave a presentation at my uni, and as I understand it, the computation engine is nothing more than the Mathematica kernel. Unlike Mathematica however Alpha was designed to be used by non-scientists, hence they focused development on natural language interpretation.

 

[olivermsun]

Alpha is natural language interpreter, which is why it does this.
But using CAS as a citation is useless. Mathematica treats "xy" as a variable xy. If you want to multiply x and y you need to input "x y" with the space. They use spaces as context markers. It's a different environment than standard mathematics, which doesn't use spacing.

As far as I know, the '^' for exponentiation is not used in "standard mathematics"; instead it's just a way to enter mathematical expressions into computers. The issue is about how the string of characters containing '^' is processed into a "standard mathematics" expression. From there on, the usual order of operations appears to be preserved.

It's interesting that there are different behaviors for 9^1/2 and 9^3/2 (as lurflurf points out).

 

[AlephZero]

It's interesting that there are different behaviors for 9^1/2 and 9^3/2 (as lurflurf points out).

This type of input is called DWIM (do what I mean), in contrast with the derogatory interpretation of WYSIWYG as "what you see is all you get".

Of course the problem with DWIM is that different people sometimes have different ideas about "meaning". Ideally, there should be a way to get the computer system to tell you exactly how it interpreted your input, e.g. by adding parentheses or whatever.

 

[olivermsun]

One good thing about Alpha is that it shows you "what it thought you meant" (in "pretty" form) as an intermediate step.

 

[glappkaeft]

Of course the problem with DWIM is that different people sometimes have different ideas about "meaning". Ideally, there should be a way to get the computer system to tell you exactly how it interpreted your input, e.g. by adding parentheses or whatever.

On of my favourite programming books says something like this (I'm paraphrasing):
"In C there are 23 rules of precedence. The smart programmer knows that multiplication and division comes before addition and subtraction and then puts parenthesis around everything else".