Common misconception

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It still doesn't stop textbooks being in error, though - I've seen examples where a student is asked to simplify something innocent-looking, like

(x^2y^4)^{1/2}

and the "answer" is given as

xy^2

without the required modulus sign around the x.

Guilty as charged, officer. I'll try to be more careful in the future though. Good point.

Reply 21

RichE

Original post by atsruser

I'm afraid you're being too subtle for me here. Isn't a function single-valued on its Riemann surface, in which case I'm not sure what ambiguity can arise. Or have I forgotten too much complex analysis?

I'm not really trying to make a subtle point - I'm just saying that in some cases it's fairly common to use the radical symbol to denote multifunctions. One could either deal with that multifunction by taking a branch or by treating all possible values at once (via the Riemann surface).

My only point was that this notation is common in this regard.

(edited 9 years ago)

Reply 22

Dilzo999

Seems you learn something everyday :biggrin:

Reply 23

atsruser

Original post by RichE

Oh OK. Along those lines, I may retract my claim about the notation being ambiguous 100 years ago - ISTR that even some of the old style A level textbooks (extant in the 70s say) would happily put up a picture of both branches of

y=\sqrt{x}, x \ge 0

in their graphing chapters, and claim that was a graph of the function.

Reply 24

davros

Original post by atsruser

y=\sqrt{x}, x \ge 0

in their graphing chapters, and claim that was a graph of the function.

I've just dug out my treasured copy of Hardy's A Course of Pure Mathematics (10th edition) hoping for some clarity. Here's what the great man had to say (page 52, the author's italics):

It should be noticed that there is an ambiguity of notation involved in such an equation as

y = \sqrt{x}

. We have up to the present regarded

\sqrt{2}

, for example, as denoting the positive square root of 2, and it would be natural to denote by

\sqrt{x}

, where x is any positive number, the positive square root of x, in which case

y = \sqrt{x}

would be a one-valued function of x. It is however often more convenient to regard

\sqrt{x}

as standing for the two-valued function whose two values are the positive and negative square roots of x.

So that's all clear, then :biggrin:

Reply 25

TenOfThem

Original post by davros

y = \sqrt{x}

. We have up to the present regarded

\sqrt{2}

, for example, as denoting the positive square root of 2, and it would be natural to denote by

\sqrt{x}

, where x is any positive number, the positive square root of x, in which case

y = \sqrt{x}

would be a one-valued function of x. It is however often more convenient to regard

\sqrt{x}

as standing for the two-valued function whose two values are the positive and negative square roots of x.

So that's all clear, then :biggrin:

I may well nip down to the cellar and take a peek in Dakin and Porter

Reply 26

ghostwalker

Under the heading of common misconceptions, one I often see on here is along the lines of:

Solve: (x-3)(x-4) = 0

Conclusion: x = 3, and 4.

NO.

x= 3, or 4. It is either one of the other, but it cannot be both at the same time.

Reply 27

ThatPerson

Original post by davros

y = \sqrt{x}

. We have up to the present regarded

\sqrt{2}

, for example, as denoting the positive square root of 2, and it would be natural to denote by

\sqrt{x}

, where x is any positive number, the positive square root of x, in which case

y = \sqrt{x}

would be a one-valued function of x. It is however often more convenient to regard

\sqrt{x}

as standing for the two-valued function whose two values are the positive and negative square roots of x.

So that's all clear, then :biggrin:

So much for standard notation removing ambiguity then :tongue:

Even when doing an innocuous question in C1 such as

4^{1/2}

, I had to pause for a second before writing "2". The mark schemes don't do much to alleviate any confusion by also accepting

\pm 2

Reply 28

username1162972

basically mathematicians always like using the positive solutions and tend to ignore the negative ones ie when drawing square root graphs.

Posted from TSR Mobile

Reply 29

james22

Original post by ghostwalker

Unless you are working in the trivial ring.

Reply 30

ghostwalker

Original post by james22

Unless you are working in the trivial ring.

Would you get a 3 or a 4 in a trivial ring?

However, my comment was aimed at a more elementary level. Someone knowing what a ring is is highly unlikely to make the mistake I was highlighting.