Hey there! Sign in to join this conversationNew here? Join for free
    Offline

    11
    ReputationRep:
    (Original post by davros)
    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.
    Offline

    15
    ReputationRep:
    (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.
    Offline

    16
    ReputationRep:
    Seems you learn something everyday .
    Offline

    11
    ReputationRep:
    (Original post by RichE)
    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.
    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.
    • Study Helper
    Offline

    16
    ReputationRep:
    (Original post by atsruser)
    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.
    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
    Offline

    16
    ReputationRep:
    (Original post by davros)
    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
    I may well nip down to the cellar and take a peek in Dakin and Porter
    • Study Helper
    Online

    13
    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.
    Offline

    14
    ReputationRep:
    (Original post by davros)
    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
    So much for standard notation removing ambiguity then

    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 .
    Offline

    18
    ReputationRep:
    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
    Offline

    15
    ReputationRep:
    (Original post by 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.
    Unless you are working in the trivial ring.
    • Study Helper
    Online

    13
    (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.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    What newspaper do you read/prefer?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.