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Why there is no answer when negative value is being square root?

  1. Jan 17, 2012 #1
    why there is no answer when negative value is being square root?
    e.g: square root of -9
    when i try to find answer from calculator, ''math error '' appears..
    so is there an explanation for this question??
    this question may looks so weird..but i m juz asking out of curiousity..
  2. jcsd
  3. Jan 17, 2012 #2


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    to our intuition we see that there is no number that if we multiplied by itself will be negative , since if it is negative then squaring it will result in a positive number , and the same for positive numbers , precisely we see that from the properties of the order field of real numbers that if x > 0 and y > 0 , the xy > 0 , and if x<0 and y<0 the xy<0 and we can see why x^2 > 0 or = 0 for any real number x , and hence there is no real number such that its square is negative . This led to what is Called by complex numbers , where the square root of -9 is 3i , where i is called imaginary unit and defined informally by square root of -1 .
  4. Jan 17, 2012 #3
    Well take a number [itex]a \geq 0[/itex] and square it, obviously two positive numbers multiplied gives a positive number. Take the second case of [itex]a < 0[/itex] and write it as [itex]-a[/itex] for [itex]a \geq 0[/itex] and square this [itex](-1)a \times (-1)a = (-1)(-1) \times a^2 \Rightarrow a^2 [/itex] which is again positive.
    Now there is a special number called [itex]i[/itex] which is defined by [itex]i^2 = -1[/itex] so you can in fact find the square root of a negative number.

    EDIT: sniped :)
  5. Jan 17, 2012 #4


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    Science Advisor

    The product of two positive numbers is positive and the product of two negative numbers is also positive. And, of course, the product of 0 with itself is 0. That is, for x positive, negative, or 0, [itex]x^2[/itex] is never negative.
  6. Jan 17, 2012 #5


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    Science Advisor

    Your calculator is designed for real numbers only. If you had a calculator which works with complex numbers, you would get an answer.
  7. Jan 18, 2012 #6
    so is it the same reason for the statement '' square root of x is always positive'' ??
  8. Jan 18, 2012 #7


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    Homework Helper

    That's a convention in that [itex]\sqrt{x}[/itex] is defined to be the positive solution y of the equation [itex]y^2 = x[/itex], where [itex]x \in \mathbb{R}^+[/itex].

    So when you just use the √ symbol, people assume you're referring to the positive value. To refer to the negative value, you need to put the minus sign in front.
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