(-1)^(2/3) = 1 ?

Replies

• 

  shalini_goel14

  Hey tashirosgt,
  
  Good observation kept before. Can you tell the reason clearly behind it, why it happens so? Is there any rule/logic made by our old mathematicians for this ? 😕
• 

  Kaustubh Katdare

  Ha! Looks interesting. I'll look into this later today.
  
  [PS: Thread moved to mathematics section]
• 

  tashirosgt

  shalini,
  
  I haven't studied the history of mathematics very much so I don't how the truly old mathematicians thought about this subject (for example the ancient Greeks). The thinking of modern mathematicians (e.g. "Principles Of Mathematical Analysis" Second Edition (1964) , by Walter Rubin) is that the simplest way to do things (in the axioms for the real numbers) is to give a definition for x [sup](1/ n} [/sup] when n is a positive integer and x > 0. Nothing is defined about the case x < 0, so exponentiation by 1/n remains undefined in that situation. Then there is no problem with the law (x [sup] a [/sup] ) [sup] b [/sup] = x [sup] (ab) [/sup] since this law is asserted only in the case when all the exponential operations are defined.
  
  Rudin's book is a college text. We see that at a secondary school level, the authors of books and web pages are not so careful about logical consistency.
  
  People's confusion about (-1)[sup] (2/3} [/sup] can come from a very old fashioned idea about mathematics. This is the idea that a string of symbols has some "true" meaning and that a mathematical definition is merely an explanation of that "true" meaning expressed in words. The modern view would be that definitions are arbitrary and established by custom. They are not "true" or "false". The interpretation of the a string of symbols comes from the mathematical definition, not vice versa. It is permissible to give a mathematical definition for a string of symbols that extends an existing definition, but you can't assume the laws that applied to the symbols in the old definition will automatically apply to the symbols in the new definition.
• 

  Saandeep Sreerambatla

  Good Discussion 😀
  
  
  My question very general but can some one solve my doubt.
  
  Why - * - = + ??
  minus*minus = Plus , but why?
• 

  Saandeep Sreerambatla

  English-Scared

  Good Discussion 😀
  
  
  My question very general but can some one solve my doubt.
  
  Why - * - = + ??
  minus*minus = Plus , but why?

  There is a number line concept explaining this !!!
  
  IS anyother explanation also there??
• 

  tashirosgt

  To prove (-A)(-B) = AB rigorously, you must state the axioms for the real numbers and prove various other things. But the rough idea is:
  
  Suppose the following facts are true for all real numbers A,B,C:
  A(B + C) = AB + AC
  A(0) = 0
  B + (-B) = B - B = 0
  A(-B) = -(AB)
  
  The you must have (-A)(0) = 0 = (-A) (B + (- B)) = (-A)B + (-A)(-B) = -AB + (-A)(-B) = 0
  
  The quantity that when added to -(AB) produces 0 is the additive inverse of AB
  So (-A)(-B) is the additive inverse of (-AB). The addtive inverse of -AB is AB. So (-A)(-B) must be AB.
  
  (This assumes you have previously established that a number has one and only one additive inverse.)
  
  
  Strictly speaking, you must distinguish betwen the phrase "a minus" and the phrase "a negative number". If X = -7 then -X is a positive number.
• 

  Saandeep Sreerambatla

  Thanks for the explanation dude 😀
  
  There is one more number line explanation too..
  
  If we want 2 * -3
  
  The expression -3 says starting at zero, go left 3 steps. The expression 2 x means do it twice.
  
  Where do you finish? -6
  
  If you see why you finish on the left side, you have your answer.
  
  What about minus x minus. Let's calculate (-2) x (-3).
  
  Again -3 says starting at zero go left 3 steps. The expression -2 x means, no, go the other way instead -- twice.
  
  Where do you finish? +6. Minus x minus = plus.
  
  
  Source:Wiki Answers.