- #1
sankalpmittal
- 785
- 25
The square root yields only one value !?
One guy advanced and asked me the question with an intention to judge my skill - " What is the value of x , if x2=4 " I , without any reluctance replied rather confidently - " Ha , its simple x will have 2 values : +2 and -2 "
He laughed hard and said " Don't you know , the value of x will be only positive ie +2 . Square root never yield negative values ie 41/2 = 2 and not -2 . "His monotonous and disdained voice made me feel uneasy . I replied " How can you prove it ? "He then gave me the following theorems :
1. To prove roots yield only positive values :1=1x1
1=-1x-1
So ;
1x1=-1x-1
On square root both sides :
(1x1)1/2 =(-1x-1)1/2
1=i2
1=-1 ?
or
http://www.artofproblemsolving.com/Wiki/index.php/Square_root
2. Because of the discontinuous nature of the square root function in the complex plane, the law √zw = √z√w is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of √z√w at all, leading to the equality's failure. A similar problem appears with the complex logarithm and the relation log z + log w = log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that –1 = 1:
http://en.wikipedia.org/wiki/Square_root#Algebraic_formula
3.
√-1/√-2 does not equal to √(-1/-2) if nos < 1 .
Sorry I forgot what he gave for this .
_____________________________________________________________________
Can anyone draw conclusion for this conversation ? Do roots really yield only positive values ? Please anyone illustrate these theorems more comprehensively .
:(
Thanks :)
( I am 14 years , class or year 10 . )
One guy advanced and asked me the question with an intention to judge my skill - " What is the value of x , if x2=4 " I , without any reluctance replied rather confidently - " Ha , its simple x will have 2 values : +2 and -2 "
He laughed hard and said " Don't you know , the value of x will be only positive ie +2 . Square root never yield negative values ie 41/2 = 2 and not -2 . "His monotonous and disdained voice made me feel uneasy . I replied " How can you prove it ? "He then gave me the following theorems :
1. To prove roots yield only positive values :1=1x1
1=-1x-1
So ;
1x1=-1x-1
On square root both sides :
(1x1)1/2 =(-1x-1)1/2
1=i2
1=-1 ?
or
http://www.artofproblemsolving.com/Wiki/index.php/Square_root
2. Because of the discontinuous nature of the square root function in the complex plane, the law √zw = √z√w is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of √z√w at all, leading to the equality's failure. A similar problem appears with the complex logarithm and the relation log z + log w = log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that –1 = 1:
http://en.wikipedia.org/wiki/Square_root#Algebraic_formula
3.
√-1/√-2 does not equal to √(-1/-2) if nos < 1 .
Sorry I forgot what he gave for this .
_____________________________________________________________________
Can anyone draw conclusion for this conversation ? Do roots really yield only positive values ? Please anyone illustrate these theorems more comprehensively .
:(
Thanks :)
( I am 14 years , class or year 10 . )
Last edited by a moderator: