Originally posted by mikenayif you solve by squaring both sides and substitute x=2, you will get "2 = -2" which is nonsense.
I don't see why this ends up with nonsense. As far as I can tell x=2 and the other solution to the expression would be sqrt(4x - x^2) = 2. Am I missing something?
it's because the operation "square root" only returns positive values, and squaring both sides introduces "extraneous" roots which will not work in the original equation.
see it?
Originally posted by BarefootChessPlayerWell, I see it but that makes your initial equation gibberish - if you're going to obey the convention of taking the positive square root then sqrt(x)=-y is false for all positive, real y. Thanks for the clarification though.
if you solve by squaring both sides and substitute x=2, you will get "2 = -2" which is nonsense.
it's because the operation "square root" only returns positive values, and squaring both sides introduces "extraneous" roots which will not work in the original equation.
see it?
Fiathahel, that is why I said the equation had 2 solutions. If sqrt(x)=-2 then sqrt(x)=2 is also true. Rather than always take the positive root I tend to accept there are two answers.
- Mike
O := {1,3,5,7,...}
E := {0,2,4,6,...}
I := {0,1,2,3,...}
Z := {...-2,-1,0,1,2,...}
Q := {all fractions (note: 3 is a fraction too: 3/1)}
R := {all real numbers}
C := {all complex numbers: a + bi, with a,b reals, and i=sqrt(-1)}
It appears to me he's not a genius then...over here in Holland we call O and E 'gelijkmachtig', wich can be translated of equal power. As the post above this one says, that means there is a one-on-one correspondence between O and E.
But there is also such a correspondence between O and I (2n+1 <-->n) and also between E and I (2n <--> n). 'Of equal power with' is usually denoted by '~'.
It gets more interesting when you see that;
I ~ Z
and I~Q
and R~P(I) (P(I) denotes the powerset of I, the collection of all subcollections of I).
and R ~ (0, 1) ((0, 1) denotes all numbers BETWEEN 0 and 1)
The term used by math-heads to compare "size" of infinite sets is "cardinality". To compare two infinite sets, you try to find a mapping between them. If you can find a one-to-one mapping, then the two sets are of the same cardinality.
For example: Let A: all positive integers
Let B: all negative integers.
Let f(x)=-x
Here we have a one-to-one function between A and B, which shows us that they are of the same cardinality.
This way one can have some kind of measure on the "size" of a set without being able to count the elements.
Originally posted by mikenayRules of sqrt;
I don't see why this ends up with nonsense. As far as I can tell x=2 and the other solution to the expression would be sqrt(4x - x^2) = 2. Am I missing something?
-sqrt(x) only gives a real as result when a real greater or equal to zero is substituted for x.
-sqrt(x) only gives reals greater or equal to zero as result.
so sqrt(4) = 2, but the equation x^2 = 4 has two solutions for x, namely sqrt(4) and -sqrt(4)