Square root of a positive real number:
The square root of a real number x is s which satisfies the equation x = s2. If x = s2 then x = (-s)2 is also satisfied because (-s)2 = s2. So, the square root of x is ±s and it is read as plus or minus s. Square root of x is symbolically denoted by √x. Thus the square root is mathematically defined as follows.
√x = ± s ↔ x = s2
Examples:
1) √9 =√32 = ±3
2) √25 = √52 = ±5
So, every square root gives two values, one with positive sign and with negative sign. Here, the positive value is called the principal square root. In general, the square root of a number is taken as its principal square root.
There is a reason for talking about the principal square root of a number. If we take the function f(x)=x2 then its inverse function g(x)=± √x. Thus every value of x is mapped onto two values. Then g(x) cannot be a function. Thus, if we define the square root of x to be the principal square root then g(x) becomes a function. This is the reason why the principal square root is defined.
Square root of a negative number:
Let x be a negative real number and s be its square root. Then by the definition of square root
x = s2 and x = (-s)2= (is)2 (because i2=-1)
So √x = ±is.
Example:
1) √-9 =√(i3)2 = ±3i
2) √25 = √(i5)2 = ±5i
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