Square root of 1

The square root of a number is another number that, when multiplied by itself, gives the initial number. Mathematically, it is written using radical sign i.e. √. The square root of a number can also be represented by raising the given number to an exponential power of ½. For example, the square root of 4 which is 2 can be represented mathematically as

√4 = √(2.2) = 2

(4)^1/2 = (2.2)^1/2 = 2

Therefore, in general, the square root of any number, say x, is written as √ x or (x)^1/2.

In the case of 1, more than one square root value is observed. The square root of 1 is mathematically represented as √1 or (1)^1/2. This implies

1 . 1= 1

Also, (-1). (-1) = 1

The two equations above indicate that the number 1 can have two square roots. The number 1 multiplied by itself gives 1. Similarly, the number (-1) when multiplied by itself also gives 1 as the product which means that the square of 1 and (-1) is 1. Hence, the converse of the statement also follows i.e.

√ (1. 1) = 1

√ [(-1). (-1)] = 1

which implies √1 = ± 1

For a clear understanding of finding the square root of 1 let us consider a simple equation. For finding the square root of one we consider a quadratic equation as below

Let √1 = x

1 = x² or

x² = 1

x² – 1 = 0….. (1)

The equation (1) now represents a quadratic equation. To solve the equation and obtain its roots, make the following substitution.

x² + 0 . x – 1 = 0….. (2)

On comparing equation (2) with the standard quadratic equation,

ax² + bx + c = 0

we obtain the following values

a = 1 ; b = 0 ; c = -1

Solve equation (2) using quadratic formula i.e.

x = [-b ± √ (b² – 4ac)] / 2a

x = [0 ± √-4 . 1 . (-1)] / 2

x = ± √4 / 2

x = ±2 / 2

x = ±1….. (3)

From equation (3) we can infer that the square root of 1 is both +1 as well as -1.

Nature of the roots

From the above example, the square root of 1 was observed to be +1 and -1. This represents a positive real root and a negative real root. Since √1 = +1 is positive, it is known as the real root of 1. Whereas, √1 = -1 is a negative root of 1.

Concept of Imaginary Roots

The concept of √-1 is totally different from √1 = -1. This happens because a negative number is never taken under the radical sign or we can say that the square root of a negative number is not possible. However, a negative square root value as for in the case of √-1 can be written in the form of ‘i’ which represents negative square root numbers i.e. complex numbers. In general, the value of i is -1.