1 is the most important element of Mathematics. One or unity in Mathematics is used to represent a single entity in a number, measurement, or calculation. The number ‘1’ has a few peculiar properties which are very important in Mathematical calculations. They are:

‘1’ is the number used to represent a single identity.

‘1’ is added to any integer to get the immediate successive integer.

When ‘1’ is subtracted from any integer, the immediate preceding integer is obtained.

1 is the multiplicative identity of any number. i.e. When any number is multiplied by itself, the number itself is obtained as the product.

The multiplicative inverse of any number is the value obtained when ‘1’ is divided by the number.

When any number is divided by ‘1’, the answer is the number itself.

When the number is divided by itself, the answer obtained is one.

The value of any number raised to the power zero is equal to unity.

It is very important to know how to find the square root of 1 because it gives a clear understanding of finding the square root of other integers. A positive value of one can be written as 1 x 1 or 12.

So, square root of 1 can be calculated as:

√1 = √12 = ±1

The formula for finding the roots of a quadratic equation can also be used to find the square root of 1.

Let the square of the number ‘x’ be equal to ‘1’. This can be written as:

x2 = 1

x = √1→ (1)

The above equation is a quadratic equation which can be represented in standard form as:

x2 + 0 x - 1 = 0

The above equation is of the form ax2 + bx + c = 0. So, a = 1, b = 0 and c = -1.

The value of ‘x’ can be found using the formula:

\[x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \frac{{ - 0 \pm \sqrt {{0^2} - 4x \times 1 \times - 1} }}{{2 \times 1}} = \pm \frac{{\sqrt 4 }}{2} = \pm \frac{2}{2} = \pm 1\] → (2)

Comparing equations (1) and (2), we can infer that the value of under root 1 is equal to either positive or negative unity.

Most commonly, the value of under root 1 is taken as positive unity or + 1.

Root value of ‘-1’ does not exist in theory. It is an imaginary number represented as ‘i’. Root of -1 is generally used to represent complex numbers which include both the real part and the imaginary part. With the knowledge of the square root of negative unity, the root value of any negative number can be found. Square root of -1 is a positive or negative imaginary unit ‘i’. However, in most cases, the value of the root of -1 is taken as a positive imaginary unit ‘i’.

These values of square root 1 to 10 are depicted on the number line as a square root spiral. (image will be updated soon)

Solve for p if p2 + 8 = 3

Solution:

p2 + 8 = 3

p2 = 3 - 8

p2 = - 5

p = √-5 = √-1 . √5

p = √5i

Find the value of 7√1 - 5√1 + 2√1 using the value of under root 1.

Solution:

Value of √1 = 1

7√1 - 5√1 + 2√1 = 7 (1) - 5 (1) + 2 (1)

= 7 - 5 + 2 = 4.

‘I’ is the first unit of imaginary numbers. It is equivalent to number ‘1’ in real numbers.

When negative unity is raised to the power of odd numbers the answer is -1 and when negative unity is raised to the power of even numbers, the answer is + 1.

The value of root 1 to any power is equal to 1.

FAQ (Frequently Asked Questions)

1. What are Squares and Square Roots?

Square of a number is the value obtained when a number is multiplied by itself once. Square of a number is the number raised to the power 2. Square of a number ‘x’ is represented as ‘x^{2}’. Square root of a number is that value which when multiplied by itself gives the number as the product. Square root of a number ‘x’ is denoted as ‘√x’ or ‘x^{1/2}’. Perfect square numbers are those numbers that have an integer value as their square roots. Every number has two square roots: a positive and a negative root. In most of the cases, the positive roots are taken into account.

2. What is the Value of Root 1?

Square root of any number is the number raised to the power ½. Square root of the positive value of 1 is either a positive value of 1 or negative value of 1. This is true because, 1 x 1 = 1 and -1 x -1 = 1. Square root of the negative value of one does not exist in theory. However, the square root of -1 is considered to be an imaginary number unit ‘i’. Square root of -1 is either a positive value of ‘i’ or negative value of ‘i’. Imaginary roots of unity are used in representing complex numbers and in calculations involving complex numbers.