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Value of Root 3

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Last updated date: 09th Apr 2024
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An Introduction

The value of root 3 is a real number when multiplied by itself gives the number 3. It's not a natural number but a fraction. The root of three is denoted by √3. The root basically, gives a worth which, when multiplied by itself gives the first (original) number. Hence, it's the basis of the first/original number.


To find the root of any real, without employing a calculator, we'd like to see first if the amount under the basis may be a perfect square or not. If variety may be a perfect square, then we will easily find its root using the prime factorization method. For instance, the root of 4 is ± 2, since 22 = 4. But, here 3 is an imperfect square. Hence, we cannot use the factorization method. Instead, we'll be using the division method to seek out the worth of the root of three. The actual value of √3 is 1.732.


About Square Root 3

The history of squares and square roots of numbers in Mathematics dates back to the 1900s when Babylonians invented various methods to determine the square root of numbers. 

 

However, Indian Mathematicians have recorded several methods to find the square root of numbers in their works such as ‘Sulabha Sutra’ by Baudhayana and ‘Aryabhatiya’ by Aryabhatta. The number below gives ‘What is the value of root 3’ determined up to 4 decimal places.

 

Properties of Square Root of a Number:

  • Only perfect square numbers have perfect square roots.

  • An even perfect square always has an even root.

  • An odd perfect square always has an odd root.

  • All perfect squares are positive integers and hence the square root of negative numbers cannot be explained with the concept of real numbers.

  • Square roots of 2 numbers can be multiplied and the product is the square root of the product of two numbers. Example: The product of square root 3 and square root 2 is square root 6. i.e. √3 x √2 = √6

  • The square root of any number is inverse operations. Therefore, the square root of the square of a number is the number itself.

The square root of any positive integer can be found using various methods. A few notable ones are :

  • Repeated subtraction method

  • Prime Division or Prime Factorization

  • Average method

  • Number Line method

  • Long division method

The first two methods can be used to find the square roots of perfect square numbers only. However, the other three methods can be employed to find the square root of any positive integer. 

 

Example:

  1. Find the square root of 100 using the repeated subtraction method.

Solution:

100 - 1 = 99

99 - 3 = 96

96 - 5 = 91

91 - 7 = 84

84 - 9 = 75

75 - 11 = 64

64 - 13 = 51

51 - 15 = 36

36 - 17 = 19

19 - 19 = 0


How to find Square Root of 3 using Average Method?

Ans: The basic principle of finding the square root of a number is the idea of finding the average of two numbers. 

 

Step 1: 

Determine the two perfect square numbers which are very close to the given number on either side. 

 

For the number ‘3’, the immediate perfect square preceding ‘3’ is ‘1’ and the immediate perfect square succeeding  3 is ‘4’.

 

Step 2:

Note down the square roots of the perfect squares identified in Step 1. 

The square root of ‘1’ is ‘1’ and the square root of ‘4’ is ‘2’.

 

Step 3:

It can be inferred that the square root of a given number lies between the square roots of numbers determined in step 2.

Square root 3 is any number between 1 and 2.

 

Step 4: Divide the number whose square root is determined by any of the numbers obtained in Step 2.

 

‘3’ can be divided either by ‘1’ or ‘2’. 

Let us divide ‘3’ by ‘2’

 

Step 5: Find the average of the quotient and divisor in Step 4.

The average of 2 and 1.5 is

 

Step 6:To get an accurate value of the square root, an average of the answer in step 5 and divisor of step 4 can be found. Finding of average can be continued as many times as required to get a precise value.

 

However, the answer in step 5 is approximately equal to the value of root 3 calculated using the calculator (1.7321).   

 

How to find Square Root of 3 by Number Line Method?

To find the square root of any number using a number line, a clear understanding of the basic concept of the “Pythagorean Theorem” is required.

Pythagorean theorem states that “In a right triangle, the square on the longest side (hypotenuse) is equal to the sum of the squares on the other two sides (base and perpendicular)”.

A series of steps to be followed to determine the square root of any number using a number line is explained below.

Step 1:

Construct a number line with a minimum of two units on either side

Step 2:

Label unit 1 as A. From the point, ‘A’ draw a perpendicular AB of unit length. 

Step 3:

Join OB. According to the Pythagorean theorem, the length of OB is calculated as 

Step 4:

From the point, ‘B’ draw a perpendicular BC of 1 unit to the line OB. Join OC. The length of OC is calculated using Pythagoras theorem as 

Length of the line OC gives the measure of the square root of 3.

 

The Root of 3 using Long Division 

We can only use the long-division approach to figure the value of the root of 3 since we all know it's non-terminating. 

  • First, write 3 as 3.00000000 and group the 0s following the decimal point in couplets of 2 from left to right, as seen below. (twain the integers to the left of the decimal point from right to left.) 

  • Consider a number that, when multiplied by itself, is a lower measure than or equal to three. In this case, the number will be 1 because 2 x 2 = 4. 

  • Taking 3 by 1 and dividing by 1, we gain a residual of 2. 

  • To make the dividend 200, drag a couple of 0s down and fill it coming to a 2. 

  • The divisor, which is 1 in this case, is multiplied by itself and published below. Now that we have 2K as the new divisor, we must find several K similar to the product of 2K. The K is lower than or equal to 200. In this case, the necessary value is 27. 

  • After a decimal place, the number 7 should be added to the quotient. During this case, the new divisor for the following division is 2K + K, which is 27 + 7, which equals 34. 


Fun Quiz:

  • What would be the length of the diagonal of one face of a cuboid whose length is 2 cm and width is 1cm?

  1. The square root of 2

  2. The square root of 5

  3. The square root of 3

 

  • What is the value of root 3 raised to power two?

  1. Nine

  2. 3

  3. The square numbers root of 3

FAQs on Value of Root 3

1. What is a Square Root of A Number?

  • The product obtained by multiplying a number by itself is called a square number. Square root of a number is that number which when multiplied by itself gives the answer as the number whose square root is to be determined.

  • Perfect square numbers are those numbers which are obtained by squaring integers.

  • Computing the square and square roots are inverse operations . Therefore, the square root of the square of a number is the number itself.

  • Square root of a perfect square is always a positive or negative integer. Square root of an even perfect square is even and that of an odd square number is odd.

2. Does the Square Root of Any Number Have Two Answers?

Square root of any number is that number which when multiplied by itself gives the product as in the question. For example, when the positive integer 3 is multiplied by itself, the product is 9. At the same time, the product still remains 9 when the negative integer ‘-3’ is multiplied by itself. This is due to a known fact that  two negative numbers are multiplied the product remains positive. So it can be  inferred that the square root of any number has two values either positive or negative. If ‘y’ is the square root of a number ‘x’, it can be represented as:

 

x = ± √y 

3. What are the properties of the square root?

Properties of Square Root of a Number are:

  • Only perfect square numbers will have perfect square roots.

  • An even perfect square will always have an even root.

  • An odd perfect square will always have an odd root.

  • The square root of negative numbers cannot be explained with the concept of real numbers, as all the perfect squares are positive integers.

  • If the square roots of two numbers are multiplied by each other, then the product is the square root of the product of two numbers. Example: √3 x √2 = √6

  • Square and the square root of any number follow inverse operations. Therefore, the square root of the square of the same number is the number itself.

4. What is the square root of 3?

The square root of a number when multiplied by itself, gives results in the form of the original number. For example, the square root of 25 is 5, which is 5 times 5 results in 25. Therefore, there are square roots of certain numbers that don't affect whole numbers, for example, let us take 3. We can express the square root of 3 in different ways, 


In the Decimal form, the number 3 is denoted as 1.732.

In the Radical form, the number 3 is denoted as √3

In the Exponent form, the number 3 is denoted as 31/2

5. How to find the square root of 3 using the long division method?

First, write 3 as 3.00000000 and group the 0s following the decimal point in couplets of 2 from left to right, as seen below. ( twain the integers to the left of the decimal point from right to left.) Consider a number that, when multiplied by itself, is a lower measure than or equal to three. In this case, the number will be 1 because 2 x 2 = 4. Taking 3 by 1 and dividing by 1, we gain a residual of 2. To make the dividend 200, drag a couple of 0s down and fill it coming to a 2. The divisor, which is 1 in this case, is multiplied by itself and published below. Now that we have 2K as the new divisor, we must find several K similar to the product of 2K. The K is lower than or equal to 200. In this case, the necessary value is 27. After a decimal place, the number 7 should be added to the quotient. During this case, the new divisor for the following division is 2K + K, which is 27 + 7, which equals 34.

6. What is the square root of numbers from 1 to 10?

The Square Root (√) of numbers 1 to 10 is listed below,

  • The square root of 1 is 1.000

  • The square root of 2 is1.414

  • The square root of 3 is 1.732

  • The square root of 4 is 2.000

  • The square root of 5 is 2.236

  • The square root of 6 is 2.449

  • The square root of 7 is 2.646

  • The square root of 8 is 2.828

  • The square root of 9 is 3.000

  • The square root of 10 is 3.162

7. How to find the root of any number without a calculator?

To find the root of any real, without employing a calculator, we'd like to see first if the amount under the basis may be a perfect square or not. If variety may be a perfect square, then we will easily find its root using the prime factorization method. For instance, the root of 4 is ± 2, since 22 = 4. But, here 3 is an imperfect square. Hence, we cannot use the factorization method. Instead, we'll be using the division method to seek out the worth of the root of three. The actual value of √3 is 1.732.