What is Square Root of 2
We already know that the square root of a number is a value that, when multiplied by itself, gives us the original number. Hence, finding the square root is the converse of finding the square of a number.
The square root of 2 is written as \(\sqrt 2\), with the radical sign ‘\(\sqrt~\) ‘ and the radicand being 2. The square root of 2 has a value that is nearly equal to 1.41421……, and this value is a non-terminating and a non-repeating value, showing us that it is an irrational number.
How did we obtain the value of \(\sqrt 2\) as 1.41421…? We use the long division method to find this value.
Deriving the Square Root of 2
We can apply the following steps to determine the square root of 2 using the long division method.
Step 1: Rewrite the number as shown below.
\(\overline 2.~\overline{00}~\overline{00}~\overline{00}\)
Step 2: Take a number whose square is less than or equal to 2.
\(1^2=1\), which is less than 2. So we will take 1.
Step 3: Write the number 1 as the divisor and 2 as the dividend. Now divide 2 by 2.
Here, Quotient = 1 and Remainder = 1.
Step 4: Bring down 00 and write it after 1, so the new dividend is 100 and add the quotient 1 to the divisor, that is, 1 + 1 = 2.
Step 5: Add a digit right next to 2 to get a new divisor such that the product of a number with the new divisor is less than or equal to 100.
24 x 4 = 96, which is less than 100.
Subtract the product 96 from 100 to get the remainder.
Here, Quotient = 1.4 and Remainder = 4
Step 6: Repeat the previous two steps to obtain the quotient up to three decimal places.
Therefore, the value of the square root of 2, that is, \(\sqrt 2\) is approximately 1.4142.
Positive and Negative Square Roots of a Number
Each number has two square roots, one is positive and other is negative. Let us understand this with an example, multiply \(-\sqrt 2\) by itself.
(\(-\sqrt 2\)) x (\(-\sqrt 2\)) = 2 [Product of two negatives is a positive]
Also, multiply \(\sqrt 2\) by itself
\(\sqrt 2\) x \(\sqrt 2\) = 2
From the above the square of \(-\sqrt 2\) is 2 and square of \(\sqrt 2\) is also 2. This leads to the fact that square roots of 2 are both \(\sqrt 2\) and \(-\sqrt 2\).
In general,
- \(\sqrt y\) represents the positive square root of y.
- \(-\sqrt y\) represents the negative square root of y.
- \(\pm~\sqrt y\) represents both the square roots of y.
Perfect Squares and Imperfect Squares
The square root of a perfect square number is always an integer. On the other hand, the value of the square root of imperfect squares is a non-integer, that is, it contains decimals or fractions.
For example, 26 is the square root of 676, which is a perfect square, and 5.099 is the square root of 26, which is an imperfect square.
Rapid Recall
Note: This table lists the approximate values of the square roots of some numbers that can be memorized to determine the square roots of higher imperfect square numbers.
Number | Square Root |
|---|---|
2 | 1.414 |
3 | 1.732 |
5 | 2.236 |
7 | 2.646 |
Solved Square Root of 2 Examples
Example 1: Find the square root of 13 by the approximation of the prime number square roots.
Solution:
Find the prime factorization of the number 13.
2 = 2 x 1
\(\sqrt 2\) = \(\sqrt {2~\times~1}\) [Apply square root both side]
⇒ \(\sqrt 2\) x \(\sqrt 1\) [Use property \(\sqrt ab\) = \(\sqrt a~\times~\sqrt b\)]
⇒ 1.414 x 1 [Substitute 1 for \(\sqrt 1\) and 1.414 for \(\sqrt 2\)]
⇒ 1.414 [Multiply]
So, the value of the square root of 2 is 1.414.
Example 2: Find the square root of 50.
Solution:
Use the factor tree method to find the prime factors of 50.
50 = 2 x 5 x 5
\(\sqrt {50}=\sqrt{2~\times~5~\times~2}\) [Apply square root both side]
\(\sqrt {50}=\sqrt{2~\times~5^2}\) [Write 5 x 5 as \(5^2\)]
\(\sqrt {50}=\sqrt 2~\times~\sqrt{5^2}\) [Use property \(\sqrt ab\) = \(\sqrt a~\times~\sqrt b\)]
\(\sqrt {50}\) = 1.414 x 5 [Substitute 5 for \(\sqrt{5^2}\) and 1.414 for \(\sqrt{2}\)]
\(\sqrt {50}\) = 7.07 [Multiply]
Hence, the square root of 50 is 7.07.
Example 3: Find the total distance covered by Larry, as he completes the five rounds of a \(100\sqrt{2}\) meter track.
Solution:
The distance covered in one round of track = \(100\sqrt{2}\) meters
The distance covered in five rounds of track = 5 x \(100\sqrt{2}\) meter
⇒ 500 x 1.414 [substitute 1.414 for \(\sqrt{2}\)]
⇒ 707 meter
So, Larry covers 707 meter in five rounds of a track.
No, the square root of numbers may or may not be rational numbers. Square roots of non-perfect square numbers are irrational numbers. For example, the square root of 3, is an irrational number. However, the square root of a perfect square number is rational. For example, the square root of 9, when calculated, results in 3, which is a rational number.
Yes, the square root of any decimal number can be calculated by the long division method.
Yes, the product of two perfect squares is always a perfect square.
For example: Let’s have two perfect square numbers, 36 and 16. When we multiply 36 and 16 we get 576 as the product, and the square root of 576 is 26.
A number is said to be a perfect square if the square root of the number is an integer. For example: 121 is a perfect square because the square root of 121 is 11, which is an integer.