## About Square Root of 2

- Square Root of 2
- Deriving the Square Root of 2
- Positive and Negative Square Roots of a Number
- Perfect Squares and Imperfect Squares
- Rapid Recall
- Solved Examples
- Frequently Asked Questions

## 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.

** Read More:**

square root of 11 | square root of 10 | square root of 3 |

square root of 20 | square root of 7 | square root of 144 |

square root of 12 | square root of 4 | square root of 24 |

square root of 8 | square root of 1024 | square root of 120 |

square root of 5 | square root of 6 | square root of 576 |

## 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.

Frequently Asked Questions on Square Root of 2

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.