Finding the value of √2 correct up to two decimal places:

√2 is an irrational number, which means it cannot be expressed as a ratio of two integers and has an infinite number of non-repeating decimal places. However, we can approximate the value of √2 to a certain degree of accuracy using different methods. Here's one way to find the value of √2 correct up to two decimal places:

Method 1: Long Division Method

One method to find the value of √2 is by using long division. Here are the steps:

1. Start by writing down 2 as the dividend and 1 as the divisor.
2. Divide 1 into 2 and write down the quotient (2) and remainder (0).
3. Bring down the next digit (0) to the right of the remainder to form 20.
4. Double the divisor (1 x 2 = 2) and write it below the 20 to form 22.
5. Find the largest digit that can be multiplied by 2 to give a result less than or equal to 20. In this case, it is 1 (2 x 1 = 2).
6. Write down 1 as the next digit of the quotient and subtract 2 from 20 to get a new remainder of 18.
7. Bring down the next digit (0) to the right of the remainder to form 180.
8. Double the divisor (1 x 2 = 2) and write it below the 180 to form 182.
9. Find the largest digit that can be multiplied by 2 to give a result less than or equal to 180. In this case, it is 9 (2 x 9 = 18).
10. Write down 9 as the next digit of the quotient and subtract 18 from 180 to get a new remainder of 162.
11. Repeat steps 7 to 10 until you have the desired degree of accuracy (in this case, two decimal places).

After several iterations, we get the following quotient and remainder:

√2 ≈ 1.41

This is the value of √2 correct up to two decimal places.

Method 2: Using a Calculator

Another method to find the value of √2 is by using a calculator. Here are the steps:

1. Press the square root (√) button on your calculator.
2. Type in 2.
3. Press the equals (=) button.

The calculator should display the value of √2, which is approximately 1.41421356. To round this value to two decimal places, we need to look at the third digit after the decimal point (4). If this digit is 5 or greater, we round up the second digit (1) by adding 1 to it. If this digit is less than 5, we leave the second digit as it is. In this case, the third digit is 4, which means we leave the second digit (1) as it is. Therefore, the value of √2 correct up to two decimal places is 1.41.

Conclusion

In conclusion, we can use different methods to find the value of √2 correct up to a certain degree of accuracy. The long division method and using a calculator are two such methods