How to find the decimal expansion of √2

Step 1: Understanding the concept
To find the decimal expansion of √2, we need to first understand that the square root of 2 is an irrational number, which means it cannot be expressed as a simple fraction. Therefore, its decimal expansion will be non-terminating and non-repeating.

Step 2: Using the long division method
We can use the long division method to find the decimal expansion of √2. Start by dividing 2 by 1 to get the initial approximation of √2. This will give you 1.4 as the first digit after the decimal point.

Step 3: Iterating the process
Next, bring down the next two zeros and double the previous result (1.4) to get 2.8. This will be the new dividend. Now, divide 2.8 by the divisor to get a new approximation of √2. Continue this process iteratively to get more decimal places.

Step 4: Calculating more decimal places
By repeating the long division process, you can calculate more decimal places of √2. Keep track of the remainders and carry them forward to get more accurate results.

Step 5: Checking for patterns
As √2 is an irrational number, its decimal expansion will not repeat. However, you can look for any patterns or sequences in the decimal places to verify the accuracy of your calculations.
By following these steps and being patient with the long division method, you can find the decimal expansion of √2 to as many decimal places as you desire.