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Square Root of 2




What is square root of 2? Let’s explore the exact steps how to find square root of a number based on the below example

  1. Initial Estimate: Start with an estimate for the square root of 2. A logical starting point might be 1, as 12=1 and 22=4, so the square root of 2 is somewhere between 1 and 2.
  2. Divide the Number by the Estimate: Divide 2 by your initial estimate. If you start with 1.5, calculate 2÷1.5≈1.33.
  3. Calculate the Average: Find the average of the result from step 2 and your estimate. Add your initial estimate (1.5) to the result (1.33) and then divide by 2, giving (1.5+1.33)÷2≈1.42.
  4. Use the Average as a New Estimate: Take the average calculated in step 3 and use it as your new estimate.
  5. Repeat the Process: Repeat steps 2 through 4 using the new estimate. Each iteration should bring you closer to the actual square root of 2.
  6. Check for Accuracy: After a few iterations, check the accuracy of your estimate by squaring it. If the square of your estimate is close to 2, then your estimate is accurate.
  7. Continue Until Satisfied: Continue iterating until the squared estimate is as close to 2 as you need it to be, depending on your required precision.
  8. Final Estimate: Once you’ve reached the desired level of accuracy, your final estimate is the square root of 2 to that precision.

Representing the Square Root of 2 Mathematically:

  1. As a Radical: If the square root cannot be simplified into a rational number, which is the case with 2, represent it with the radical symbol: √2.
  2. As an Approximate Decimal: If a decimal approximation is sufficient, you could use the result from your iterations, for example, √2≈1.42.
  3. Using the Approximation Symbol: If representing it as an approximation, you might use the symbol, like √2≈1.42.
  4. Scientific Notation or Interval Notation: These are less common for a number like 2 but can be used in more advanced or specific mathematical contexts.

In practical scenarios, especially when high precision is not critical, the square root of 2 is often left in its radical form, √2, or approximated to a few decimal places. This is because the square root of 2 is an irrational number, meaning it cannot be expressed exactly as a fraction and its decimal representation is non-repeating and non-terminating.

Explore variety of examples below:

General rules: How to Find Square Root of a Number

Practical examples:

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