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How to Find Square Root of Decimal Without Calculator

Reviewed by Calculator Editorial Team

Finding the square root of a decimal number without a calculator can be challenging but is possible using several reliable methods. This guide explains the most effective techniques, including long division and estimation, with step-by-step instructions and practical examples.

Methods to Find Square Root of Decimal

There are several methods to find the square root of a decimal number without a calculator. The two most common and reliable methods are:

  1. Long Division Method - A systematic approach that provides an accurate result.
  2. Estimation Method - A quicker approach that provides an approximate result.

Both methods can be used depending on the required level of precision. The long division method is generally more accurate, while the estimation method is faster but less precise.

Long Division Method

The long division method is a systematic approach to finding the square root of a decimal number. It involves a series of steps that can be followed to achieve a precise result.

Step-by-Step Instructions

  1. Set Up the Problem: Write the decimal number under the square root symbol.
  2. Pair the Digits: Pair the digits of the decimal number from right to left, starting from the decimal point.
  3. Find the Largest Square: Find the largest perfect square that is less than or equal to the first pair of digits.
  4. Subtract and Bring Down: Subtract the perfect square from the first pair and bring down the next pair of digits.
  5. Double the Divisor: Double the current divisor and find a digit to place next to it that, when multiplied by the new number, is less than or equal to the current remainder.
  6. Repeat the Process: Continue the process until the desired level of precision is achieved.

Example: Find the square root of 2.25 using the long division method.

  1. Pair the digits: 2 and 25.
  2. Find the largest square less than or equal to 2: 1 (since 1² = 1).
  3. Subtract 1 from 2 to get 1, then bring down 25 to make 125.
  4. Double the divisor (1) to get 2, then find a digit (d) such that (20 + d) × d ≤ 125. The digit is 1 (since 21 × 1 = 21).
  5. Subtract 21 from 125 to get 104, then bring down 00 to make 10400.
  6. Double the divisor (11) to get 22, then find a digit (d) such that (220 + d) × d ≤ 10400. The digit is 1 (since 221 × 1 = 221).
  7. Subtract 221 from 10400 to get 10179, then bring down 00 to make 1017900.
  8. Double the divisor (111) to get 222, then find a digit (d) such that (2220 + d) × d ≤ 1017900. The digit is 1 (since 2221 × 1 = 2221).

The square root of 2.25 is approximately 1.5.

The long division method provides a precise result but requires careful attention to each step. It is particularly useful when an exact answer is needed.

Estimation Method

The estimation method is a quicker approach to finding the square root of a decimal number. It involves estimating the square root based on known perfect squares and refining the estimate.

Step-by-Step Instructions

  1. Identify Known Squares: Identify two perfect squares between which the decimal number lies.
  2. Estimate the Square Root: Estimate the square root based on the known squares.
  3. Refine the Estimate: Refine the estimate by testing numbers around the initial estimate.

Example: Find the square root of 1.44 using the estimation method.

  1. Identify known squares: 1 (1²) and 4 (2²).
  2. Estimate the square root: Since 1.44 is between 1 and 4, the square root is between 1 and 2.
  3. Refine the estimate: Test 1.2 (1.2² = 1.44).

The square root of 1.44 is exactly 1.2.

The estimation method is faster but less precise than the long division method. It is useful when an approximate answer is sufficient.

Comparison of Methods

Both methods have their advantages and disadvantages. The long division method provides a precise result but requires more time and effort. The estimation method is faster but less accurate.

Method Precision Time Required Complexity
Long Division High Moderate Moderate
Estimation Low Low Low

Choose the long division method when an exact answer is needed, and the estimation method when a quick approximation is sufficient.

Frequently Asked Questions

Can I find the square root of a decimal number without a calculator?
Yes, you can use methods like long division and estimation to find the square root of a decimal number without a calculator.
Which method is more accurate?
The long division method is more accurate than the estimation method.
How do I pair the digits in the long division method?
Pair the digits from right to left, starting from the decimal point.
Can the estimation method give an exact answer?
No, the estimation method provides an approximate answer. For an exact answer, use the long division method.
What if the decimal number is not a perfect square?
Use the long division method to find an approximate square root.