Square Root of Decimal Numbers
Finding the square root of decimal numbers extends the square root concept to numbers that are not whole. The method involves converting the decimal to a fraction, or using the long division method adapted for decimals.
For example, √0.49 = 0.7 because 0.7 × 0.7 = 0.49. Similarly, √1.44 = 1.2 because 1.2 × 1.2 = 1.44.
The key idea is to pair digits from the decimal point — two digits to the left and two digits to the right — and proceed with the long division method as usual.
What is Square Root of Decimal Numbers?
Definition: The square root of a decimal number is a value that, when multiplied by itself, gives the original decimal number.
√(decimal) = √(numerator/denominator) = √numerator / √denominator
Methods:
- Method 1: Convert decimal to fraction, find square root of numerator and denominator separately.
- Method 2: Use long division method with decimal pairing.
Methods
Method 1: Fraction method
- Convert the decimal to a fraction.
- Find the square root of the numerator and denominator separately.
- Write the result as a decimal.
Example: √0.0016
- 0.0016 = 16/10000
- √16 = 4, √10000 = 100
- √0.0016 = 4/100 = 0.04
Method 2: Long division method for decimals
- Place a bar over every pair of digits, starting from the decimal point — pairs go left for the integer part and right for the decimal part.
- If the decimal part has an odd number of digits, add a zero at the end to make it even.
- Proceed with the long division method as for whole numbers.
- Place the decimal point in the answer when you bring down the first pair after the decimal.
Counting decimal places rule:
- If the number has 2 decimal places, the square root has 1 decimal place.
- If the number has 4 decimal places, the square root has 2 decimal places.
- General: decimal places in √N = (decimal places in N) ÷ 2.
Solved Examples
Example 1: Example 1: Simple decimal
Problem: Find √0.49.
Solution:
- 0.49 = 49/100
- √49 = 7, √100 = 10
- √0.49 = 7/10 = 0.7
Answer: √0.49 = 0.7.
Example 2: Example 2: Four decimal places
Problem: Find √0.0081.
Solution:
- 0.0081 = 81/10000
- √81 = 9, √10000 = 100
- √0.0081 = 9/100 = 0.09
Answer: √0.0081 = 0.09.
Example 3: Example 3: Mixed number decimal
Problem: Find √2.25.
Solution:
- 2.25 = 225/100
- √225 = 15, √100 = 10
- √2.25 = 15/10 = 1.5
Answer: √2.25 = 1.5.
Example 4: Example 4: Larger decimal
Problem: Find √5.76.
Solution:
- 5.76 = 576/100
- √576 = 24 (since 24² = 576), √100 = 10
- √5.76 = 24/10 = 2.4
Answer: √5.76 = 2.4.
Example 5: Example 5: Six decimal places
Problem: Find √0.000169.
Solution:
- 0.000169 = 169/1000000
- √169 = 13, √1000000 = 1000
- √0.000169 = 13/1000 = 0.013
Answer: √0.000169 = 0.013.
Example 6: Example 6: Long division method
Problem: Find √3.24 by long division.
Solution:
- Pair the digits: 3 . 24 → pairs are (3) and (24).
- Largest number whose square ≤ 3 is 1 (1² = 1). Write 1. Remainder = 3 − 1 = 2.
- Bring down 24. New dividend = 224. Place decimal point in answer.
- Double the quotient: 2 × 1 = 2. Find digit d such that (20+d) × d ≤ 224.
- 28 × 8 = 224. Exactly! So d = 8.
- √3.24 = 1.8
Answer: √3.24 = 1.8.
Example 7: Example 7: Decimal with integer part
Problem: Find √12.96.
Solution:
- 12.96 = 1296/100
- √1296 = 36 (since 36² = 1296)
- √100 = 10
- √12.96 = 36/10 = 3.6
Answer: √12.96 = 3.6.
Example 8: Example 8: Word problem
Problem: The area of a square plot is 6.25 m². Find the side length.
Solution:
- Side = √Area = √6.25
- 6.25 = 625/100
- √625 = 25, √100 = 10
- Side = 25/10 = 2.5 m
Answer: Side = 2.5 m.
Example 9: Example 9: Very small decimal
Problem: Find √0.000004.
Solution:
- 0.000004 = 4/1000000
- √4 = 2, √1000000 = 1000
- √0.000004 = 2/1000 = 0.002
Answer: √0.000004 = 0.002.
Example 10: Example 10: Decimal needing trailing zero
Problem: Find √0.9 (approximate to 2 decimal places).
Solution:
- 0.9 has 1 decimal place — add a zero: 0.90
- 0.90 = 90/100
- √90 ≈ 9.487, √100 = 10
- √0.90 ≈ 0.9487 ≈ 0.95
Note: 0.9 is NOT a perfect square decimal, so the answer is approximate.
Answer: √0.9 ≈ 0.95 (approx.).
Real-World Applications
Real-world applications:
- Measurement: Finding side lengths when areas are given in decimal form.
- Science: Calculating standard deviations and other statistical measures involving square roots of decimals.
- Finance: Interest rate calculations sometimes require square roots of decimal quantities.
- Engineering: Precision measurements often involve decimal square roots.
Key Points to Remember
- √(a/b) = √a / √b.
- Number of decimal places in the square root = half the decimal places in the number.
- Convert decimal to fraction for easy calculation if numerator and denominator are perfect squares.
- In long division, pair digits from the decimal point — left for integers, right for decimals.
- Add a trailing zero if the decimal part has an odd number of digits.
- Common: √0.25 = 0.5, √0.04 = 0.2, √1.44 = 1.2.
- Not all decimals have exact square roots — some are approximate.
- A perfect square decimal has an even number of decimal places.
Practice Problems
- Find √0.36.
- Find √0.0064.
- Find √7.84.
- Find √0.000225.
- The area of a square is 1.96 cm². Find its side.
- Find √10.24 by the fraction method.
- Find √0.0144.
- Estimate √0.5 to two decimal places.
Frequently Asked Questions
Q1. How do you find the square root of a decimal?
Convert the decimal to a fraction (e.g., 0.49 = 49/100), find the square root of numerator and denominator separately, then convert back to decimal.
Q2. Can you use long division for decimal square roots?
Yes. Pair digits from the decimal point outward. Place the decimal in the answer when you start working on the decimal pairs.
Q3. How many decimal places does the square root have?
The square root has half the number of decimal places as the original number. √0.0049 (4 decimal places) = 0.07 (2 decimal places).
Q4. What if the decimal doesn't have an even number of places?
Add a trailing zero. For example, 0.9 becomes 0.90 for pairing purposes. But note that √0.9 is not a perfect square.
Q5. Is √0.04 equal to 0.2 or 0.02?
√0.04 = 0.2, because 0.2 × 0.2 = 0.04. The number of decimal places halves: 2 decimal places → 1.
Q6. Can a decimal number be a perfect square?
Yes, if it has an even number of decimal places and the digits (without the decimal) form a perfect square. Example: 0.0081 = 81/10000, both are perfect squares.
