Square Root by Long Division
The long division method is a systematic procedure for finding the square root of any number — whether it is a perfect square or not. While the prime factorisation method works only for perfect squares, the long division method can find the square root of large numbers, decimals, and even non-perfect squares (to any desired number of decimal places).
This method is similar to the regular long division you already know, but with some important differences in the procedure. It involves grouping digits into pairs, finding suitable divisors, and bringing down pairs of digits at each step.
In Class 8, the long division method is taught as part of "Squares and Square Roots". It is especially useful for numbers that are too large for the prime factorisation method or when you need the square root of a number that is not a perfect square.
What is Square Root by Long Division?
Definition: The long division method for square roots is a step-by-step division procedure where the number is grouped into pairs of digits, and at each step, a suitable digit is found that, when combined with the divisor and multiplied, does not exceed the current dividend.
Key features:
- Works for all positive numbers — perfect squares and non-perfect squares
- Can find the square root to any number of decimal places
- Groups digits into pairs starting from the decimal point
- Each step produces one digit of the square root
Square Root by Long Division Formula
Steps of the Long Division Method:
Step 1: Group digits into pairs from the decimal point
Step 2: Find the largest number whose square is less than or equal to the first group
Step 3: Write this as the first digit of the square root and the first divisor
Step 4: Subtract and bring down the next pair
Step 5: Double the quotient so far to form the new divisor's tens part
Step 6: Find a digit x such that (new divisor tens part x 10 + x) x x does not exceed the current dividend
Step 7: Repeat steps 4-6 until all pairs are used
For decimal numbers:
- Pair digits to the left of the decimal point from right to left.
- Pair digits to the right of the decimal point from left to right.
- Add zeros in pairs to the right if needed for more decimal places.
Derivation and Proof
Why does the long division method work?
The method is based on the algebraic identity:
- (a + b)² = a² + 2ab + b² = a² + b(2a + b)
Explanation with an example:
Suppose we want sqrt(1521).
Step 1: Group: 15 | 21
Step 2: The largest integer whose square ≤ 15 is 3 (since 3² = 9 and 4² = 16). So a = 3 (representing 30). Remainder = 15 - 9 = 6.
Step 3: Bring down 21. Current dividend = 621.
Step 4: Double the quotient: 2 x 3 = 6. This represents 2a = 60.
Step 5: Find b such that (60 + b) x b ≤ 621. Try b = 9: (60 + 9) x 9 = 69 x 9 = 621. Exact.
So sqrt(1521) = 39.
The algebra: 39 = 30 + 9. 39² = 30² + 9(2 x 30 + 9) = 900 + 9 x 69 = 900 + 621 = 1521.
Each step of the long division method mirrors the identity (a + b)² = a² + b(2a + b), where a is the quotient found so far and b is the next digit.
Types and Properties
The long division method is applied in several types of problems:
1. Square root of a perfect square (integer):
- Group digits into pairs from right.
- The answer is an exact integer with no remainder.
2. Square root of a large number:
- Same procedure but with more pairs of digits.
- Useful when prime factorisation is too tedious.
3. Square root of a decimal number:
- Group digits to the left of the decimal from right to left.
- Group digits to the right of the decimal from left to right.
- The decimal point in the answer goes directly above the decimal in the number.
4. Square root of a non-perfect square (approximate):
- After all digit pairs are used, add pairs of zeros after the decimal point.
- Continue the process to get the answer to the required decimal places.
5. Finding the number of digits in the square root:
- Count the number of pairs. The number of digit-pairs = number of digits in the square root.
- A number with n digits has a square root with n/2 digits (if n is even) or (n+1)/2 digits (if n is odd).
Solved Examples
Example 1: Example 1: Square root of 1521
Problem: Find sqrt(1521) by long division.
Solution:
Step 1: Group: 15 | 21
Step 2: Largest number whose square ≤ 15: 3 (3² = 9). Write 3 as quotient.
Subtract: 15 - 9 = 6. Bring down 21. Dividend = 621.
Step 3: Double the quotient: 2 x 3 = 6_. Find digit x: 6x x x ≤ 621.
Try x = 9: 69 x 9 = 621. Exact match.
Subtract: 621 - 621 = 0.
Answer: sqrt(1521) = 39.
Example 2: Example 2: Square root of 5625
Problem: Find sqrt(5625) by long division.
Solution:
Step 1: Group: 56 | 25
Step 2: Largest number whose square ≤ 56: 7 (7² = 49). Write 7.
Subtract: 56 - 49 = 7. Bring down 25. Dividend = 725.
Step 3: Double: 2 x 7 = 14_. Find x: 14x x x ≤ 725.
Try x = 5: 145 x 5 = 725. Exact.
Remainder = 0.
Answer: sqrt(5625) = 75.
Example 3: Example 3: Square root of 17424
Problem: Find sqrt(17424) by long division.
Solution:
Step 1: Group: 1 | 74 | 24 (start from right, so 24, 74, and the remaining 1)
Step 2: Largest number whose square ≤ 1: 1 (1² = 1). Write 1.
Subtract: 1 - 1 = 0. Bring down 74. Dividend = 74.
Step 3: Double: 2 x 1 = 2_. Find x: 2x x x ≤ 74.
Try x = 3: 23 x 3 = 69 ≤ 74. Try x = 4: 24 x 4 = 96 > 74. So x = 3.
Subtract: 74 - 69 = 5. Bring down 24. Dividend = 524.
Step 4: Double: 2 x 13 = 26_. Find x: 26x x x ≤ 524.
Try x = 2: 262 x 2 = 524. Exact.
Remainder = 0.
Answer: sqrt(17424) = 132.
Example 4: Example 4: Square root of a decimal — sqrt(2.56)
Problem: Find sqrt(2.56) by long division.
Solution:
Step 1: Group: 2 . 56 (one group before decimal, one after)
Step 2: Largest number whose square ≤ 2: 1 (1² = 1). Write 1.
Subtract: 2 - 1 = 1. Bring down 56. Dividend = 156.
Place decimal point in the answer.
Step 3: Double: 2 x 1 = 2_. Find x: 2x x x ≤ 156.
Try x = 6: 26 x 6 = 156. Exact.
Remainder = 0.
Answer: sqrt(2.56) = 1.6.
Example 5: Example 5: Square root of a non-perfect square — sqrt(3) to 2 decimal places
Problem: Find sqrt(3) correct to 2 decimal places.
Solution:
Step 1: Write 3 as 3.00 00 00 (add pairs of zeros).
Group: 3 . 00 | 00 | 00
Step 2: Largest number whose square ≤ 3: 1 (1² = 1). Write 1.
Subtract: 3 - 1 = 2. Bring down 00. Dividend = 200. Place decimal.
Step 3: Double: 2 x 1 = 2_. Find x: 2x x x ≤ 200.
Try x = 7: 27 x 7 = 189 ≤ 200. Try x = 8: 28 x 8 = 224 > 200. So x = 7.
Subtract: 200 - 189 = 11. Bring down 00. Dividend = 1100.
Step 4: Double: 2 x 17 = 34_. Find x: 34x x x ≤ 1100.
Try x = 3: 343 x 3 = 1029 ≤ 1100. Try x = 4: 344 x 4 = 1376 > 1100. So x = 3.
Subtract: 1100 - 1029 = 71. Bring down 00. Dividend = 7100.
Step 5: Double: 2 x 173 = 346_. Find x: 346x x x ≤ 7100.
Try x = 2: 3462 x 2 = 6924 ≤ 7100. So x = 2.
So sqrt(3) = 1.732...
Answer: sqrt(3) ≈ 1.73 (correct to 2 decimal places).
Example 6: Example 6: Square root of 7744
Problem: Find sqrt(7744) by long division.
Solution:
Step 1: Group: 77 | 44
Step 2: Largest number whose square ≤ 77: 8 (8² = 64). Write 8.
Subtract: 77 - 64 = 13. Bring down 44. Dividend = 1344.
Step 3: Double: 2 x 8 = 16_. Find x: 16x x x ≤ 1344.
Try x = 8: 168 x 8 = 1344. Exact.
Remainder = 0.
Answer: sqrt(7744) = 88.
Example 7: Example 7: Square root of 106929
Problem: Find sqrt(106929) by long division.
Solution:
Step 1: Group: 10 | 69 | 29
Step 2: Largest square ≤ 10: 3 (3² = 9). Write 3.
Subtract: 10 - 9 = 1. Bring down 69. Dividend = 169.
Step 3: Double: 2 x 3 = 6_. Find x: 6x x x ≤ 169.
Try x = 2: 62 x 2 = 124. Try x = 3: 63 x 3 = 189 > 169. So x = 2.
Subtract: 169 - 124 = 45. Bring down 29. Dividend = 4529.
Step 4: Double: 2 x 32 = 64_. Find x: 64x x x ≤ 4529.
Try x = 7: 647 x 7 = 4529. Exact.
Answer: sqrt(106929) = 327.
Example 8: Example 8: Finding number of digits
Problem: Without finding the square root, determine how many digits the square root of 1522756 has.
Solution:
Group the digits into pairs from the right: 1 | 52 | 27 | 56
Number of pairs (groups) = 4
Each group produces one digit of the square root.
Answer: sqrt(1522756) has 4 digits.
(Verification: sqrt(1522756) = 1234, which indeed has 4 digits.)
Example 9: Example 9: Largest 4-digit perfect square
Problem: Find the largest 4-digit number that is a perfect square.
Solution:
The largest 4-digit number is 9999. Find sqrt(9999) by long division.
Group: 99 | 99
Largest square ≤ 99: 9 (9² = 81). Subtract: 99 - 81 = 18. Bring down 99. Dividend = 1899.
Double: 2 x 9 = 18_. Try x = 9: 189 x 9 = 1701. Subtract: 1899 - 1701 = 198.
sqrt(9999) = 99 with remainder 198. So 9999 is not a perfect square.
The largest 4-digit perfect square = 99² = 9801.
Answer: The largest 4-digit perfect square is 9801 (= 99²).
Example 10: Example 10: Square root of 54756
Problem: Find sqrt(54756) by long division.
Solution:
Step 1: Group: 5 | 47 | 56
Step 2: Largest square ≤ 5: 2 (2² = 4). Write 2.
Subtract: 5 - 4 = 1. Bring down 47. Dividend = 147.
Step 3: Double: 2 x 2 = 4_. Find x: 4x x x ≤ 147.
Try x = 3: 43 x 3 = 129. Try x = 4: 44 x 4 = 176 > 147. So x = 3.
Subtract: 147 - 129 = 18. Bring down 56. Dividend = 1856.
Step 4: Double: 2 x 23 = 46_. Find x: 46x x x ≤ 1856.
Try x = 4: 464 x 4 = 1856. Exact.
Answer: sqrt(54756) = 234.
Real-World Applications
The long division method for square roots has many practical uses:
- Engineering: Engineers often need square roots for calculations involving Pythagoras theorem, stress analysis, and electrical circuits. The long division method works when calculators are not available.
- Competitive Exams: Many entrance exams (NTSE, Olympiad, JEE Foundation) test the long division method for square roots.
- Approximation: Finding sqrt(2), sqrt(3), sqrt(5), etc. to several decimal places is needed in trigonometry and coordinate geometry.
- Number Theory: Determining how many digits the square root of a large number has, and finding the largest perfect square less than a given number.
- Land Measurement: Finding the side of a square plot from its area requires square roots. For large areas, the long division method is practical.
- Computer Science: The long division algorithm for square roots is the basis of many software implementations of the sqrt() function.
Key Points to Remember
- The long division method works for all positive numbers, not just perfect squares.
- Always group digits into pairs starting from the decimal point — left side from right to left, right side from left to right.
- The number of digit pairs = the number of digits in the square root.
- At each step: double the quotient, find the appropriate next digit, subtract, and bring down the next pair.
- For non-perfect squares, add pairs of 00 after the decimal point to continue.
- Place the decimal point in the answer directly above the decimal point in the number.
- This method is based on the identity (a + b)² = a² + b(2a + b).
- The method can find the square root to any number of decimal places.
- It is more practical than prime factorisation for large numbers.
- The remainder at any step should always be non-negative. If the chosen digit makes the product exceed the dividend, try a smaller digit.
Practice Problems
- Find sqrt(2025) by long division.
- Find sqrt(9801) by long division.
- Find sqrt(119025) by long division.
- Find sqrt(5) correct to 3 decimal places.
- Find sqrt(2.89) by long division.
- How many digits are in sqrt(998001)?
- Find the smallest 3-digit perfect square.
- Find sqrt(21316) by long division.
Frequently Asked Questions
Q1. What is the long division method for square roots?
It is a step-by-step procedure where digits are grouped into pairs, and at each step a suitable digit is found for the quotient by testing what value satisfies the division condition.
Q2. Why do we group digits into pairs?
Because a 2-digit number has a 1-digit square root, a 4-digit number has a 2-digit square root, and so on. Each pair of digits produces one digit of the square root.
Q3. Can this method find the square root of a non-perfect square?
Yes. For non-perfect squares, add pairs of zeros after the decimal point and continue the process to get the answer to as many decimal places as needed.
Q4. How do you handle decimal numbers?
Group digits to the left of the decimal from right to left, and digits to the right from left to right. Place the decimal in the answer above the decimal in the number.
Q5. What does 'double the quotient' mean?
At each step, take the quotient obtained so far and multiply it by 2. This doubled value forms the beginning of the new divisor, and the next digit is appended to complete it.
Q6. When is this method better than prime factorisation?
For large numbers (difficult to factorise), non-perfect squares (which have no integer square root), and decimal numbers.
Q7. How do you find the number of digits in a square root?
Group the number into pairs from the right. The number of groups (including a possible single digit on the left) equals the number of digits in the square root.
Q8. What is the algebraic basis of this method?
The identity (a + b)² = a² + b(2a + b). At each step, a is the quotient so far and b is the next digit being determined.
Q9. What if the remainder is not zero at the end?
If there are no more digit pairs and the remainder is not zero, the number is not a perfect square. Place a decimal point and add pairs of zeros to continue finding the approximate value.
Q10. How accurate can this method be?
As accurate as you need. Each additional pair of zeros after the decimal point gives one more decimal place of accuracy.
