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How do you find a square root by repeated subtraction?
To find the square root of a perfect square, you successively subtract consecutive odd numbers (1, 3, 5, 7, and so on) from the original number until you reach zero. The total number of subtractions you perform gives you the square root.
Here is a step-by-step example of finding the square root of 25:
25 − 1 = 24 (1st subtraction)
24 − 3 = 21 (2nd subtraction)
21 − 5 = 16 (3rd subtraction)
16 − 7 = 9 (4th subtraction)
9 − 9 = 0 (5th subtraction)
Since it takes exactly 5 steps to reach zero, the square root of 25 is 5.
This works exactly the same way for larger perfect squares, such as 36 or 49. For 36, you would perform a 6th subtraction (11 − 11 = 0). So it takes 6 steps, meaning the square root is 6.
Why does subtracting odd numbers work?
This method works because the sum of the first n odd natural numbers is always exactly n². For example, 1 + 3 + 5 = 9, which is 3². By subtracting odd numbers starting from 1, you are running this addition process in reverse. When you finally hit zero, you have accounted for all the odd numbers that sum up to your original number, and counting those steps reveals the square root.
This relationship is a staple of the Class 8 mathematics curriculum, which introduces it to build intuition before moving on to faster division methods. Just like noticing the pattern in sums like 9 + 99 + 999 + …, grasping why numbers behave this way helps students see maths as a connected system rather than a list of arbitrary rules. You can also spot this sequence in physics: an object falling from rest covers distances proportional to 1, 3, 5, and 7 in equal time intervals, which links back to the distance in the nth second formula.
How do odd numbers build a square algebraically?
The connection between odd numbers and squares is not just a coincidence; it is deeply rooted in algebra. If you take any integer n and square it to get n², you can find the next perfect square by evaluating (n + 1)².
When you expand (n + 1)², the result is n² + 2n + 1. The difference between the new square and the old square is exactly 2n + 1. In mathematics, the expression 2n + 1 defines an odd number. Therefore, to get from one perfect square to the next, you must add an odd number.
For instance, to move from 4² (16) to 5² (25), you calculate the difference: 25 − 16 = 9. Here, 9 is the 5th odd number. Because every step from zero to a perfect square is a step of one odd number, adding up the first n odd numbers will always land you on n². The repeated subtraction method simply walks backward down this exact staircase, removing one odd number at a time until you reach the bottom.
What happens with non-perfect squares?
The method is perfectly accurate, but it has two main limits:
It only stops at zero for perfect squares. If you try to find the square root of a non-perfect square like 20, your subtractions will eventually jump past zero into negative numbers (for example, 11 − 11 = 0 is replaced by 11 − 13 = −2). If this happens, you immediately know the original number is not a perfect integer square.
It is slow for large numbers. Finding the square root of 10,000 would take 100 manual subtractions. For larger values, students move on to prime factorisation or long division algorithms that scale much more efficiently. While repeated subtraction is not the most practical method for a mathematics exam, it remains an excellent teaching tool for visualising numbers.
Why is active calculation important for learning?
While a calculator gives you the square root instantly, writing out the subtractions by hand forces you to engage with the structure of the numbers. The 2014 PNAS meta-analysis by Scott Freeman and colleagues found that students in traditional lecture courses are 1.5 times more likely to fail than students in active learning classes. Calculating by hand, drawing diagrams, and finding patterns are all forms of active learning that help cement concepts in a student's memory.
If you are a student preparing for exams, taking the time to understand why a method works is the key to retaining it. When you get stuck on a new topic, Fermi's AI tutor teaches rather than just printing the answer. It notices where you are stuck, decides whether to explain, ask or show, and works out what you already know to build a plan for your session. You can also explore definitions and formulas across mathematics in the Fermi Concept Library.
