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Square Root Progression

The methods outlined below are intended as aids to mental arithmetic in multiplication.

The simple rule for progressing from one square root to another is as follows:-

  1. Double the Starting Square. [Result 1].
  2. Add the difference between the Starting and Ending Square to Result 1 [Result 2].
  3. Multiply Result 2 by the difference. [Result 3]
  4. Add Result 3 to the original square to get the final square root.
Example 1 : To move from the square of 9 to the square of 15.
  • Difference between 15 and 9 = 6.
  • Double starting square: 9+9 and add difference (6) giving us 24.
  • Multiply 24 by difference 6 = 144.
  • The square of 9 = 81 + 144 = 225 (the square of 15).
Example 2 : The simple case of moving from the square of 11 to the square of 12.
  • The square of 11 is 121
  • Add 11+11+1= 23 to 121 = 144 (the square of 12)

Knowledge of square root progression could be important in determining the factors of some numbers.

For instance, say you had a number like 391. On the face of it, it isn't obvious that 391 has prime factors. But we know that if you add 9 (a perfect square) to 391 we get 400 (another perfect square).

Those facts tell us 391 must have prime factors. See Prime Factors Of A Number. Given 20 is the square root of 400 and 3 is the square root of 9, the answer lies 20 ± 3 which gives 17 and 23 - the factors of 391.

Of course, the smaller the numbers the easier the mental maths!

Input the appropriate figures below to explore other numbers.

Square Root Progression

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As an extension of the application of the above method please see Mental Multiplication.


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