Square Root

What is the Odd relationship between n and n[SUP]2[/SUP] ? You re right. That is the straight forward relationship. I am afraid that it was a toungue-in-cheek quetion with a play on the word 'Odd'.
(Hint: It has to do with odd numbers)


  • Ramani Aswath
    Ramani Aswath
    In the early sixties we had only hand operated mechanicl calculators and slide rules for numerical work. While mechanical calculators had great accuracy they could not do square roots, which figure in most engineering calculations. We used an odd property of square numbers to extract square roots using these mechanical marvels.

    The square root of any number is equal to the number of ODD numbers starting with 1, whose sum equals the number.
    That is, n[SUP]2[/SUP] = Sum of first n odd numbers.

    e.g. 25 = (1+3+5+7+9)

    100 = (1+3+5+7+9+11+13+15+17+19)
    We set the number on the calculator and go on subtracting successive odd numbers starting with one. The calculator had a dial to count the number of revolutions. When the main number got reduced to zero, the number of revolutions gave the square root.
    Nice trick sir. Great to know how people used to calculate in 60s. I have got another approach. I have recently started employing vedic mathematics here.

    Its easy, although I mostly trust my 991 ES calculator, it is just for fun the method is cool 😀
  • Ramani Aswath
    Ramani Aswath
    Vedic mathematics and astronomy are fascinating. The astronomers used a form of interesting mnemonics to remember complicated planetary data.
    An aside:
    Try multiplying the number: 12345679 by 1x9 to 9x9 (9 to 99). Gives interesting answers. Multiplying it by 999999999 (nine nines) gives a very satisfying result.

    123456789x9999999999 = 1.23456789x10[SUP]18 [/SUP]​the answers are cool 😀
  • Ramani Aswath
    Ramani Aswath
    123456789X9=1111111101 123456789x9999999999 = 1.23456789x10[SUP]18 [/SUP]​the answers are cool 😀
    @Issue: Please note: there is no '8'. The number is 12345679.

    It is best if the whole answer number is seen not as seen on a calculator.

    12345679 x 999999999 = 12345678987654321

    12345679 x (1x9) = 111 111 111
    12345679 x (2x9) = 222 222 222 and so on to
    12345679 x (9x9) = 999 999 999
    oh yes, thanks for reminding about absence of 8.

    Another interesting method but this time for finding the squares or rather multiplying two numbers is as follows.

    Suppose you want to square 45.
    so multiply 5 and 5=25
    and for the number in 10's place add one and multiply with previous number. means 4+1= 5 and multiply this number with 4=20.

    So we get 45X45= ( 5X4_5X5)=2025
    similarly 35X35=(4X3_5X5)=1225

    now the method can be used for any number whose 10s place is same and the addition of unit's place is 10. In other words you can multiply 74X76 here the 10's place 7 is same while in unit's place 6+4=10.

    I hope by this method the multiplication become's a child's play (~As my grandfather used to say 👍) 😀
  • Ramani Aswath
    Ramani Aswath
    @Issue: Have you seen this:
    No sir the book is not getting downloaded, but I have noted down its title will see to download it from another source soon.

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