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Math Help - Finding Square

  1. #1
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    Finding Square

    Hi,

    I there any easy way to find suare of numbers between 40 and 50, 60 and 90?

    Thanks!
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  2. #2
    Super Member Quacky's Avatar
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    Well 6\times6=36 and 10\times10=100, that leaves you to try 7,8 and 9.

    Algebraically?

    I suppose you could say that
    40<x^2<50
    \sqrt{40}<x<\sqrt50
    6.32<x<7.07
    So 7 squared lies between 40 and 50, and adapt this for the other numbers.
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  3. #3
    MHF Contributor

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    He was asking for squares, not square roots.

    ManuLi, you can do things like 47= 40+ 7 so 47^2= 40^2+ 2(40)(7)+ 7^2= 1600+ 560+ 49 but I don't know that that is any simpler than just multiplying 47 times 47.
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  4. #4
    Super Member Quacky's Avatar
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    Oh I misunderstood, evidently. Sorry for the useless post .
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  5. #5
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    I'd try a spreadsheet. Easy to set up a sequence, and they are all there instantly.
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  6. #6
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    Hello, ManuLi!

    Is there any easy way to find squares of numbers between 40 and 50, 60 and 90?

    There is a trick for squaring numbers "near 50",



    Let N \:=\:50 \pm d .where d is the "deviation from 50".

    Then: . N^2 \:=\:(50 \pm d)^2 \:=\:2500 \pm 100d + d\:\!^2 \:=\:100(25 \pm d) + d\:\!^2

    So, in the hundreds-place is: . 25 \pm d
    . . . followed by d\:\!^2, the deviation-squared.



    Example: . 53^2

    The deviation is: . d = +3

    The answer is: . 25 + 3 \:=\:28
    . .followed by: . 3^2 \:=\:09

    Therefore: . 53^2 \:=\:2809



    Example: . 44^2

    The deviation is: . d = -6

    The answer is: . 25 - 6 \:=\:19
    . .followed by: . (\text{-}6)^2 \:=\:36

    Therefore: . 44^2 \;=\;1936


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    Consecutive squares differ by consecutive odd numbers.

    . . \begin{array}{cccccccccccccc}<br />
25 && 36 && 49 && 64 && 81 && 100 && \hdots\\<br />
& 11 && 13 && 15 && 17 && 19 && \hdots \end{array}



    If we know one square, we can crank out the next few squares.


    For example: . 60^2 \,=\,3600

    Double the 60 and add 1: . 2(60) + 1 \,=\,121
    That is the odd number we will add.


    . . \begin{array}{ccc}\;3600 &=& 60^2 \\<br />
+ 121 \\<br />
\;3721 &=& 61^2 \\<br />
+ 123 \\<br />
\;3844 &=& 62^2 \\<br />
+ 125 \\<br />
\;3969 &=& 63^2 \\<br />
+ 127 \\<br />
\;4096 &=& 64^2 \\<br />
\vdots && \vdots<br />
\end{array}

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