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Thread: Mental calculus tips..

  1. #1
    Moo
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    Mental calculations tips..

    Hello,

    1.*1.

    This has nothing to do with extraordinary technique, but you gotta think about it (I'm confident that some of you know about it)..

    $\displaystyle (10+a)(10+b)=10(10+a)+b(10+a)=10(10+a)+10b+ab=10(1 0+a+b)+ab$

    So multiplying two such numbers is equivalent to :

    - sum one of the numbers with the unit digit of the other one
    - multiply it by 10
    - add the product of the two unit digits.

    $\displaystyle 19 \times 17= ?$

    - $\displaystyle 17+9=26$

    - $\displaystyle 26 \times 10=260$

    - $\displaystyle 7 \times 9=63$

    - $\displaystyle 19 \times 17=260+63=323$


    I hope this would help at least one person in the future

    Enjoy calculations !
    Last edited by Moo; Aug 21st 2008 at 07:02 AM. Reason: calculation, not calculus...
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello

    I didn't know about this hint. Usually, I simply develop $\displaystyle (10+a)(10+b)=100+10(a+b)+ab$.

    In the case of $\displaystyle 19 \times 17$, where 9 and 7 are close to 10, I'd rather do $\displaystyle 19\times17=(20-1)(20-3)=400-20(1+3)+3$ which avoids manipulating too large numbers. (yes, I consider 63 as large )
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  3. #3
    Moo
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    Well, this goes too for 12x15 for example, which involves smaller numbers
    I only gave an example with random numbers... Sometimes, there are easier ways of calculating
    When I'm talking about mental calculations, I consider it as being without any paper & pen... Unfortunately, (20-1)(20-3) for some people needs some writing :-)

    Another thing :

    The square of a number whose unit digit is 5.

    If you're looking for the square of a number N=X5 (with X representating all the digits of the number, meaning that actually, $\displaystyle N=10X+5$), don't worry, this can be shortened ^^

    First of all, calculate $\displaystyle X(X+1)=Y$

    $\displaystyle N^2$ will simply be $\displaystyle Y25 \ (100Y+25)$

    For example :

    $\displaystyle 35^2 = ?$

    $\displaystyle X=3$
    $\displaystyle X(X+1)=3 \times 4={\color{red}12}$

    Hence $\displaystyle 35^2={\color{red}12}25$

    Another example :

    $\displaystyle 115^2= ?$

    $\displaystyle X=11$
    $\displaystyle X(X+1)=11 \times 12={\color{red}132}$

    Hence $\displaystyle 115^2={\color{red}132}25$
    Last edited by Moo; Apr 18th 2008 at 12:52 PM. Reason: calculation, not calculus...
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  4. #4
    GAMMA Mathematics
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    Good hint, but it hardly deals with Calculus concepts... more with arithmetic and algebra!
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  5. #5
    Moo
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    I'm sorry, I don't know that much about the different categories :s
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  6. #6
    Moo
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    Using binomial to avoid long calculations...

    If you're asked to multiply two numbers, you can rapidly go through this checking.
    See if the numbers are symmetric with respect to a number N.
    If so, the numbers a and b can be written as : $\displaystyle a=N-x$ and $\displaystyle b=N+x$
    The product $\displaystyle ab$ will be :

    $\displaystyle ab=(N-x)(N+x)$

    By using a binomial formula, you get :

    $\displaystyle ab=N^2-x^2$

    Which can sometimes be more sympathical to calculate..


    For example :

    $\displaystyle 58 \times 42 = ?$

    Notice that these numbers are symmetric with respect to 50.

    $\displaystyle 58=50+8$
    $\displaystyle 42=50-8$

    $\displaystyle 58 \times 42 =(50+8)(50-8)=50^2-8^2=2500-64=2436$




    I know that these methods look elementary, but it sometimes helps when you are in a hurry and not allowed to use any calculators
    Last edited by Moo; Apr 18th 2008 at 12:52 PM. Reason: claculation not calculus...
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  7. #7
    Bar0n janvdl's Avatar
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    Moo, I believe the word you are looking for is calculations, not calculus.

    Other than that, very nice, good to see someone besides Colby and TPH post some interesting tutorials on things.
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  8. #8
    Moo
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    Quote Originally Posted by janvdl View Post
    Moo, I believe the word you are looking for is calculations, not calculus.

    Other than that, very nice, good to see someone besides Colby and TPH post some interesting tutorials on things.
    Thanks, I've corrected the mistakes... And I've learnt one more thing !
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  9. #9
    Moo
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    Check for the unit digits (and cubic roots of numbers)

    In general cases, when you gotta check rapidly if your calculation seems correct, look at the digit number.

    Let $\displaystyle X=x*10+a$ and $\displaystyle Y=y*10+b$

    $\displaystyle XY=(ay+x+bx+y)*10+ab$

    Hence the unit digit of XY will be the unit digit of ab, which is easier to get.

    For example :

    $\displaystyle 46 \times 73$

    Here, $\displaystyle a=6$ and $\displaystyle b=3$. So $\displaystyle ab=1{\color{blue} 8}$
    Hence the unit digit of $\displaystyle 46 \times 73$ is $\displaystyle {\color{blue} 8}$
    If you find a result whose digit number is not 8, you can directly calculate again.




    Here is only a beginning for cubic roots... And shall be known

    $\displaystyle 1^3={\color{red}1}$
    $\displaystyle 2^3={\color{red}8}$
    $\displaystyle 3^3=2{\color{red}7}$
    $\displaystyle 4^3=6{\color{red}4}$
    $\displaystyle 5^3=12{\color{red}5}$
    $\displaystyle 6^3=21{\color{red}6}$
    $\displaystyle 7^3=34{\color{red}3}$
    $\displaystyle 8^3=51{\color{red}2}$
    $\displaystyle 9^3=72{\color{red}9}$
    $\displaystyle 10^3=100{\color{red}0}$

    Notice that each number appears once in the unit digit.

    This means that if, for example, you're looking for the cubic root of a number whose unit digit is 6, its cubic root necessarily ends with a 6.
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  10. #10
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Moo View Post
    Hello,

    1.*1.

    This has nothing to do with extraordinary technique, but you gotta think about it (I'm confident that some of you know about it)..

    $\displaystyle (10+a)(10+b)=10(10+a)+b(10+a)=10(10+a)+10b+ab=10(1 0+a+b)+ab$

    So multiplying two such numbers is equivalent to :

    - sum one of the numbers with the unit digit of the other one
    - multiply it by 10
    - add the product of the two unit digits.

    $\displaystyle 19 \times 17= ?$

    - $\displaystyle 17+9=26$

    - $\displaystyle 26 \times 10=260$

    - $\displaystyle 7 \times 9=63$

    - $\displaystyle 19 \times 17=260+63=323$


    I hope this would help at least one person in the future

    Enjoy calculations !
    Hi Moo,

    I was jus wondering if you do this....say I solve an equation and get $\displaystyle x=\frac{288}{128}$...my mind works like this $\displaystyle x=\frac{200+88}{100+28}=\frac{4(50+22)}{4(25+7)}=\ frac{77}{37}$ that is how my mind simplifies things...its much quicker in my mind though...is that how you process things?
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  11. #11
    Moo
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    Hm...
    It was a reflex for me to divide by 4...

    I see that 288=280+8 rather than 200+88
    Then 120+8 rather than 128..
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  12. #12
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    Quote Originally Posted by Moo View Post
    Hello,

    1.*1.

    This has nothing to do with extraordinary technique, but you gotta think about it (I'm confident that some of you know about it)..

    $\displaystyle (10+a)(10+b)=10(10+a)+b(10+a)=10(10+a)+10b+ab=10(1 0+a+b)+ab$

    So multiplying two such numbers is equivalent to :

    - sum one of the numbers with the unit digit of the other one
    - multiply it by 10
    - add the product of the two unit digits.

    $\displaystyle 19 \times 17= ?$

    - $\displaystyle 17+9=26$

    - $\displaystyle 26 \times 10=260$

    - $\displaystyle 7 \times 9=63$

    - $\displaystyle 19 \times 17=260+63=323$


    I hope this would help at least one person in the future

    Enjoy calculations !

    well, pretty good, but I knew the mental multiplication tricks, all you have to do is dismantle the multiplication steps and add
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  13. #13
    Moo
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    I know it's a basic thing, but it simplifies steps when you directly apply it
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