Number Representation : 32-bit Single-Precision Floating-Point Numbers

Manostaxx The text that follows is owned by the site above referred. Here is only a small part of the article, for more please follow the link SOURCE: https://www3.ntu.edu.sg/home/ehchua/programming/java/datarepresentation.html In computers, floating-point numbers are represented in scientific notation of fraction (F) and exponent (E) with a radix of 2, in the form of F×2^E. Both […]

Formulas aproximadas de calculo de pi

Manostaxx

Aqui ficam algumas formulas para o calculo aproximado de pi:

      22 / 7       = 3.142857142857143...
     201 / 64      = 3.140625 
     333 / 106     = 3.141509433962264...
     355 / 113     = 3.141592920353983...
   75948 / 24175   = 3.141592554291623...
  100798 / 32085   = 3.141592644537946...
  103993 / 33102   = 3.141592653011902...
  312689 / 99532   = 3.141592653618936...
  833719 / 265381  = 3.141592653581078...
 4272943 / 1360120 = 3.141592653589389...
 5419351 / 1725033 = 3.141592653589815...
        PI         = 3.141592653589793.....

 

Curiosidade Matematica – Formulas para Calcular Senos

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SOURCE: http://erichware.info/en/tutorial/math/sines.htm

                                                                __________
             ___     ___     ___     ___       ___     ___       /       ___
           \/ 30 + \/ 10 - \/ 6  - \/ 2  - ( \/ 6  - \/ 2  ) * \/ 10 + \/ 20
sin(03) = -------------------------------------------------------------------
                                        16

              ____________      ___________
             /         ___     /        ___
           \/ 30 - 6*\/ 5  - \/ 6 + 2*\/ 5
sin(06) = ----------------------------------
                           8

              _________      _________
             /      ___     /      ___
           \/ 3 + \/ 5  - \/ 5 - \/ 5
sin(09) = -----------------------------
                        4

              ____________
             /         ___     ___     ___
           \/ 10 + 2*\/ 5  + \/ 3  - \/ 15
sin(12) = ---------------------------------
                          8

              _________
             /      ___
           \/ 2 - \/ 3
sin(15) = --------------
                2

              _________
             /  ___
           \/ \/ 5  - 1
sin(18) = --------------
                4

                 ____________________________________________________________
                /             _______________________________________________
               /             /                                   ____________
           \  /          \  /       ___            ___          /         ___
            \/  2 - 1/2 * \/  7 - \/ 5  + 1/2 * (\/ 5  - 1) * \/ 30 + 6*\/ 5
sin(21) = --------------------------------------------------------------------
                                            2

                ___________________________
               /               ____________
           \  /       ___     /         ___
            \/  7 + \/ 5  - \/ 30 + 6*\/ 5
sin(24) = ----------------------------------
                          4

                _________________________
               /             ____________
           \  /             /         ___
            \/  2 - 1/2 * \/ 10 - 2*\/ 5
sin(27) = --------------------------------
                         2


sin(30) = 1/2

                 ________________________________________
                /             ___________________________
               /             /               ____________
           \  /          \  /       ___     /         ___
            \/  2 - 1/2 * \/  7 + \/ 5  - \/ 30 + 6*\/ 5
sin(33) = ------------------------------------------------
                                 2

              ____________
             /         ___
           \/ 10 - 2*\/ 5
sin(36) = -----------------
                  4

                _____________________________________________
               /               ____________
           \  /               /         ___     ___     ___
            \/  2 - 1/4 * ( \/ 10 + 2*\/ 5  + \/ 3  - \/ 15 )
sin(39) = ----------------------------------------------------
                                   2

                ________________________________________________
               /                                    ____________
           \  /       ___             ___          /         ___
            \/  9 + \/ 5  -  1/2 * (\/ 5  - 1) * \/ 30 + 6*\/ 5
sin(42) = -------------------------------------------------------
                                    4

             ___
           \/ 2
sin(45) = -------
             2

                ________________________________________________
               /                                    ____________
           \  /       ___             ___          /         ___
            \/  7 - \/ 5  +  1/2 * (\/ 5  - 1) * \/ 30 + 6*\/ 5
sin(48) = -------------------------------------------------------
                                    4

                _____________________________________________
               /               ____________
           \  /               /         ___     ___     ___
            \/  2 + 1/4 * ( \/ 10 + 2*\/ 5  + \/ 3  - \/ 15 )
sin(51) = ----------------------------------------------------
                                   2

              ___________
             /        ___
           \/ 6 + 2*\/ 5
sin(54) = ----------------
                 4

                 ________________________________________
                /             ___________________________
               /             /               ____________
           \  /          \  /       ___     /         ___
            \/  2 + 1/2 * \/  7 + \/ 5  - \/ 30 + 6*\/ 5
sin(57) = ------------------------------------------------
                                 2

             ___
           \/ 3
sin(60) = -------
             2

                _________________________
               /             ____________
           \  /             /         ___
            \/  2 + 1/2 * \/ 10 - 2*\/ 5
sin(63) = --------------------------------
                         2

                ___________________________
               /               ____________
           \  /       ___     /         ___
            \/  9 - \/ 5  + \/ 30 + 6*\/ 5
sin(66) = ----------------------------------
                          4

                 ____________________________________________________________
                /             _______________________________________________
               /             /                                   ____________
           \  /          \  /       ___            ___          /         ___
            \/  2 + 1/2 * \/  7 - \/ 5  + 1/2 * (\/ 5  - 1) * \/ 30 + 6*\/ 5
sin(69) = --------------------------------------------------------------------
                                            2

              ____________
             /         ___
           \/ 10 + 2*\/ 5
sin(72) = -----------------
                  4

              _________
             /      ___
           \/ 2 + \/ 3
sin(75) = --------------
                2

                ________________________________________________
               /                                    ____________
           \  /       ___             ___          /         ___
            \/  9 + \/ 5  +  1/2 * (\/ 5  - 1) * \/ 30 + 6*\/ 5
sin(78) = -------------------------------------------------------
                                    4

                _________________________
               /             ____________
           \  /             /         ___
            \/  2 + 1/2 * \/ 10 + 2*\/ 5
sin(81) = --------------------------------
                         2

                ___________________________
               /               ____________
           \  /       ___     /         ___
            \/  7 + \/ 5  + \/ 30 + 6*\/ 5
sin(84) = ----------------------------------
                          4

                 ________________________________________
                /             ___________________________
               /             /               ____________
           \  /          \  /       ___     /         ___
            \/  2 + 1/2 * \/  7 + \/ 5  + \/ 30 + 6*\/ 5
sin(87) = ------------------------------------------------
                                 2



sin(90) = 1

 

Difficult conversations: Seek first to understand, then to be understood

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SOURCE: http://www.lifecoach-directory.org.uk/lifecoach-articles/difficult-conversations-seek-first-to-understand-then-to-be-understood

Written by: Karin Mueller

Karin MuellerAll conversations work best when both parties feel respected and able to say what they feel needs to be said. However, creating an atmosphere where this is possible isn’t always easy. And without a doubt, the higher the stakes and the bigger the anticipated disagreement, the harder it gets.

Preparing for difficult conversations is key but when the time has come to talk, keeping in mind a few crucial points will help you to have a respectful and constructive conversation. And if common ground exists, it’ll be much easier to find.

  • Be genuinely curious: Many disagreements stem from inaccurate assumptions and interpretations. Meeting the other person with a genuinely curious and explorative mindset changes the dynamics. Ask them what their views are. Be prepared to listen, and be possibly surprised, as you might learn things you didn’t know or expect. ‘Seek first to understand and then to be understood’ is a great phrase to keep in mind.
  • Avoid blame: Whatever has happened, looking for whose fault it is will only block the path to finding a common way forward. As the one blamed feels backed into a corner, you can expect defensiveness and possibly aggression. So instead of pointing the finger, concentrate on finding common ground and discussing where to go from here.
  • Always play the ball, never the player: Have you ever witnessed an argument that started around who’s taking out the rubbish and quickly escalated into a series of nuclear attacks and counter attacks on everything the other person stood for? Using a small argument as an opportunity for an all-encompassing personal attack creates winners and losers, but hardly leads to an agreed way forward. So never lose sight of what’s prompted the conversation, and stick to it.
  • Acknowledge emotions: Difficult conversations are bound to get emotional from time to time. And whether it’s anger, cynicism, tears or rage – yours or that of the other person – emotions tell you that something’s going on that matters. So don’t ignore them, but acknowledge them.
  • Breathe: When things get tough, try to pause and breathe calmly. Not only will it help you to reset and refocus, but it’ll also buy you a little time. Silence is your friend and might well be a welcome mini break for the other person too.
  • Summarise outcomes: In an ideal conversation, you will reach a way forward that, even if not universally liked, will have been discussed and acknowledged by everyone involved. Make sure you summarise these outcomes, actions and agreements.
  • Follow up: And finally, when all is said and done and the issue is off the table, make sure to have another unrelated interaction with this person before too long. Show them that just because one issue needed sorting out, it hasn’t impacted on your relationship.

The 6174 Trick

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SOURCE: http://www.learn-with-math-games.com/math-tricks-for-kids.html

Math Skills: This trick is good for helping students practice four-digit subtraction.

What You Need: Piece of paper and pencil

Performing the Math Trick

  • Have someone choose any four-digit number.  (At least one digit should be different from the other digits)
  • Arrange the digits in order from largest to smallest and write that number down.

Example

  • The person chooses the number 4,081
  • Rearrange digits in order from largest to smallest:

8410

  • Now have them reverse the digits of this number and subtract it from the last number they wrote down.
  • After they subtract and get the answer, have them arrange the numbers from greatest to least if they are not already.
  • Now have them subtract the reverse of this number from itself (same procedure as above).
  • Keep repeating these same steps and they will soon arrive at the magic number: 6174!  

Not at the Magic Number yet, so they keep going.

And just like magic…..they will reach the Magic Number! (See below)

Pretty Cool!  It works every time.

Note: It might be a good idea to make sure they are doing the subtractions correctly, otherwise they may not get to the Magic Number.

Easy Rules for Divisibility

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SOURCE: http://www.wisebread.com/11-useful-genius-math-tricks-that-are-actually-easy

If you need to be able to decide quickly if 408 slices of pie can be evenly split by 12 people, here are some useful shortcuts. These rules works for all numbers without fractions and decimals.

  • Divisible by 2 if the number’s last digit is divisible by 2 (e.g. 298).
  • Divisible by 3 if the sum of the digits of the number are divisible by 3 (501 is because 5 + 0 + 1 equals 6, which is divisible by 3).
  • Divisible by 4 if the last two digits of the number are divisible by 4 (2,340 because 40 is a multiple of 4).
  • Divisible by 5 if the last digit is 0 or 5 (1,505).
  • Divisible by 6 if the rules of divisibility for 2 and 3 work for that number (408).
  • Divisible by 9 if the sum of digits of the number are divisible by 9 (6,390 because 6 + 3 + 9 + 0 equals 18, which is divisible by 9).
  • Divisible by 12 if the rules of divisibility for 3 and 4 work for that number (e.g. 408).

Multiplication Shortcut for 2 Digit Numbers

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SOURCE: https://www.easycalculation.com/math-facts/multiplication-shortcut-for-two-digit-numbers.html

This trick can be applied when

– First digit of both the numbers (multiplicand and multiplier) are same.

– Sum of their second digits equals 10.

Example: 23 x 27

In the above example, as first digit of both the numbers are 2 and sum on second digits (3+7) is 10 we can use this trick.

Follow the simple steps to learn the shortcut trick

Step 1: Take the first digit, add it with one and multiply the resultant value with the same.

= (2+1) x 2 = 6

Step 2: Multiply the second digits of both the numbers.

= 3×7 = 21

Step 3: Now join and write the step 1 and step 2 result together. That is the answer. So 23 x 27 = 621

With few more examples,

Example 1: 92 x 98

(9 + 1) x 9 = 90

2 x 8 = 16

Answer is 92 x 98 = 9016

Example 2: 77 x 73

(7 + 1) x 7 = 56

7 x 3 = 21

Answer is 77 x 73 = 5621

Example 3: 85 x 85

(8 + 1) x 8 = 72

5 x 5 = 25

Answer is 85 x 85 = 7225

Finding out the missing digits

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SOURCE: https://www.easycalculation.com/math-facts/finding-out-the-missing-digits.html

Here is an interesting trick to find the missing number. Follow the steps below to learn this simple trick.

Step 1: Let your friend to write down 4 to 5 digit numbers.

For example, 39861

Step 2: Ask your friend to sum up the digits.

3 + 9 + 8 + 6 + 1 = 27

Step 3: Now ask him/her to subtract the step 2 result from the main value

39861 – 27 = 39834

Step 4: Ask you friend to leave one digit and say all other digit from the result.

Let us keep that he left 8 and says the remaining number 3 9 x 3 4 to you.

Step 5: You can find the left out digit with the simple trick

Just add up all the digits given by your friend

3 + 9 + 3 + 4 = 19 = 1 + 9 = 10 = 1+0 = 1

Now subtract from 9, the result is 9-1 = 8

Note: How it works. Any number subtracted from the sum of its digits will be a multiple of Nine. That’s the magic of number nine.

Vedic math shortcut for the multiple of nine series

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SOURCE: https://www.easycalculation.com/math-facts/vedic-math-shortcut-for-the-multiple-of-nine-series.html

This shortcut technique is useful for the multiplication of nine series.

Let us see this shortcut with an example,

456 x 999

Step 1: Take the first value and subtract with 1.

456 – 1 = 455

Step 2: Subtract the first number from the base value. Here its 1000.

1000 – 456 = 544

Step 3: Combine the results of step 1 and step 2. So, the answer is 456 x 999 = 455544

Note: Use this shortcut when when both number have same number of digits.