Mentally Squaring Two-Digit Numbers

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Squaring a number means multiplying a number by itself. Learning how to do it mentally can be very encouraging, and can let you impress your friends. Additionally, practising it improves working memory and concentration skills.

In this post, you are going to learn how to mentally square a number of two digits. The technique explained here does not build on previous mental math techniques, so you can study this technique on its own. In future posts, I will explain how to square larger numbers, so you better practise this one while preparing for more complex ones.

The Technique

It is very simple. The best way to learn it is by going through the technique with an example.

Let’s say we have the number 64. The first step is to find the number that when subtracted to the number to be squared, rounds down  to the closest number that ends in zero.

The closest number to 64 that ends in zero is 60.

64 – 4 = 60 => hence 4 is the number we are looking for. We’ll call it the base.

If we add and subtract the base to the number to square we obtain the following two numbers:

64 + 4 = 68

64 – 4 = 60

Then the first part of the square will be obtained by multiplying those two numbers between them.

That multiplication is very easy to do mentally, because one of the numbers ends in zero. Hence, the multiplication is actually a multiplication of a number of two cyphers times a number of one cypher, and then add a zero at the end.

68 * 6 = 408 => we add a zero at the end => 4080

Then, to finish the square, we take the base and square it. Since it is a number of one cypher its square is straight from the multiplication tables you learned at basic school.

42 = 4 x 4 = 16

Then we add both results:

4080 + 16 = 4096

Final result: 4096

Some Examples

                   74

722  =              =  5180 + 22 = 5184

                   70

                   48

392      =              =  1440 + 92 = 1521

                   30

                   30

252  =             =  600 + 52 = 625

                   20

The Workout

I recommend doing the following workout until you master the technique. You should practise every day, taking a rest on Sundays.

  • Start the random numbers generator and configure it to generate numbers between 0 and 99 (two cyphers random number).
  • Start a timer for 1 minute. Your goal is to perform 10 squared powers in 1 minute.
  • Generate a random number and calculate its squared power as quickly as possible, using the technique explained above.
  • When calculated, write the result on the paper and go back to step number 2. Do not check now if you have correctly squared the numbers.
  • When the minute is up, check how many squared powers were correct.
  • If after 1 minute you have correctly done 10 or more, then you have mastered this technique and you can move to the next technique that squares any cyphers numbers. Otherwise, continue with this training.
  • Repeat the whole process five times.
  • Write down on your Training Sheet the number of cyphers you are able to calculate 10 squared powers in one minute.
  • Next day, try to beat that number.

Conclusion

The whole workout should take you less than 10 minutes, hence you can combine it with other mental math workouts. addition and subtraction workouts in the same training session, or with multiplication and division workouts.

For example, a good complete mental math workout can be of 30 minutes a day from Monday to Saturday. Each day include a 10 minutes of squared numbers workout. Then, on Monday, Wednesday and Friday, perform 10 minutes of addition and 10 minutes of subtraction workouts. On Tuesday, Thursday and Saturday, perform 10 minutes of multiplication and 10 minutes of division workouts.

Random number generator programs

 

 

photo credit: visualpanic via photopin cc

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