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Math world, simple mental calculations or what’s going on with education?

Today, I want to write blog post which is absolutely not related to programming. It related to math and education in general those days. During work interviews, I see a lot of people, who was absolutely unable to calculate mentally. They just can’t understand, that it’s possible to do without calculators. When my kids (2nd, 6th and 7th grade) were small I taught them to play with numbers, and until 4th grade (bigger kids) they were able do it. but then school teachers “killed” this ability. Why people should use calculator for simple math operations, if he can do it mentally? Shame you, the modern educational system. Let’s go back and try to understand how people were able to live without devil devices, such as calculators…

Following the paint of Nikolai Bogdanov-Belsky “Counting in their heads”. This painting is dated 1895.

Богданов Бельский - Устный счёт

As you can see at the painting, peasant kids trying to solve following exercise mentally:

(102 + 112 + 122 + 132 + 142) / 365

This is not very simple exercise, especially when should be solved without your favorite calculator. However, when I was 4th grade I learned to square two-digit numbers mentally (my, and I think, yours too): First, find the nearest multiple of ten, by raising or lowering your number, then add and remove the rest to each of numbers and add the square of oddment. For example

45 * 45 = (45+5) * (45-5) + (5 * 5) = 50 * 40 + 25 = (5 * 4) * 100 + 25 = 20 * 100 + 25 = 2000 + 25 = 2025
14 * 14 = (14+4) * (14-4) + (4 * 4) = 18 * 10 + 16 = 180 + 16 = 196

So, now it can be solved easily:

102 = 100
112= (11+1) * (11-1) + 1 = 12 * 10 + 1 = 121
122= (12+2) * (12-2) + 4 = 14 * 10 + 4 = 144
132= (13+3) * (13-3) + 9 = 16 * 10 + 9 = 169
142= (14+4) * (14-4) + 16 = 18 * 10 + 16 = 196

And so on… but wait, 100+121+144 already equals 365, which is our denominator. Next sequence will bring us 169+196, which is also 365. So the answer to this black board brain teaser is 2.

However, it can be rather complicated to calculate 862 for instance:

862 = (86 + 4) * (86 – 4) + (4 * 4) = 90 * 82 + 16…

Let’s try another way – multiple the difference between the number and 25 by 100, then add the square of the difference or excess of the number and 50. For example

862 = (86 – 25) * 100 + (86 – 50)2 = 61 * 100 +  362 = 6100 + (36 – 25) * 100 + (50 – 36)2 = 6100 + 1100 + 142 = 7200 + 196 = 7396

Isn’t it really simple and fun to calculate squares of numbers?

Bonus: how to calculate multiple of two digit numbers with the sum of its unity digits equals to 10?

  1. Multiply first digit of the first number by 10
  2. Add 1 to first digit of second number and multiply the result by 10
  3. Multiply results of step 1 and step 2
  4. Deduct second number and the result of step 1
  5. Multiply second digit of the first number by the result of step 4
  6. Add results of steps 3 and 5

Looks complicated? Let’s make it easier. Assuming that first number is X = 10x + z and second number is Y = 10y + (10 – z), the formula for quick multiplication calculation is: 100 * x * (y + 1) + z * (Y – 10 * x). For example:

96 * 84 = 100 * 9 * (8+1) + 6 * (84-10 * 9) = 100 * 9 * 9 + 6 * (84 – 90) = 8100 – 6 * 6 = 8100 – 36 = 8064
37 * 93 = 100 * 3 * (9+1) + 7 * (93 – 10 * 3) = 3000 + 7 * 63 = 3000 + (100 * 6 * 1 + 3 * (7 – 60)) = 3000 + 600 – 3 * 53 = 3600 – 159 = 3441

Have a nice day and be good people. Also, throw out all hardware calculators and uninstall all software :)

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8 Responses to “Math world, simple mental calculations or what’s going on with education?”

  1. Ali Says:

    Owesome. Shame on bad usage of calculators instead of using our great inner calculator.

  2. Christopher Bennage Says:

    I thought that this was only a problem with education in the US. :-(
    I have a theory (and the evidence continues to support it) that children are born smart and that we adults dumb them down (mostly for our convenience).

  3. Sergio Says:

    everithnig is much easier on thіs painting: In many exUSSR countries scholars learn squares of first 10 2-digit numbers by rote…
    10 – 100
    11 – 121
    13 – 169
    14 – 196
    15 – 225
    16 – 256
    17 – 289
    18 – 324
    19 – 361
    20 – 400

  4. Dano Says:

    Ok, I must be much younger than you and am terrible at math. I tried to comprehend your first example with 45. Still not making sense, I sure hope the impending nuclear fallout doesn’t wipe out calculators.

    I agree with Christopher and Sergio. If I would have learned some basics, perhaps I could do some harder calculations mentally.

  5. Getafix Says:

    In technologically advanced countries we can’t even do simple additions and subtractions without calculators (check out clerks in shops) whereas in technology deprived countries the shopkeepers keep track of your purchases mentally and do it faster than that we do with calculators.

    Another easy way to accomplish finding squares will be to apply the two formulas (a+b)^2 = a^2 + b^2 + 2*a*b and (a-b)^2 = a^2 – 2*a*b + b^2 as appropriate.

    These formulas can be used effectively to do mental math.

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