/anzan_training

Primary LanguagePythonGNU General Public License v3.0GPL-3.0

anzan_training

This is a simple training programm for multiplication of (up to) two-digit integers.

screenshot

This is probably not for anzan (mental arithmetic) per se, but still helps.

It has three courses:

  1. Genral
  2. Indian
  3. Mixed

Methods of calculation

In the Indian course (2), probelms are designed to work with so called Indian calculations.

There are three types.

  1. The tens place of a and b match and the sum of the units place of a and b is 10 (eg. $97 \times 93$)
  2. The tens place of a and b match and the sum of the units place of a and b is NOT 10 (eg. $78 \times 76$)
  3. the units place of a and b match and the sum of the tens place of a and b is 10 (eg. $79 \times 39$)

In case 1,

  1. Multiply the tens place of a with the sum of the tens place of b and 1 ($9 \times (9 + 1)= 90$)
  2. Join this value with the product of the units place of a and that b ($7 \times 3$)

eg. $97 \times 93 = 9021$ as $9 \times (9 + 1) = 90$ is followed by $7 \times 3 = 21$

$$ \begin{align} (10p + q) \times (10p + (10-q)) &= 100 \cdot p^2 + 10 p (q + 10 - q) + q(10-q) \\ &= 100 p^2 + q(10-q) \end{align} $$

In case 2,

  1. Add the unit place of b to a to gain c ($78 + 6 = 84$)
  2. Multiply this value c ($84$) with the tens place of b (9) ($84 \times 7 = 588$)
  3. Obain the product of the units place of a and b ($8 \times 6$) to 10-fold of the product above ($588 \times 10$)

eg. $78 \times 76 = 5928$ as $(78 + 6) \times 7 = 588$ and add $8 \times 6$ to $588\times10$

$$ \begin{align} (10p + q) \times (10p + r) &= 100 p^2 + 10 p (q + r) + qr \\ &= (10 p + q + r) 10 p + qr \end{align} $$

In case 3,

  1. Add the unit places of b to the product of the tens place of a and b ($7 \times 3 + 9$)
  2. Join the product of the units place of a and b.

eg. $79 \times 39 = 3081$ as $7 \times 3 + 9 = 30$ is followed by $9 \times 9 = 81$

$$ \begin{align} (10p + q) \times (10(10-p) + q) &= 100 p(10-p) + 10q (p + (10-p)) + q^2 \\ &= 1000p -100p^2 + 100q + q^2\\ &= (10p -p^2 + q)*100 + q^2\\ &= (p(10-p) + q)*100 + q^2\\ \end{align} $$

In general,

  1. Join the product of the tens place of a and b and that of the units place of a and b
  2. Obtain the sum of the product of the tens place of a and the units place of a and b and that of the units place of a and the tens place of b
  3. Add the numbers by adjusting the digits

eg. $56 \times 79 = 4424$ as $7\times5$ is joined with $6 \times 9$ to obtain $3554$, and $(5 \times 9 + 6 * 7) \times 10 = 87 \times 10$ is added.

$$ \begin{align} (10p + q) \times (10r + s) &= 100 pr + qs + 10(ps +qr) \\ \end{align} $$