User:Eas4200c.f08.nine.r

Albuquerque's Multiplication method:
'''This method involves a different approach to solving multiplication problems. All that is required is basic knowledge of single digit multiplication (ex: 4x2, 5x8).'''

Altough this method seems complicated, once you become familiar with it's procedure, you can solve problems really fast!

'''This method is not practical for large numbers multiplied by large numbers, however, it offers an alternate way of solving a multiplication problem. '''

First step
We need to identify the number of digits we are multiplying:

ex: 43 x 87  (2 digits by 2 digits) ex: 9844 x 9485 (4 digits by 4 digits) If the digits are not the same, we look at the number with the larger number of digits:

ex: 738484 x 94 = 738484 x 000094 (6 digits by 6 digits) ex: 37 x 9 = 37 x 09 (2 digits by 2 digits)

Therefore any number multiplied by any other number takes the form:

$$\begin{bmatrix} x_1&x_2 &x_3  &...\ x_n  &\times   &y_1  &y_2  &y_3 &...\ y_n \\ \end{bmatrix}$$

where n is the number of digits.

ex: $$ \begin{bmatrix} 1&3 & \times &  4&5 \\ x_1&x_2 &\times  &y_1  & y_2 \end{bmatrix}$$

ex: $$ \begin{bmatrix} 5&7 &8  &0  &\times   &4  &2  &1 &9 \\ x_1&x_2 &x_3  &x_4  &\times  &y_1  &y_2  &y_3  &y_4 \end{bmatrix}$$

Second step
Apply general formula:

$$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red} [(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\ \ \color {blue} [(x_1 \times y_4) + (x_2 \times y_3) + (x_3 \times y_2) + (x_4 \times y_1)]\ \color {black} ....$$

Each term is the number of n, so for n=2 you only need the first two terms of this formula, for n=3 you need first three terms, etc...

ex: 13 x 45 (2 x 2 digits) (n=2)

Formula becomes: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)] \color {black}$$

ex: 134 x 984 (3 x 3 digits) (n=3)

Formula becomes: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red} [(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\color {black}$$

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'''Now you take your last term, ignore the first component and copy remaining components while adding 1 to each y term. Keep doing this untill you have a single component.'''

ex: 13 x 45 (2 x 2 digits) (n=2)

Formula: $$\ \ \ \ \color {red}[(x_1\times y_1)]\ \ \color {blue} \underbrace{[(x_1 \times y_2) + (x_2 \times y_1)]}_{Last \ Term} \color {black}$$

Take last term: $$\color {blue} \underbrace{[(x_1 \times y_2)}_{First \ Component} +\underbrace{(x_2 \times y_1)]}_{Remaining \ Component}\color {black}$$

Ignore first component, and copy remaining components while adding 1 to the y terms:

First component ignored: $$\color {blue}(x_1 \times y_2)\color {black}$$, Remaining components: $$\color {blue}(x_2 \times y_1)\color {black}$$

Copy remaining component while adding 1 to y term: $$\color {red}(x_2 \times y_1) \color {black}\rightarrow \color {red}(x_2 \times y_2)\color {black}$$

So your final formula for becomes:  $$

\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red} [(x_2 \times y_2)]\color {black}$$

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ex: 134 x 984 (3 x 3 digits) (n=3)

Formula: $$\color {red}[(x_1\times y_1)]\ \ \color {blue}[(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red} [(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\color {black}$$

Last term: $$\color {red}[(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\color {black}$$

Ignoring first component and repeating remaining components while adding 1 to y terms: $$\color {blue}[(x_2 \times y_3) + (x_3 \times y_2)]\color {black}$$

Repeat step again, ignore first term and repeat remaining term while adding 1 to y term: $$\color {red}[(x_3 \times y_3)]\color {black}$$

The reason we repeat the step is because we want to obtain a single component.

So final formula becomes:

$$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red}[(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\ \ \color {blue}[(x_2 \times y_3) + (x_3 \times y_2)]\ \ \color {red}[(x_3 \times y_3)]\color {black}$$

Third step
Now we start plugging in the known values and solving for each component:

ex: 13 x 45 (2 x 2 digits) (n=2)

Formula: $$

\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red} [(x_2 \times y_2)]\color {black}$$

Becomes: $$

\color {red} [(1\times 4)]\ \color {blue}[(1 \times 5) + (3 \times 4)]\ \color {red} [(3 \times 5)]\color {black} = \color {red}[4]\ \color {blue}[5 + 12]\ \color {red} [15] \color {black} = \color {red} [4]\ \color {blue} [17]\ \color {red} [15] \color {black} $$

Now we have to leave each term with only a single digit, except the first term, which can have up to two digits.

So we start working from right to left making sure terms have only one digit, so in this case:

$$ \color {red} [4]\ \color {blue}[17 + 1]\ \color {red} [5]\color {black} $$ since we moved the 1 in [15] to [17]

Now we have: $$ \color {red} [4]\ \color {blue} [18]\ \color {red} [5] \color {black}$$ but we need to move the 1 in [18] to [4]

So we have: $$ \color {red} [4 + 1]\ \color {blue} [8]\ \color {red} [5] \color {black} = \color {red} [5]\ \color {blue} [8]\ \color {red} [5] \color {black} = 585 $$

Final answer: 13 x 45 = 585!

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134 x 984 (3 x 3 digits) (n=3)

Formula: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red}[(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\ \ \color {blue}[(x_2 \times y_3) + (x_3 \times y_2)]\ \ \color {red}[(x_3 \times y_3)]\color {black}$$

So we have: $$\color {red}[(1\times 9)]\ \color {blue}[(1 \times 8) + (3 \times 9)]\ \color {red}[(1 \times 4) + (3 \times 8) + (4 \times 9)]\ \color {blue}[(3 \times 4) + (4 \times 8)]\ \color {red}[(4 \times 4)]\color {black}$$

Simplifying: $$ \color {red}[9]\ \color {blue}[8 + 27]\ \color {red}[4 + 24 + 36]\ \color {blue}[12 + 32]\ \color {red}[16]\color {black} = \color {red}[9]\ \color {blue}[35]\ \color {red}[64]\ \color {blue}[44]\ \color {red}[16]\color {black} $$

Now we start leaving each term with a single digit (except the first term), moving from right to left:

$$ \color {red}[9]\ \color {blue}[35]\ \color {red}[64]\ \color {blue}[44 + 1]\ \color {red}[6] \color {black} \rightarrow \color {red}[9]\ \color {blue}[35]\ \color {red}[64 + 4]\ \color {blue}[5]\ \color {red}[6] \color {black}\rightarrow \color {red}[9]\ \color {blue}[35 + 6]\ \color {red}[8]\ \color {blue}[5]\ \color {red}[6] \color {black}\rightarrow \color {red}[9 + 4]\ \color {blue}[1]\ \color {red}[8]\ \color {blue}[5]\ \color {red}[6]\color {black} \rightarrow \color {red}[13]\ \color {blue}[1]\ \color {red}[8]\ \color {blue}[5]\ \color {red}[6] \color {black} = 131856 $$

'''Final answer: 134 x 984 = 131856! '''

General cases
I have here a list of the formulas for different numbers: ____________________________________________________________________________

(2 x 2 digits)

Formula: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red} [(x_2 \times y_2)]\color {black}$$ ____________________________________________________________________________ (3 x 3 digits)

Formula: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red}[(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\ \ \color {blue}[(x_2 \times y_3) + (x_3 \times y_2)]\ \ \color {red}[(x_3 \times y_3)]\color {black}$$ ____________________________________________________________________________ (4 x 4 digits)

Formula: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red}[(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\ \ \color {blue}[(x_1 \times y_4) + (x_2 \times y_3) + (x_3 \times y_2) + (x_4 \times y_1)]\ \ \color {black} $$

$$ \color{red} [(x_2 \times y_4) + (x_3 \times y_3) + (x_4 \times y_2)]\ \ \color {blue} [(x_3 \times y_4) + (x_4 \times y_3)]\ \ \color {red} [(x_4 \times y_4)] \ \color {black} $$ ____________________________________________________________________________ (5 x 5 digits)

Formula: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red}[(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\ \ \color {blue}[(x_1 \times y_4) + (x_2 \times y_3) + (x_3 \times y_2) + (x_4 \times y_1)]\color {black} $$

$$ \color{red} [(x_1 \times y_5) + (x_2 \times y_4) + (x_3 \times y_3) + (x_4 \times y_2) + (x_5 \times y_1)]\ \ \color {blue} [(x_2 \times y_5) + (x_3 \times y_4) + (x_4 \times y_3) + (x_5 \times y_2)]\ \ \color {red} [(x_3 \times y_5) + (x_4 \times y_4) + (x_5 \times y_3)]\color {black} $$

$$\color {blue} [(x_4 \times y_5) + (x_5 \times y_4)]\ \ \color {red} [(x_5 \times y_5)] \color {black} $$ ____________________________________________________________________________ (6 x 6 digits)

Formula: $$\color {red}[(x_1\times y_1)]\ \ \color {blue} [(x_1 \times y_2) + (x_2 \times y_1)]\ \ \color {red}[(x_1 \times y_3) + (x_2 \times y_2) + (x_3 \times y_1)]\ \ \color {blue}[(x_1 \times y_4) + (x_2 \times y_3) + (x_3 \times y_2) + (x_4 \times y_1)]\color {black} $$

$$ \color{red} [(x_1 \times y_5) + (x_2 \times y_4) + (x_3 \times y_3) + (x_4 \times y_2) + (x_5 \times y_1)]\ \ \color {blue} [(x_1 \times y_6) + (x_2 \times y_5) + (x_3 \times y_4) + (x_4 \times y_3) + (x_5 \times y_2) + (x_6 \times y_1)] \color {black} $$

$$\color {red} [(x_2 \times y_6) + (x_3 \times y_5) + (x_4 \times y_4) + (x_5 \times y_3) + (x_6 \times y_2)]\ \ \color {blue} [(x_3 \times y_6) + (x_4 \times y_5) + (x_5 \times y_4) + (x_6 \times y_3)]\ \ \color {red} [(x_4 \times y_6) + (x_5 \times y_5) + (x_6 \times y_4)]\color {black} $$

$$ \color {blue} [(x_5 \times y_6)+ (x_6 \times y_5)]\ \ \color {red} [(x_6 \times y_6)]\color {black} $$

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NOTES:

Make sure your answer should not have more digits then the total number of digits you're multiplying:

ex: 2 x 2 digits should have an answer with 4 or less digits!

ex: 3 x 3 digits should have an answer with 6 or less digits!

This website was created by Mechanical and Aerospace engineer Ricardo Albuquerque.

Contact: Ricardo.albuquerque@piper.com