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...

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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.

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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:
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(2 x 2 digits)

Formula: $\color {red}[(x_{1}\times y_{1})]\ \ \color {blue}[(x_{1}\times y_{2})+(x_{2}\times y_{1})]\ \ \color {red}[(x_{2}\times y_{2})]\color {black}$


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(3 x 3 digits)

Formula: $\color {red}[(x_{1}\times y_{1})]\ \ \color {blue}[(x_{1}\times y_{2})+(x_{2}\times y_{1})]\ \ \color {red}[(x_{1}\times y_{3})+(x_{2}\times y_{2})+(x_{3}\times y_{1})]\ \ \color {blue}[(x_{2}\times y_{3})+(x_{3}\times y_{2})]\ \ \color {red}[(x_{3}\times y_{3})]\color {black}$


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(4 x 4 digits)

Formula: $\color {red}[(x_{1}\times y_{1})]\ \ \color {blue}[(x_{1}\times y_{2})+(x_{2}\times y_{1})]\ \ \color {red}[(x_{1}\times y_{3})+(x_{2}\times y_{2})+(x_{3}\times y_{1})]\ \ \color {blue}[(x_{1}\times y_{4})+(x_{2}\times y_{3})+(x_{3}\times y_{2})+(x_{4}\times y_{1})]\ \ \color {black}$

$\color {red}[(x_{2}\times y_{4})+(x_{3}\times y_{3})+(x_{4}\times y_{2})]\ \ \color {blue}[(x_{3}\times y_{4})+(x_{4}\times y_{3})]\ \ \color {red}[(x_{4}\times y_{4})]\ \color {black}$


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(5 x 5 digits)

Formula: $\color {red}[(x_{1}\times y_{1})]\ \ \color {blue}[(x_{1}\times y_{2})+(x_{2}\times y_{1})]\ \ \color {red}[(x_{1}\times y_{3})+(x_{2}\times y_{2})+(x_{3}\times y_{1})]\ \ \color {blue}[(x_{1}\times y_{4})+(x_{2}\times y_{3})+(x_{3}\times y_{2})+(x_{4}\times y_{1})]\color {black}$

$\color {red}[(x_{1}\times y_{5})+(x_{2}\times y_{4})+(x_{3}\times y_{3})+(x_{4}\times y_{2})+(x_{5}\times y_{1})]\ \ \color {blue}[(x_{2}\times y_{5})+(x_{3}\times y_{4})+(x_{4}\times y_{3})+(x_{5}\times y_{2})]\ \ \color {red}[(x_{3}\times y_{5})+(x_{4}\times y_{4})+(x_{5}\times y_{3})]\color {black}$

$\color {blue}[(x_{4}\times y_{5})+(x_{5}\times y_{4})]\ \ \color {red}[(x_{5}\times y_{5})]\color {black}$


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(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