Multiplication using vedic mathematiis continues

Multiplying Two Numbers that are 'Closely Related'

We are now ready to extend the multiplication technique described above to the most general case, i.e. the multiplication of any two numbers that are 'closely related'. The precise definition of 'closely related' is:

"Numbers that are a small distance away from a 'proportional power of 10' such that the differences between the original numbers and this proportional power of 10 are simple to multiply".

This may sound very complicated, but it is actually quite simple. Firstly the 'proportional power of 10' is just a simple multiple or division of a power of 10, powers of 10 being 10, 100, 1000, 10000, etc. So, proportional powers of 10 are 10, 20, 25, 30, 40,... 80, 90,100, 200, 250, 300,... 800, 900, etc. etc. (Note that 25, 250, etc. are proportional powers of 10 in this definition because they are a simple division of a power of 10, e.g. 100/4=25).
So If both of the numbers you want to multiply are 'close' to one of these numbers (i.e. the residuals are small enough so that you can multiply them easily), then the extended technique can be used. This extended technique is simply the addition of the sub-sutra Anurupyena or Proportionately to the technique already described above. The complete description of the extended Vertically and Crosswise technique is:

1. Place the two numbers you wish to multiply one on top of the other, leave an answer line below.
2. Choose a 'working base' that is close to both numbers, this must be a 'proportional power of 10'. The 'theoretical base' is the actual power of 10 before you have multiplied or divided it to get your 'working base'. e.g. If the 'working base' is 25 then the 'theoretical base' would be 100 (the 'theoretical base' of 100 would have been divided by 4 in this case to get the 'working base' of 25). Remember the 'proportionality' of what you have done, (e.g. if you have divided by 4 or multiplied by 3 etc.), you will use this 'proportionality' correct the left hand side of the answer.
3. Subtract the 'working base' from each of the original numbers and place the results to the right of each, (remember to include the sign of the result, e.g. 21-25=-4, 28-25=+3). We will call these results the 'residuals'.
4. Vertically multiply the 'residuals' obtained above noting the sign of the result, (i.e. if the signs of the 'residuals' are different then the sign of the multiplication result will be negative, if the signs of the 'residuals' are the same then the result will be positive. We will call this the 'Vertically Result'.
5. Add or Subtract Crosswise following the sign present on the particular diagonal chosen, (you will get the same answer no matter which diagonal is selected so pick the easiest calculation to create the 'Crosswise Result'.
6. 'Correct' the 'Crosswise Result' by repeating the 'proportionality' used to create the 'working base', (e.g. if you divided by 4 to get your 'working base' then divide the 'Crosswise Result' by 4 also). Note that if the proportionality was a division, and if this resulted in a result with a fractional part, then this fractional part must be transferred to the right hand side of the answer by adding the same fractional proportion of the theoretical base to the previously calculated 'Vertically result'.
7. If there are now too many digits in 'Vertically Result' (i.e. more than the number of zeroes in the 'theoretical base'), then you have to carry the leading digit(s) to the next stage remembering to preserve the signs. We will call the result of the Vertically multiplication after any carry has been removed the 'remainder'.
8. If the 'remainder' is positive you can just place it on the answer line, if it is negative, it must be replaced with it's compliment before placing it on the answer line, (remember the compliment of a number is the result of subtracting the number from the 'theoretical base' ). If you have to compliment the 'remainder' to make it positive then you must also reduce the carry by 1, (if there is no carry yet then the carry becomes -1)
9. Add or Subtract any carry (according to it's sign) to the corrected Crosswise result above, place this result in the answer line to the left of the 'remainder' part of the answer.
10. That's it, the digits on the answer line are the results of the original multiplication.

Now, that all sounds terribly complicated, but it really is much harder to write (and read) the steps than to actually follow them! In fact you have already followed all the steps in the many previous examples, the only additional steps are the 'proportionately calculations. Some examples will clarify.

• Multiply 489x512

${\displaystyle {\begin{aligned}W.B.=500=100\times 5&\\{\begin{matrix}489&\longrightarrow &-11\\\\512&\longrightarrow &+12\\\hline \quad \end{matrix}}\quad \Rightarrow \quad &{\begin{matrix}489&&-11\\&&\downarrow \\512&&+12\\\hline &&-132\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}489&&-11\\&\swarrow &\\512&&+12\\\hline 501&&-132\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}489&&-11\\&&\\512&&+12\\\hline 5\times 501&&-132\end{matrix}}\quad \Rightarrow \quad \end{aligned}}}$

As you can see above, first a 'working base' of 500 is chosen as it is close to both numbers, in this case we use a 'theoretical base' of 100 and a 'proportionality' of x5, (we could also have used 1000 and ÷2 as will be seen in the next example). The 'residuals' are then found i.e. -11 and +12 and these are then multiplied giving -132 (the multiply by 11 rule is used here giving 132 but as the residual signs are different the answer is -132). We now subtract Crosswise 512-11 resulting in 501 (we could have chosen the other diagonal if we wanted i.e. 489+12=501, both give the same answer). Then we 'correct' the Crosswise subtraction by repeating the original proportionately step, i.e. 501x5=2505.

${\displaystyle {\begin{matrix}489&&-11\\&&\\512&&+12\\\hline 2505&&-132\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}489&&-11\\&&\\512&_{-1}&+12\\\hline 2505&&-32\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}489&&-11\\&&\\512&_{-2}&+12\\\hline 2505&&68\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}489&&-11\\&&\\512&&+12\\\hline 2503&&68\end{matrix}}}$

We can only have 2 digits in the 'remainder' result because our 'theoretical base' is 100 so we must carry the -1 leaving -32 on the answer line, however we can't put a negative number on the answer line either so we compliment -32 resulting in 68 (and then reduce the carry by 1 to -2). Finally we subtract the carry giving 2503. Thus the final answer is the contactination of both parts, i.e. 250368
This calculation is as complicated as it gets and every step has been split out into it's constituent parts intentionally so you can more easily see and understand each step, but remember that you would not write each step down like this, you would usually either do it all mentally or just write down the part answers directly.
You can see below that choosing the alternative proportionality (i.e. 1000÷2) results in the same answer:

• Multiply 489x512

${\displaystyle {\begin{aligned}W.B.=500=1000\div 2&\\{\begin{matrix}489&\longrightarrow &-11\\\\512&\longrightarrow &+12\\\hline \quad \end{matrix}}\quad \Rightarrow \quad &{\begin{matrix}489&&-11\\&&\downarrow \\512&&+12\\\hline &&-132\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}489&&-11\\&\swarrow &\\512&&+12\\\hline 501\div 2&&-132\end{matrix}}\quad \Rightarrow \quad \end{aligned}}}$

You can see above that we are performing the same multiplication (489x512) and we are still using a 'working base' of 500 but this time the 'theoretical base' is 1000 and the 'proportionality' is ÷2. The calculation proceeds as before, with the the diagonally subtraction resulting in 501, but this time the 'proportionality' is ÷2 so we have to divide 501 by 2 resulting in 250½.

${\displaystyle {\begin{matrix}489&&-11\\&&\\512&&+12\\\hline 250{\frac {1}{2}}&&-132\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}489&&-11\\&&\\512&&+12\\\hline 250&&368\end{matrix}}}$

We can't write ½ in the answer so we transfer it back to the remainder column as half of the 'theoretical base', i.e. 1000÷2=500 added to the -132 already present giving 368.
The many varied ways you can tackle a numerical problem with the Vedic techniques is a key advantage to the system. With practice you will become more confident when working with numbers and achieve a deeper understanding of arithmetic. However, one essential point in the above technique must be remembered, that is:

You must NOT process any carries until you have 'corrected' the Crosswise calculation for any initial proportionality you applied at the beginning of the process.

It is a common mistake to try to deal with the carries before 'correcting' the left hand side of the answer for proportionality. If you do this the answer will be wrong.

Reference: https://en.wikibooks.org/wiki/Vedic_Mathematics/Techniques/Multiplication