## Multidigit multiplication: vertically and crosswise

by

I think I have a new favorite way to multiply numbers: the vertically and crosswise technique. I learned about it from George Gheverghese Joseph’s Crest of the Peacock [a book that is sadly out of print], and was recently reminded of just how cool it is.

The vertically and crosswise multiplication technique was popularized by Swami Bharati Krishna Tirthaji’s Vedic Mathematics, published posthumously in 1965. Tirthaji claimed to have reconstructed the mathematical content of sixteen sutras (concise rules) from an appendix to the Atharvaveda, one of the four Vedas, texts held sacred in Hinduism; the Atharvaveda dates to as early as 1000BCE. While his claim of the Vedic origins of Tirthaji’s methods is questionable, the mathematics itself (whether ancient or modern) is quite cool.

Tirthaji’s vertically and crosswise multiplication method is essentially the same as the general multiplication method found in Trachtenberg’s system of rapid arithmetic, devised by Jakow Trachtenberg during his imprisonment during WWII.

The method also is somewhat similar to lattice or grid multiplication, although more of the steps are done mentally rather than writing down all of the initial data as is done in the lattice method.

It is easiest to describe the method in the context of several examples.

Example: Find 78 x 36

We write the numbers one above the other, aligning place values as in the usual method of multidigit multiplication. The ones place of the product is determined vertically, by computing 6×8. The result, 48, tells us the ones place of the product is an 8, and we have an additional 4 tens to keep track of.

The tens place of the product is determined crosswise, by taking both the product 6×7 and also the product 3×8, plus the extra 4 tens we have from above. This yields 42+24+4=70 tens, so the tens place of the product is 0, and we have an additional 7 hundreds to keep track of.

Finally, the hundreds place of the product is determined vertically, by computing 3×7, plus the 7 extra hundreds from above. That gives 21+7=28 hundreds.

Putting it all together, we have 2808 as the product.

Why the method works: At each stage, you are computing those partial products whose place value matches the place value of the digit you are computing.

Why the method is interesting: There are relatively few intermediate results to keep track of (and all of those involve at most two-digit numbers). With a little practice, one can do all of the arithmetic in one’s head, and when presented with a multiplication problem of any complexity proceed directly to writing down the digits of the answer.

Example: Find 978×423 Ones place: 3×8=24; write down 4, remember 2. 

Tens place: 2 + 3×7 + 2×8 = 39; write down 9, remember 3. 

Hundreds place: 3+3×9+2×7+4×8=76; write down, 6, remember 7. 

Thousands: 7+2×9+4×7=53; write down 3, remember 5. 

Ten-thousands: 5+4×9=41; write down 41. 

Example: Find 927803×46817. (Pad a zero to have two six digit numbers: 927803×046817; while that step isn’t mathematically necessary, it helps retain the geometric vertically and crosswise pattern to the various partial products.)

Ones: 7×3=21; remember 2. 

Tens: 2+7×0+1×3=5. 

100s: 7×8+1×0+8×3=80; remember 8. 

1000s: 8+7×7+1×8+8×0+6×3=83; remember 8. 

10000s: 8+7×2+1×7+8×8+6×0+4×3=105; remember 10. 

100000s: 10+7×9+1×2+8×7+6×8+4×0+0x3=179; remember 17. 

1000000s: 17+1×9+8×2+6×7+4×8+0x0=116; remember 11. 

millions: 11+8×9+6×2+4×7+0x8=123; remember 12. 

10 millions: 12+6×9+4×2+0x7=74; remember 7. 

100 millions: 7+4×9+0x2=43; remember 4. 

billions: 4+0x9=4. 

Advertisements

### 14 Responses to “Multidigit multiplication: vertically and crosswise”

1. Sherry Lewis Says:

This is amazing. It takes a little practice to get the flow of where to multiply next.

2. BRISDANE Says:

THOROUGHLY FASCINATING

3. ymberzal Says:

yeah, this was discovered by an indian! i am proud of dat.!

4. The Second Bunch of Ways to Multiply « 360 Says:

[…] Vertically and Crosswise Multiplication (which TwoPi wrote about just over a year ago).  TwoPi tells me that it should probably be called […]

5. camille Says:

TRULY FASCINATING…AND IT HAS BEEN A GREAT HELP FOR MY MATH PROJECT… HOPE YOU’LL DISPLAY MORE 😀

6. chona Says:

Big Big thanks. I’m homeschooling my children. Kisses.

7. astrnelis Says:

OMG! I run across this article because I was trying to figure out why this works and if others multiply math the way I have been doing it for ages now. I actually got in trouble when I was multiplying this way so I practiced what they taught me but I still to this day used this very same system to multiply very large numbers and do the additions in my head. I have no idea why I use this method but I am glad to know that it is a proven and true method that won’t leave me down. Thank you for sharing 🙂

8. mark Cyril s. jubay Says:

I have discovered a solution which is very similar to this but only differs to some processes. I call it Jubix method since it uses vertical and crosswise pattern. The result would be like an inverse pyramid and then when added, provides the answer. I dont know if i can publish it since it also uses vertical and crosswise pattern which is similar to the method as shown above. well i discovered the method year 2003.

9. Ξ Says:

mark — it sounds like a method I just learned about a few months ago — I’ll try to write up a post this weekend and you can see if it’s the same. [But if you have it written up, by all means, send a link.]

10. ankush Says:

it’s really amzing to solve such big multiplications in a few seconds.
I waz about to join uc mas but now i’ve got this & defeat any one from uc mas to.
thanxxx 360worldpress

11. Jeremy Says:

I know I saw this in an old (c. 1960’s?) book about ways people multiplied in the Middle Ages. I don’t know whether it’s really of Vedic origin, but it might indeed be ancient. I wish I could remember the title of the book I read. I’d like to go back to the same library and find it again.

12. TwoPi Says:

That sounds a lot like the book “Capitalism and Arithmetic” by Frank Swetz, which contains a translation with commentary of the Treviso Arithmetic of 1478.

13. deepanjan Says:

I have done a visual simplification of this method so that the patterns can be meorized easily. Have a look. 14. James Glover Says:

The Vedic method of vertically and crosswise is a general method and has the distinct advantage that the answer is obtained in one line of mental working no matter how large the numbers. An extension to this is with the use of vinculum (negative) numbers whereby any digit greater than 5 can usefully be replaced by a negative (e.g. 29 is 30 – 1). The Vedic system, as discovered by B.K.Thirtha around 1917 but only published posthumously, provides several other methods for multiplication not mentioned in the list of 25 which are used for special cases. The most notable uses deficiencies or surpluses to multiply very easily numbers close to any power of 10 or factor or multiple of a power of 10.

Youtube clips describing these can be found at,

Multiplication by All from 9 and the last from 10, below the base
https://www.youtube.com/edit?o=U&video_id=DATb5slHeMc

Multiplication by All from 9 and the last from 10, above the base
https://www.youtube.com/edit?o=U&video_id=VRsT2Q8orQw

Multiplication by All from 9 and the last from 10 with vinculums
https://www.youtube.com/edit?o=U&video_id=L9ofWNzHECQ

Multiplication by Vertically and Crosswise
https://www.youtube.com/edit?o=U&video_id=uVEv3jh6hsI

Multiplying two 3-digit numbers in one line
https://www.youtube.com/edit?o=U&video_id=xcsUTohBkvY

Multiplying two 4-digit numbers in one line
https://www.youtube.com/edit?o=U&video_id=a3j7h9dTw9k

Decimal multiplication
https://www.youtube.com/edit?o=U&video_id=_KeZSoMI_Mg

Squaring numbers ending in 5
https://www.youtube.com/edit?o=U&video_id=78yTDlqGyA0

Squaring numbers close to the power of 10
https://www.youtube.com/edit?o=U&video_id=s8mBZps1XhY

Cubing a 2-digit number
https://www.youtube.com/edit?o=U&video_id=4M8tqh5Xbmk