Multidigit multiplication: vertically and crosswise

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. [4]

Tens place: 2 + 3×7 + 2×8 = 39; write down 9, remember 3. [94]

Hundreds place: 3+3×9+2×7+4×8=76; write down, 6, remember 7. [694]

Thousands: 7+2×9+4×7=53; write down 3, remember 5. [3694]

Ten-thousands: 5+4×9=41; write down 41. [413694]

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. [1]

Tens: 2+7×0+1×3=5. [51]

100s: 7×8+1×0+8×3=80; remember 8. [051]

1000s: 8+7×7+1×8+8×0+6×3=83; remember 8. [3051]

10000s: 8+7×2+1×7+8×8+6×0+4×3=105; remember 10. [53051]

100000s: 10+7×9+1×2+8×7+6×8+4×0+0x3=179; remember 17. [953051]

1000000s: 17+1×9+8×2+6×7+4×8+0x0=116; remember 11. [6953051]

millions: 11+8×9+6×2+4×7+0x8=123; remember 12. [36953051]

10 millions: 12+6×9+4×2+0x7=74; remember 7. [436953051]

100 millions: 7+4×9+0x2=43; remember 4. [3436953051]

billions: 4+0x9=4. [43436953051]