Theory
A multiplication shortcut
How quickly can you multiply 97 by 96 in your head? The long way is fiddly. There is a technique that makes it almost instant, and it comes from the sutra Nikhilam Navatashcaramam Dashatah, usually translated 'all from 9 and the last from 10'.
This unit studies a set of such calculation techniques. Each is a shortcut, a clever, systematic way to compute, and your syllabus asks you to learn each one, work an example, and then express it in code. We begin with Nikhilam, a shortcut for multiplying numbers that sit just below a base like 100.
Watch out
How this unit is framed
The sixteen sutras studied here come from the 20th-century book 'Vedic Mathematics' (1965) by Bharati Krishna Tirtha. We treat each sutra as the computation technique it encodes, teach the method, work an example, and code it. We make no historical claims beyond the book itself, and we do not invent ancient Sanskrit sources or verse text. For the sutras' full treatment, consult the prescribed text. The value here is practical: these are neat, checkable arithmetic methods you can program.
Theory
The Nikhilam method, near base 100
To multiply two numbers just below 100, follow three steps.
1. Find each number's deficiency (how far below 100 it is). For 97 the deficiency is 3; for 96 it is 4. (The phrase 'all from 9 and the last from 10' is the quick way to find a deficiency: subtract each digit from 9 and the last from 10.)
2. Left part: subtract one number's deficiency from the other. 97 minus 4 is 93 (and 96 minus 3 is also 93, it works both ways).
3. Right part: multiply the two deficiencies. 3 times 4 is 12.
Put them together: 93 | 12 = 9312. So 97 times 96 is 9312.
Theory
Another example
Try 98 times 97, base 100.
- Deficiencies: 98 is 2 below, 97 is 3 below.
- Left part: 98 minus 3 is 95 (or 97 minus 2, also 95).
- Right part: 2 times 3 is 6, written as 06 to fill the two digit places the base 100 needs.
Result: 95 | 06 = 9506. And indeed 98 times 97 is 9506. Notice the right part is padded to two digits because the base 100 has two zeros; getting that padding right is the one detail to watch.
Practical
Nikhilam in Python (verified)
def nikhilam(a, b, base=100):
da = base - a # deficiency of a (e.g. 100-97 = 3)
db = base - b # deficiency of b (e.g. 100-96 = 4)
left = a - db # cross-subtraction (== b - da)
right = da * db # product of deficiencies
return left * base + right # combine: left | right
print(nikhilam(97, 96)) # 9312
print(nikhilam(98, 97)) # 9506
# left*base + right places 'right' in the base's digit slots (padding handled).This example runs in Gri-Learn on the web, where you can edit it and see the output.
Quiz
Using Nikhilam near base 100, multiply 97 by 96. What are the left and right parts, and the answer?
- Left 93, right 12, answer 9312 (deficiencies 3 and 4: 97-4=93, and 3x4=12)
- Left 93, right 7, answer 937
- Left 91, right 12, answer 9112
- Left 97, right 96, answer 9796
Show the answer
Left 93, right 12, answer 9312 (deficiencies 3 and 4: 97-4=93, and 3x4=12)
The deficiencies of 97 and 96 from 100 are 3 and 4. Left part: cross-subtract, 97 minus 4 equals 93 (equivalently 96 minus 3 equals 93). Right part: multiply the deficiencies, 3 times 4 equals 12. Combine as 93 | 12 = 9312, which is exactly 97 times 96. Option B forgets to multiply the deficiencies (using 3+4 or similar) and mis-sizes the answer. Option C mis-subtracts the left part. Option D just writes the original numbers, ignoring the method. The technique: deficiencies, cross-subtract for the left, multiply for the right, then combine, and it checks out against ordinary multiplication.
Think first
Why does the Nikhilam method work?
It looks like magic, but it is ordinary algebra. Why does cross-subtract-and-multiply give the right product? Then tap.
Show the answer
Because it is just the algebra of (base minus deficiency) multiplied out. Write the two numbers as (B - x) and (B - y), where B is the base (100) and x, y are the deficiencies (3 and 4). Multiply them algebraically: (B - x)(B - y) = B squared - Bx - By + xy = B(B - x - y) + xy. Now read the two pieces. The first piece, B times (B - x - y), is the base times the LEFT part, and (B - x - y) equals (B - x) - y, that is, the first number minus the second's deficiency (97 - 4 = 93), which is exactly the cross-subtraction. The second piece, xy, is the RIGHT part, the product of the deficiencies (3 times 4 = 12). Since B is 100, multiplying the left part by B just shifts it into the hundreds place, and xy fills the last two digits, giving 93 then 12, or 9312. So the method is not mystical at all; it is a tidy way of evaluating (B - x)(B - y) that avoids a full long multiplication by exploiting the base. Understanding this also tells you WHEN it helps: the closer the numbers are to a base, the smaller the deficiencies, and the easier the arithmetic, which is precisely why Nikhilam shines for numbers near 100 (or 1000). Clever bookkeeping of simple algebra.
Summary
Key takeaways
- The sixteen sutras come from the 1965 book 'Vedic Mathematics' by Bharati Krishna Tirtha; we teach each as a computation technique and code it.
- Nikhilam ('all from 9 and the last from 10') multiplies numbers near a base like 100.
- Find each number's deficiency below the base (97 -> 3, 96 -> 4).
- Left part: cross-subtract (97 - 4 = 93, same as 96 - 3); right part: multiply the deficiencies (3 x 4 = 12).
- Combine: 93 | 12 = 9312, which is 97 x 96; pad the right part to the base's digit count (98 x 97 = 95 | 06 = 9506).
- It works because (B - x)(B - y) = B(B - x - y) + xy, plain algebra exploiting the base.
- Memory hook: deficiencies, cross-subtract left, multiply right, combine.