Theory
Squaring near a base
You already have Ekadhikena for squaring numbers ending in 5. But what about squaring 98, or 103? These sit near the base 100, and the sutra Yavadunam, 'whatever the deficiency', squares them in two quick moves.
The idea reuses the deficiency you met in Nikhilam: how far the number is from the base. This lesson shows the method for numbers just below and just above 100, with verified examples, from Bharati Krishna Tirtha's 1965 book.
Theory
The method
To square a number near base 100, find its deficiency d (how far from 100, negative if below, positive if above), then:
- Left part = the number plus its deficiency d (so for a number below the base, you reduce it by the shortfall).
- Right part = d squared, padded to two digits (because base 100 has two zeros).
Example, 98 squared: deficiency d is -2 (2 below 100). Left = 98 + (-2) = 96. Right = (-2) squared = 4, padded to 04. Result: 96 | 04 = 9604. And 98 squared is 9604. Two moves: reduce by the deficiency, then square the deficiency.
Theory
Above the base too
The same rule works for numbers above the base, using a positive deficiency (an excess).
103 squared: the excess d is +3 (3 above 100). Left = 103 + 3 = 106. Right = 3 squared = 9, padded to 09. Result: 106 | 09 = 10609. And 103 squared is 10609.
Notice the left part adds the deficiency either way: below the base, d is negative so you subtract; above, d is positive so you add. One consistent rule, left is number plus d, right is d squared, covers both directions.
Practical
Yavadunam in Python (verified)
def square_near_base(n, base=100):
d = n - base # deficiency: -2 for 98, +3 for 103
left = n + d # number plus its deficiency
right = d * d # square of the deficiency
return left * base + right # combine (base places 'right' correctly)
print(square_near_base(98)) # 9604
print(square_near_base(103)) # 10609This example runs in Gri-Learn on the web, where you can edit it and see the output.
Quiz
Using Yavadunam, what is 98 squared (base 100)?
- 9204, from left 92 and right 04
- 9604, because the deficiency is -2: left = 98 + (-2) = 96, right = (-2) squared = 04
- 9640, from 96 and 40
- 9804, because 98 stays and you add 04
Show the answer
9604, because the deficiency is -2: left = 98 + (-2) = 96, right = (-2) squared = 04
The deficiency of 98 from 100 is -2. Left part: 98 + (-2) = 96 (reduce the number by its shortfall of 2). Right part: (-2) squared = 4, padded to two digits as 04. Combine: 96 | 04 = 9604, which is 98 squared. Option A over-reduces the left part (using -6 or similar). Option C mis-pads the right (4 becomes 04, not 40). Option D forgets to reduce the left part at all. The rule: left is the number plus its (negative) deficiency, right is the deficiency squared, padded to the base's digit count.
Think first
Why does 'number plus deficiency, then deficiency squared' give the square?
Why does this two-part rule produce the exact square of a number near the base? Then tap.
Show the answer
Because it is the algebra of (base + d) squared, where d is the deficiency (which can be negative). Write the number as B + d, with B the base (100) and d the deficiency (for 98, d = -2, so 100 + (-2) = 98; for 103, d = +3). Square it: (B + d) squared = B squared + 2Bd + d squared = B(B + 2d) + d squared. Now read the pieces. The first piece is B times (B + 2d); and B + 2d equals (B + d) + d, which is the NUMBER plus its deficiency (98 + (-2) = 96, or 103 + 3 = 106), that is the LEFT part. Multiplying it by B just shifts it into the higher place. The second piece is d squared, the RIGHT part (for 98 that is 4, padded to 04). Since B is 100, the left part lands in the hundreds and d squared fills the last two digits, giving 96 then 04, or 9604. So 'number plus deficiency' for the left and 'deficiency squared' for the right are simply the two terms of (B + d) squared, read off directly. This also explains the padding (d squared occupies as many digit slots as the base has zeros) and why the method works equally above or below the base (d just carries its sign). Once more, a tidy algebraic identity dressed as a memorable rule. Square the base-plus-deficiency, and the two parts fall out.
Summary
Key takeaways
- Yavadunam ('whatever the deficiency') squares a number near a base like 100.
- Find the deficiency d = number - base (negative below the base, positive above).
- Left part = number + d; right part = d squared, padded to the base's digit count.
- 98 squared: d = -2, left = 96, right = 04 -> 9604; 103 squared: d = +3, left = 106, right = 09 -> 10609.
- One rule covers both directions because d carries its sign (subtract below, add above).
- It works because (B + d) squared = B(B + 2d) + d squared; from the 1965 book.
- Memory hook: number plus deficiency on the left, deficiency squared on the right.