Theory
Putting the shortcuts to work
The word Udharan (udaharana) means a worked illustration, an example. So this lesson is a chance to consolidate: to take the shortcuts you have already learned and extract answers to fresh problems, practising how to recognise which technique fits and apply it cleanly.
So far you have two tools: Nikhilam (multiplying numbers near a base) and Ekadhikena Purvena (squaring numbers ending in 5). Here we drill them on new numbers. (As always, this material is from Bharati Krishna Tirtha's 1965 book; consult the prescribed text for the exact intended scope of this topic.)
Theory
Extracting answers with Nikhilam
Take 96 times 95, near base 100.
- Deficiencies: 96 is 4 below 100, 95 is 5 below.
- Left part: cross-subtract, 96 minus 5 is 91 (or 95 minus 4, also 91).
- Right part: multiply the deficiencies, 4 times 5 is 20.
Result: 91 | 20 = 9120, and 96 times 95 is indeed 9120. The skill being practised is recognition: seeing two numbers just below 100 and immediately reaching for Nikhilam, then extracting the answer in seconds.
Theory
Extracting squares with Ekadhikena
Now spot the pattern for numbers ending in 5 and apply Ekadhikena Purvena.
- 45 squared: front part 4, times one more (5), is 4 times 5 = 20; append 25 -> 2025.
- 95 squared: front part 9, times one more (10), is 9 times 10 = 90; append 25 -> 9025.
Both check against ordinary squaring. The point of an udaharana, a worked illustration, is exactly this fluency: you no longer derive the method each time; you recognise the shape of the problem and extract the answer.
Practical
Reusing the shortcuts on new problems (verified)
def nikhilam(a, b, base=100):
da, db = base - a, base - b
return (a - db) * base + da * db
def square_ending_in_5(n):
f = n // 10
return f * (f + 1) * 100 + 25
print(nikhilam(96, 95)) # 9120
print(square_ending_in_5(45)) # 2025
print(square_ending_in_5(95)) # 9025This example runs in Gri-Learn on the web, where you can edit it and see the output.
Quiz
For the illustration 96 times 95 using Nikhilam near base 100, what is the answer?
- 9120, from left 91 (96-5) and right 20 (4x5)
- 9130, from left 91 and right 30
- 8145, from adding the deficiencies
- 9125, because it ends in 25
Show the answer
9120, from left 91 (96-5) and right 20 (4x5)
Deficiencies of 96 and 95 from 100 are 4 and 5. Left part: 96 minus 5 (cross-subtraction) is 91. Right part: 4 times 5 is 20. Combine: 91 | 20 = 9120, which equals 96 times 95. Option B miscomputes the right part (4 times 5 is 20, not 30). Option C adds deficiencies instead of applying the method. Option D confuses this with the squares-ending-in-5 rule; 96 times 95 does not end in 25. Recognise the shape (two numbers near 100 -> Nikhilam), then extract: deficiencies, cross-subtract, multiply, combine.
Think first
Why does practising with worked examples matter?
You already know the methods. Why spend a lesson just applying them to more numbers? Then tap.
Show the answer
Because knowing a method and being FLUENT with it are different things, and worked examples (udaharana) are how you cross that gap. When you first learn Nikhilam or Ekadhikena, you follow the steps deliberately, checking each one. That is understanding, but it is slow, and under exam or real-world pressure, slow-and-deliberate is not enough. Repeatedly extracting answers from fresh problems builds two things: RECOGNITION (instantly seeing 'these numbers are near 100, so Nikhilam fits' or 'this ends in 5, so Ekadhikena') and AUTOMATICITY (running the steps quickly and accurately without re-deriving them). This is the same reason you practised sorting or loops by hand rather than just reading about them: doing consolidates. It also surfaces edge cases and common slips (like forgetting to pad the right part, or using the number itself instead of one more), so you catch your own mistakes before they matter. A worked-illustration lesson is not filler; it is the deliberate practice that turns a method you can follow into a skill you can use at speed. Recognition plus repetition equals fluency.
Summary
Key takeaways
- Udharan (udaharana) means a worked illustration; this lesson consolidates the shortcuts already learned.
- The skill practised is recognising which technique fits a problem, then extracting the answer quickly.
- Nikhilam on 96 times 95: deficiencies 4 and 5, left 91, right 20 -> 9120.
- Ekadhikena on 45 squared -> 2025, and 95 squared -> 90 | 25 = 9025.
- Each answer checks against ordinary arithmetic; the same coded functions handle new inputs.
- This is a technique-consolidation topic from the 1965 book; confirm the intended scope with the prescribed text.
- Memory hook: recognise the shape, pick the sutra, extract the answer.