What is Vedic Mathematics?
Vedic Mathematics is a system of mental and shortcut arithmetic compiled by Swami Bharati Krishna Tirthaji (1884-1960), Shankaracharya of Govardhan Math, Puri. He claimed to have reconstructed it from the parishishtas (appendices) of the Atharva Veda.
The system is condensed into 16 Sutras and 13 Sub-sutras (Upa-sutras) — concise aphorisms that capture powerful arithmetic principles.
The 16 Sutras (Selected)
| Sutra | Meaning | Application |
|---|---|---|
| Ekadhikena Purvena | "By one more than the previous" | Squaring numbers ending in 5 |
| Nikhilam Navatashcaramam Dashatah | "All from 9 and the last from 10" | Subtraction, multiplication near base |
| Urdhva-Tiryagbhyam | "Vertically and crosswise" | General multiplication |
| Paravartya Yojayet | "Transpose and apply" | Division |
| Shunyam Saamyasamuccaye | "When sum is the same, that sum is zero" | Equations |
| (Anurupye) Shunyam Anyat | "If one is in ratio, the other is zero" | Simultaneous equations |
| Sankalana-Vyavakalanabhyam | "By addition and subtraction" | Factoring |
Example 1 — Squaring Numbers Ending in 5 (Ekadhikena Purvena)
Rule: For any number ending in 5, multiply the leading digit by (itself + 1), append 25.
Find 75²:
- Leading digit = 7
- 7 × (7+1) = 7 × 8 = 56
- Append "25"
- Answer: 5625
Find 105²:
- Leading "10"; 10 × 11 = 110
- Answer: 11025Example 2 — Multiplication near a Base (Nikhilam Sutra)
Multiply 97 × 96 (base = 100):
Number Deficiency
────── ──────────
97 03 ← (100 − 97)
96 04 ← (100 − 96)
── ──
Cross-subtract: 97 − 4 = 93 (or 96 − 3 = 93) ← left part
Multiply: 3 × 4 = 12 ← right part
Answer: 9312Verification: 97 × 96 = 9312 ✓
Example 3 — Urdhva-Tiryagbhyam (Vertically & Crosswise)
Multiply 23 × 41:
2 3
4 1
───────────
Step 1 (right): 3 × 1 = 3 → ___3
Step 2 (cross): (2×1) + (3×4) = 14 → __ 4 (carry 1)
Step 3 (left): 2 × 4 = 8 → _9_
Answer: 943This method easily extends to multi-digit multiplication and is mentally faster than the conventional algorithm.
Why Vedic Mathematics Matters
- Speed: Calculations once done with paper can be done mentally in seconds.
- Multiple Methods: A single problem can be solved in many ways — develops mathematical creativity.
- Pattern Recognition: Builds intuition for number relationships.
- Education: Adopted by CBSE and several state boards as supplementary curriculum.
Authentic Vedic Mathematics — A Caveat
Modern scholars debate whether Tirthaji's "16 Sutras" come directly from the Atharva Veda or are his own systematisation. Regardless of origin, the techniques are mathematically valid and pedagogically powerful, and they continue an authentic Indian tradition of mental computation (e.g., the ancient Kuttaka and Sankalita methods).
Key Terms — Lesson 3.3 (Vedic Mathematics)
"Name the sutra for this operation" is the standard question; pair each sutra with its use.
Vedic Mathematics — Tirthaji's system of 16 Sutras + 13 Upa-sutras for rapid mental computation. Sutra — A short aphorism encoding an arithmetic principle. Upa-sutra — A sub-aphorism / corollary (13 in number). Ekadhikena Purvena — "By one more than the previous"; squares numbers ending in 5. Nikhilam Navatashcaramam Dashatah — "All from 9 and the last from 10"; multiplication near a base. Urdhva-Tiryagbhyam — "Vertically and crosswise"; the general multiplication method. Paravartya Yojayet — "Transpose and apply"; used for division. Base — The reference power of ten (10, 100, …) from which deficiencies are measured in the Nikhilam method.
Worked Example
Multiply 98 × 97 by the Nikhilam sutra (base = 100).
- Deficiencies from 100: 100 − 98 = 2, 100 − 97 = 3.
- Left part (cross-subtract): 98 − 3 = 95 (equivalently 97 − 2 = 95).
- Right part (multiply deficiencies): 2 × 3 = 06 (padded to two digits, since base 100 has two zeros).
- Join the parts: 9506.
Check: 98 × 97 = 9506. ✓ The whole product is found mentally in a single line, with no long multiplication.