Ancient Indian Mathematics
Vedic Maths Tricks for Squares & Cubes — With Sanskrit Sutra Origins
Lightning-fast mental calculation using ancient Indian methods — with the original Sanskrit sutras and their meanings explained for students and competitive exam takers.
What Are the Vedic Maths Sutras?
सूत्र (Sūtra) means “thread” in Sanskrit — a concise, aphoristic rule designed to carry maximum meaning in minimum words. In Vedic Mathematics, each sutra is a Sanskrit phrase that encodes a mental calculation technique. There are 16 main sutras and 13 sub-sutras (उपसूत्र — upasūtra).
| Sanskrit Sutra | IAST | Meaning | Used For |
|---|---|---|---|
| एकाधिकेन पूर्वेण | Ekādhikena Pūrveṇa | By one more than the previous one | Squaring numbers ending in 5; finding recurring decimals |
| निखिलं नवतश्चरमं दशतः | Nikhilaṃ Navataścaramaṃ Daśataḥ | All from 9, last from 10 | Fast multiplication near bases of 10, 100, 1000 |
| ऊर्ध्वतिर्यग्भ्याम् | Ūrdhva-tiryagbhyāṃ | Vertically and crosswise | General multiplication of any two numbers |
| परावर्त्य योजयेत् | Parāvartya Yojayet | Transpose and apply | Division and algebraic equations |
| आनुरूप्येण | Āṇurūpyeṇa | Proportionately | Proportional calculations and ratios |
Trick 1: Square of a Number Ending in 5
Sutra: एकाधिकेन पूर्वेण (Ekādhikena Pūrveṇa)— “By one more than the previous one.”
Method: For any number ending in 5 (like 35, 75, 125):
- Take the tens digit (for 35, that's 3)
- Multiply it by the next number: 3 × 4 = 12
- Append 25 at the end
- Answer: 35² = 1225
More examples: 65² = 6×7 | 25 = 4225 · 85² = 8×9 | 25 = 7225 · 95² = 9×10 | 25 = 9025
Trick 2: Multiply Numbers Near 100 (Nikhilam Sutra)
Sutra: निखिलं नवतश्चरमं दशतः— “All from 9, last from 10.” Used for multiplying numbers close to 100.
Example: 97 × 96
- Deficiencies from 100: 97 is −3, 96 is −4
- Cross subtraction: 97 − 4 = 93 (left part)
- Product of deficiencies: 3 × 4 = 12 (right part)
- Answer: 9312
For numbers above 100: 104 × 103 → surplus +4 and +3 → 104+3 = 107 | 4×3 = 12 → 10712
Trick 3: Cube of Any Two-Digit Number
Example: 12³
- Write down a ratio: a : b = 1 : 2 (for 12, a=1, b=2)
- First row: a³, a²b, ab², b³ → 1, 2, 4, 8
- Double middle terms: 1, 4, 8, 8
- Add diagonally carrying: 1 | 4+2=6 | 8+4=12 → carry 1 | 8+1=9 | → carry forward
- Answer: 1728
Frequently Asked Questions
What are the 16 Vedic Maths sutras?
The 16 Vedic Maths sutras (sutras = concise rules in Sanskrit) were compiled by Bharati Krishna Tirthaji in "Vedic Mathematics" (1965). Key sutras include: Ekādhikena Pūrveṇa (By one more than the previous one), Nikhilaṃ Navataścaramaṃ Daśataḥ (All from 9, last from 10), Ūrdhva-tiryagbhyāṃ (Vertically and cross-wise), and Āṇurūpyeṇa (By the proportionately). Each sutra is a Sanskrit phrase encoding a calculation principle.
How do you find the square of a number near 100 using Vedic Maths?
Use the Nikhilam sutra (All from 9, last from 10). For 97²: find the deficiency (97 is 3 less than 100). Left part = 97 − 3 = 94. Right part = 3² = 09. Answer = 9409. For 104²: surplus = +4. Left = 104 + 4 = 108. Right = 4² = 16. Answer = 10816. This works for any number near a base (10, 100, 1000).
What does "sutra" mean in Sanskrit?
Sūtra (सूत्र) means "thread" or "string" in Sanskrit — from the root siv meaning "to sew." In ancient India, a sutra was a memorizable, minimally worded rule that could be threaded into memory like beads on a string. Panini's Ashtadhyayi has ~4,000 sutras, the Yoga Sutras has 196, and Vedic Maths has 16 sutras + 13 sub-sutras. All encode maximum meaning in minimum words.
Is Vedic Maths actually from the Vedas?
The term is somewhat misleading. The tricks compiled by Bharati Krishna Tirthaji (published 1965) are not directly in the Vedas. However, many techniques are genuinely ancient — India's Sulba Sutras (800–200 BCE) contain geometric algebra, and the Aryabhatiya (499 CE) describes numerical methods. The modern "Vedic Maths" system systematizes ancient Indian mathematical intuitions into teachable techniques, even if the specific sutra-compilation is modern.
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