The 16 Vedic Sutras
Each sutra is a concise Sanskrit aphorism encoding a complete mathematical technique. Click any sutra to explore animated examples, proofs, and interactive practice.
एकाधिकेन पूर्वेण
Ekadhikena Pūrveṇa
By one more than the previous one
When the last digit is 5, the prefix does all the work — multiply it by itself plus one.
निखिलं नवतश्चरमं दशतः
Nikhilaṃ Navataścaramaṃ Daśataḥ
All from 9 and the last from 10
Find how far each number is from the nearest power of 10. Those distances do the multiplication for you.
ऊर्ध्व-तिर्यग्भ्याम्
Ūrdhva-Tiryagbhyāṃ
Vertically and crosswise
Multiply digits vertically, then crosswise, then vertically again — the pattern forms an expanding diamond.
परावर्त्य योजयेत्
Parāvartya Yojayet
Transpose and apply
Division by a near-10 divisor: transpose the sub-digits of the divisor (negate them) and use them as a flag that helps you divide step by step.
शून्यं साम्यसमुच्चये
Śūnyaṃ Sāmyasamuccaye
When the sum is the same, that sum is zero
If the same expression appears on both sides of an equation with equal sums, it equals zero — the equation solves instantly.
आनुरूप्ये शून्यमन्यत्
Ānurūpye Śūnyamanyat
If one is in ratio, the other is zero
In simultaneous equations, if the ratio of coefficients matches, one variable vanishes and the other solves directly.
संकलन-व्यवकलनाभ्याम्
Saṅkalana-vyavakalanābhyāṃ
By addition and subtraction
Add the two equations to get one answer; subtract them to get another. Two equations → two answers in two steps.
पूरणापूरणाभ्याम्
Pūraṇāpūraṇābhyāṃ
By the completion or non-completion
Complete the square (or cube) by adding and subtracting the missing term — Vedic style completion method.
चलन-कलनाभ्याम्
Calanā-Kalanābhyāṃ
Differences and similarities
Use calculus-like derivative thinking: find roots of a polynomial by examining its rate of change pattern.
यावदूनम्
Yāvadūnaṃ
Whatever the deficiency
Square a number near a base: subtract (or add) the deficiency to get the left part, square the deficiency for the right part.
व्यष्टिसमष्टि
Vyaṣṭisamaṣṭhi
Part and whole
Factor a quadratic by finding two numbers whose sum is the middle coefficient and product is the last — the Vedic way to factor trinomials.
शेषाण्यङ्केन चरमेण
Śeṣāṇyaṅkena Caramaṇa
The remainders by the last digit
Express numbers in terms of a reference base and use remainders (vinculum) to simplify complex calculations.
सोपान्त्यद्वयमन्त्यम्
Sopāntyadvayamantyam
The ultimate and twice the penultimate
In certain sum-of-fractions equations, the answer combines the last and second-to-last terms in a specific ratio.
एकन्यूनेन पूर्वेण
Ekanyūnena Pūrveṇa
By one less than the previous one
Multiply by a string of 9s: subtract 1 from the multiplicand for the left part; find the complement of the multiplicand for the right part.
गुणितसमुच्चयः
Guṇitasamuccayaḥ
The product of the sum is the sum of the products
The digit sum of a product equals the digit sum of the product of the individual digit sums — a verification tool.
गुणकसमुच्चयः
Guṇakasamuccayaḥ
The factors of the sum is the sum of the factors
The factors of an expression when evaluated at a value equal the expression's value — use this to verify factorizations.