Mathematical Sūtras & Upasūtras

The entire mathematical structure of Vedic Mathematics is contained in 16 Sūtras and 16 Upasūtras. These sūtras and upasūtras provide the complete set of rules governing Vedic Mathematics, and its application.
 
A 'sūtra' is a rule (or a formula) that works on specific mathematical situations (commonly known as the domain of a mathematical formula), and an 'upasūtra' is a corollary - also sometimes referred as 'Sub-Sūtra'. An upasūtra may be a corollary of a specific sūtra, or a general rule. We present here, a list of all 16 sūtras and 16 upasūtras used in Vedic Mathematics. Each sūtra, or upasūtra, include working examples, tutorials, and the logic of its working.
 
 
List of Vedic Mathematical Sūtras and Upasūtras:
 

Sūtra (सूत्र)

1. एकाधिकेन पूर्वेण
   (ekādhikena pūrveṇa)
   By one more than the one before »
2. निखिलं नवतश्चरमं दशत:
   (nikhilaṃ navataścaramaṃ daśataḥ)
   All from 9, and the last from 10 »
3. ऊर्ध्वतिर्यग्भ्यां
   (ūrdhva tiryagbhyāṃ)
   Vertically and crosswise »
4. परावर्त्य योजयेत्
   (parāvartya yojayet)
   Transpose and adjust »
5. शून्यं साम्यसमुच्चये
   (śūnyaṃ sāmyasamuccaye)
   If the samuccaya is same, it is zero »
6. आनुरूप्ये शून्यमन्यत्
   (ānūrupye śūnayamanyat)
   If one is in ratio, the other is zero »
7. संकलन व्यवकलनाभ्यां
   (saṅkalana vyavakalanābhyāṃ)
   By addition, and by subtraction »
8. पूरणापूरणाभ्यां
   (pūraṇāpūraṇābhyāṃ)
   By completion, or non-completion »
9. चलन कलनाभ्यां
   (calana kalanābhyāṃ)
   Sequential motion »
10. यावदूनम्
    (yāvadūnāṃ)
    By the deficiency »
11. व्यष्टिसमष्टिः
    (vyaṣṭisamaṣṭiḥ)
    Specific and general »
12. शेषाण्यंकेन चरमेण
    (śeṣāṇyaṅkena carameṇa)
    The remainders, by the last digit »
13. सोपान्त्यद्वयमन्त्यं
    (sopāntyadvayamantyaṃ)
    The ultimate, and twice the penultimate »
14. एकन्यूनेन पूर्वेण
    (ekanyūnena pūrvena)
    By one less than the one before »
15. गुणितसमुच्चय:
    (guṇitasamucchayaḥ)
    The product of the sum »
16. गुणकसमुच्चय:
    (guṇakasamucchayaḥ)
    All the multipliers »

Upasūtra (उपसूत्र)

1. आनुरूप्येण
   (ānurūpyeṇa)
   Proportionately »
2. शिष्यते शेषसंञः
   (śiṣyate śeṣasṃjñaḥ)
   The remainder remains constant »
3. आधमाधेनान्त्यमन्त्येन
   (ādyamādyenāntyamantyena)
   First by the first, and last by the last »
4. केवलैः सप्तकं गुणयात्
   (kevalaiḥ saptakaṃ guṇyāt)
   For 7, the multiplicand is 143 »
5. वेष्टनम्
   (veṣṭanaṃ)
   By osculation »
6. यावदूनं तावदुनं
   (yāvadūnaṃ tāvadūnaṃ)
   Lessen by the deficiency »
7. यावदूनं तावदूनीकृत्य वर्गं च योजयेत्
   (yāvadūnam tāvadūnīkṛtya vargañca yojayet)
   Lessen by the deficiency, and set up square of the deficiency »
8. अन्त्ययोर्दशकेऽपि
   (antyayordaśake'pi)
   Last totalling 10 »
9. अन्त्ययोरेव
   (antyayoreva)
   Only the last terms »
10. समुच्चयगुणितः
    (samuccayaguṇitaḥ)
    The sum of the products »
11. लोपनस्थापनाभ्यां
    (lopanasthāpanābhyāṃ)
    By alternate elimination and retention »
12. विलोकनं
    (vilokanaṃ)
    By mere observation »
13. गुणितसमुच्चयः समुच्चयगुणितः
    (guṇitasmuccayaḥ samuccayaguṇitaḥ)
    The product of the sum, is the sum of the products »
14. ध्वजांक
    (dhvajāṅka)
    On the flag »
15. शुद्ध
    (śūddha)
    Purification »
16. द्वन्द्वयोग
    (dvaṅdvayoga)
    Sum of pairs »

 
(* Some of the above links are disabled, because of the ongoing work on the respective web pages. We hope to have them live, as soon as possible.)
 
Note that, the last 3 upasūtras are not included in many references, but these upasūtras are part of Vedic Mathematics. We have deliberately included them in this section to list the complete set of rules governing Vedic Mathematics.
 
In order to understand Vedic Mathematics and before plunging into the Sūtras and Upasūtras:

• One must have an understanding of the terms, that are used throughout the Sūtras and Upasūtras presented here for reference.
• The principles of Vedic Mathematics usually deals with the individual digits of a number, rather than the entire number. That is how it breaks the calculations into simple steps.
• As in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications.