Sūtras of Vedic Mathematics
Sixteen sūtras
एकाधिकेन पूर्वेण ॥१॥ 1. By one more than the one before.
निखिलं नवतश्चरमं दशतः ॥२॥ 2. All from nine and the last from ten.
ऊर्ध्वतिर्यग्भ्याम् ॥३॥ 3. Vertically and crosswise.
परावर्त्य योजयेत् ॥४॥ 4. Transpose and apply.
शून्यं साम्यसमुच्चये ॥५॥ 5. If the Sāmuchchaya is the same, it is zero.
आनुरूप्ये शून्यमन्यत् ॥६॥ 6. If one is the ratio, the other is zero.
संकलनव्यवकलनाभ्याम् ॥७॥ 7. By addition and subtraction.
पूरणापूरणाभ्याम् ॥८॥ 8. By the completion or non-completion.
चलनकलनाभ्याम् ॥९॥ 9. Differential calculus.
यावदूनम् ॥१०॥ 10. By the deficiency.
व्यष्टिसमष्टिः ॥११॥ 11. Specific and general.
शेषाण्यङ्केन चरमेण ॥१२॥ 12. The remainders by the last digit.
सोपान्त्यद्वयमन्त्यम् ॥१३॥ 13. The ultimate and twice the penultimate.
एकन्यूनेन पुर्वेण ॥१४॥ 14. By one less than the one before.
गुणितसमुच्चयः ॥१५॥ 15. The product of the sum.
गुणकसमुच्चयः ॥१६॥ 16. All the multipliers.
Sixteen upasūtras
आनुरूप्येण ॥१॥ 1. Proportionately.
शिष्यते शेषसंज्ञः ॥२॥ 2. The remainder remains constant.
आद्यमाद्ये नान्त्यमन्त्येन ॥३॥ 3. The first by the first and the last by the last.
केवलैः सप्तकं गुण्यात् ॥४॥ 4. For 7, the multiplier is 143.
कलौ क्षुद्रससैः ॥४क॥ 4b. For 13, the multiplier is 077.
कंसे क्षामदाहखलैर्मलैः ॥४ख॥ 4c. For 17, the multiplier is 05882353.
वेष्टनम् ॥५॥ 5. By osculation.
यावदूनं तावदूनम् ॥६॥ 6. Lessen by the deficiency.
यावदूनं तावदूनीकृत्य वर्गं च योजयेत् ॥७॥ 7. Whatever the deficiency, lessen by that amount, and set up the square of the deficiency.
अन्त्ययोर्दशकेऽपि ॥८॥ 8. Last totalling ten.
अन्त्ययोरेव ॥९॥ 9. Only the last terms.
समुच्चयगुणितः ॥१०॥ 10. The sum of the products.
लोपनस्थापनाभ्याम् ॥११॥ 11. By alternate elimination and retention.
विलोकनम् ॥१२॥ 12. Mere observation.
गुणितसमुच्चयः समुच्चयगुणितः ॥१३॥ 13. The product of the sum is the sum of the products.
ध्वजाङ्क ॥१४॥ 14. On the flag.
द्वन्द्वयोग ॥१५॥ 15. Duplex combination process.
शुद्धः ॥१६॥ 16. Purification.
Kaṭapayādi Sāṅkhya

कादि नव टादि नव पादि पञ्चक याद्यष्टक क्षः शून्यम्

  1. and the following 8 letters
  2. and the following 8 letters
  3. and the following 4 letters
  4. and the following 7 letters
  5. क्ष for zero
क 1ख 2ग 3घ 4ङ 5
च 6छ 7ज 8झ 9
ट 1ठ 2ड 3ढ 4ण 5
त 6थ 7द 8ध 9
प 1फ 2ब 3भ 4म 5
य 1र 2ल 3व 4
स 5श 6ष 7ह 8
क्ष 0
Sixteen sūtras transliterated
ekādhikena pūrveṇa (1) 1. By one more than the one before.
nikhilaṃ navataśhcharamaṃ daśhataḥ (2) 2. All from nine and the last from ten.
ūrdhvatiryagbhyām (3) 3. Vertically and crosswise.
parāvartya yojayet (4) 4. Transpose and apply.
śhūnyaṃ sāmyasamuchchaye (5) 5. If the Sāmuchchaya is the same, it is zero.
ānurūpye śhūnyamanyat (6) 6. If one is the ratio, the other is zero.
saṃkalanavyavakalanābhyām (7) 7. By addition and subtraction.
pūraṇāpūraṇābhyām (8) 8. By the completion or non-completion.
chalanakalanābhyām (9) 9. Differential calculus.
yāvadūnam (10) 10. By the deficiency.
vyaṣhṭisamaṣhṭiḥ (11) 11. Specific and general.
śheṣhāṇyaṅkena charameṇa (12) 12. The remainders by the last digit.
sopāntyadvayamantyam (13) 13. The ultimate and twice the penultimate.
ekanyūnena purveṇa (14) 14. By one less than the one before.
guṇitasamuchchayaḥ (15) 15. The product of the sum.
guṇakasamuchchayaḥ (16) 16. All the multipliers.
Sixteen upasūtras transliterated
ānurūpyeṇa (1) 1. Proportionately.
śhiṣhyate śheṣhasaṃjñaḥ (2) 2. The remainder remains constant.
ādyamādye nāntyamantyena (3) 3. The first by the first and the last by the last.
kevalaiḥ saptakaṃ guṇyāt (4) 4. For 7, the multiplier is 143.
kalau kṣudrasasaiḥ (4b) 4b. For 13, the multiplier is 077.
kaṃse kṣāmadāhakhalairmalaiḥ (4c) 4c. For 17, the multiplier is 05882353.
veṣhṭanam (5) 5. By osculation.
yāvadūnaṃ tāvadūnam (6) 6. Lessen by the deficiency.
yāvadūnaṃ tāvadūnīkṛitya vargaṃ cha yojayet (7) 7. Whatever the deficiency, lessen by that amount, and set up the square of the deficiency.
antyayordaśhake'pi (8) 8. Last totalling ten.
antyayoreva (9) 9. Only the last terms.
samuchchayaguṇitaḥ (10) 10. The sum of the products.
lopanasthāpanābhyām (11) 11. By alternate elimination and retention.
vilokanam (12) 12. Mere observation.
guṇitasamuchchayaḥ samuchchayaguṇitaḥ (13) 13. The product of the sum is the sum of the products.
dhvajāṅka (14) 14. On the flag.
dvandvayoga (15) 15. Duplex combination process.
śhuddhaḥ (16) 16. Purification.
Kaṭapayādi Sāṅkhya transliterated

kādi nava ṭādi nava pādi pañcaka yādyaṣṭaka kṣaḥ śūnyam

  1. ka and the following 8 letters
  2. ṭa and the following 8 letters
  3. pa and the following 4 letters
  4. ya and the following 7 letters
  5. kṣa for zero
ka 1kha 2ga 3gha 4ṅa 5
ca 6cha 7ja 8jha 9ña
ṭa 1ṭha 2ḍa 3ḍha 4ṇa 5
ta 6tha 7da 8dha 9na
pa 1pha 2ba 3bha 4ma 5
ya 1ra 2la 3va 4
sa 5śa 6ṣa 7ha 8
kṣa 0