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.
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 seven, the multiplier is 143.
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. ↖