History of Hindu Mathematics: A Source Book v2.c17.s3: 4. Laws of Signs
History of Hindu Mathematics: A Source Book v2.c17.s3
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4. Laws of Signs
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**Addition.** Brahmagupta (628) says:
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“The sum of two positive numbers is positive, of two negative numbers is negative; of a positive and a negative number is their difference.”⁴
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⁴ *BrSpSi*, xviii. 30.
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Mahâvîra (850):
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“In the addition of a positive and a negative number (the result) is (their) difference. The addition of two positive or two negative numbers (gives) as much positive or negative numbers respectively."¹
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Śrīpati (1039):
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"In the addition of two negative or two positive numbers the result is their sum; the addition of a positive and a negative number is their difference."² then they must be added together."¹ Mahâvîra:
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¹ *BrSpSi*, xviii. 31-2.
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"A positive number to be subtracted from another number becomes negative and a negative number to be subtracted becomes positive."²
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² *GSS*, i. 51.
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Śrîpati:
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"A positive (number) to be subtracted becomes negative, a negative becomes positive; (the subsequent operation is) addition as explained before."³
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³ *SiŚe*, xiv. 3.
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Bhâskara II:
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"A positive (number) while being subtracted becomes negative and a negative becomes positive; then addition as explained before."⁴
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⁴ *BBi*, p. 3.
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Nârâyaṇa:
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"Of the subtrahend affirmation becomes negation and negation affirmation; then addition as described before."⁵
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⁵ *NBi*, I, R. 9.
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**Multiplication.** Brahmagupta says:
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"The product of a positive and a negative (number) is negative; of two negatives is positive; positive multiplied by positive is positive."⁶
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⁶ *BrSpSi*, xviii. 33.
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Mahâvîra:
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"In the multiplication of two negative or two positive numbers the result is positive; but it is negative in the case of (the multiplication of) a positive and a negative number."⁷
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⁷ *GSS*, i. 50.
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Śrîpati:
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"On multiplying two negative or two positive
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numbers (the product is) positive; in the multiplication of positive and negative (the result is) negative.”¹
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¹ *SiŚe*, xiv. 4.
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Bhâskara II:
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“The product of two positive or two negative (numbers) is positive; the product of positive and negative is negative.”²
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² *BBi*, p. 3.
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The same rule is stated by Nârâyaṇa.³
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³ *NBi*, I, R. 9.
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**Division.** Brahmagupta states:
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“Positive divided by positive or negative divided by negative becomes positive. But positive divided by negative is negative and negative divided by positive remains negative.”⁴
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⁴ *BrSpSi*, xviii. 34.
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Mahâvîra:
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“In the division of two negative or two positive numbers the quotient is positive, but it is negative in the case of (the division of) positive and negative.”⁵
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⁵ *GSS*, i. 50.
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Śrîpati:
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“On dividing negative by negative or positive by positive, (the quotient) will be positive, (but it will be) otherwise in the case of positive and negative.”⁶
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⁶ *SiŚe*, xiv. 4.
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Bhâskara II simply observes: “In the case of division also, such are the rules (*i.e.*, as in the case of multiplication).”⁷ Similarly Nârâyaṇa remarks, “What have been implied in the case of multiplication of positive and negative numbers will hold also in the case of division.”⁸
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⁷ *BBi*, p. 3.
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⁸ *NBi* I, R. 10.
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**Evolution and Involution.** Brahmagupta says:
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“The square of a positive or a negative number is
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positive. . . . The (sign of the) root is the same as was that from which the square was derived.”1
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As regards the latter portion of this rule the commentator Pṛthûdakasvâmî (860) remarks, “The square-root should be taken either negative or positive, as will be most suitable for subsequent operations to be carried on.”
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Mahâvîra:
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“The square of a positive or of a negative number is positive: their square-roots are positive and negative respectively. Since a negative number by its own nature is not a square, it has no square-root.”2
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Śrîpati:
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“The square of a positive and a negative number is positive. It will become what it was in the case of the square-root. A negative number by itself is non-square, so its square-root is unreal; so the rule (for the square-root) should be applied in the case of a positive number.”3
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Bhâskara II:
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“The square of a positive and a negative number is positive;
Linked texts (4)
- Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
- Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
- Bījagaṇita — Bhāskara II, Sanskrit
- Siddhāntaśekhara — Śrīpati, Sanskrit