Moonlight of Mathematics, Part 1 (1936) - Page 20

Now, the rule for Operations involving Zero.

A quantity added to or subtracted from zero

remains unchanged; when multiplied by zero, it becomes zero.

The product and other results of zero are zero.

In addition, zero becomes equal to the quantity added to it. ॥ 30 ॥

original: "rāśiḥ khena yutono-'vikṛtas..." The text uses two words for zero: "kha" (literally meaning space or sky) and "shunya" (the void).

Example.

What is the result when one hundred is added to zero or subtracted from it?

And what is that hundred multiplied by zero? Tell me quickly, what is

zero multiplied by a hundred, and what is zero divided by zero?

What is the square, the square root, the cube, and the cube root of zero?

And what are eight added to zero? O learned one, tell me if your

proficiency in operations with zero is truly great! ॥ 112 ॥

Statement. The quantity: 100. This quantity, added to or subtracted from zero, remains unchanged as 100. Statement. 100. This multiplied by zero becomes 0. Or zero multiplied by this [100] becomes 0. Zero divided by zero is 0. The author provides $0 \div 0 = 0$. Note that in modern mathematics, this is considered "undefined," but historical Indian mathematicians explored various results for this operation. Its square is 0. Its square root is 0. Its cube is 0. Its cube root is 0. Eight (8) added to this [zero] becomes equal to the added quantity: 8.

In this work on Arithmeticoriginal: "pāṭīgaṇite." Arithmetic or "board-calculation.", since the concept of division by zerooriginal: "khahara." Literally "zero-divisor." is not used in worldly transactions or common understanding, it has not been mentioned here. In our work on Algebraoriginal: "bījagaṇite." Literally "seed-calculation" or the mathematics of elements/origins., division by zero is discussed because of its utility in algebraic methods.

Thus end the basic operations.

Now, the rule for Concurrence.

The sum of two quantities is written twice;

the difference is added to one and subtracted from the other.

These two results, when halved, are the two quantities.

This calculation is called Concurrence or Transition. ॥ 31 ॥

original: "saṅkramaṇa." This is a classic algebraic rule for finding two unknown numbers (x and y) if you know their sum (s) and their difference (d). The formulas are x = (s+d)/2 and y = (s-d)/2.

1. To understand why zero divided by zero is not always zero, one should look into the principles of Calculus. original: "calanakalana." This refers to the study of motion/fluxions, an Indian precursor to modern calculus concepts.
2. This rule is exactly according to the explanation of Concurrence given by Bhāskarācārya. Bhāskara II (1114–1185) was the most famous Indian mathematician of the medieval period.