### **Invaluable Zero**

Children do you know the concept of zero was first formulated by an Indian Mathematician Brahmagupta during 628AD. Zero is strangest number in the universe and continuing its role as one of the great paradoxes of human thought. It is both nothing and everything.

Zero is an even number, because it is divisible by 2. By most definitions 0 is a natural number, and then the only natural number not to be positive. The number 0 is the smallest non-negative integer.

The number 0 is neither positive nor negative and appears in the middle of number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors and cannot be composite because it cannot be expressed by multiplying prime numbers (0 must always be one of the factors).

Initially, the zero as a number was not available. There was the idea of empty space, which may be thought conceptually similar to zero. Babylonians around 700 BC used three hooks to denote an empty place in the positional notation.

Almost during the same time, Greek mathematicians made some unique contributions to mathematics. Euclid wrote a book on number theory named Elements, but that was completely based on geometry and no concept of zero was mentioned.

Around AD 650, the use of zero as a number came into Indian mathematics. The Indians used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty placeholder in positional numbers from as early as 200AD in India. Around 500 AD Aryabhata devised a number system, which had no zero, as a positional system, but used to denote empty space. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. For example, to represent ‘100’ it would be two dots after 1.

In 628 AD, Brahmagupta wrote Brahmasphutasiddhanta (The Opening of the Universe), and attempted to give the rules for arithmetic involving zero and negative numbers. He explained that given a number, if you subtract it from itself you obtain a zero. He gave the following rules for addition, which involve zero: The sum of zero and a negative number is negative, the sum of a positive number and zero is positive; the sum of zero and zero is zero. Similarly, he gave the correct rules for subtraction also. But when it comes to division by zero, he gave some rules that were not correct.

In 830, Mahavira wrote Ganita Sara Samgraha (Collections of Mathematics Briefings), which was designed as an update of Brahmagupta’s book. He correctly stated the multiplication rules for zero, but again gave incorrect rule for division by zero.

After 500 years of Brahmagupta, Bhaskara tried to solve the problem of division by stating that any number divided by zero was infinity. Well, conceptually though it is still incorrect (division by zero is indeterminate not infinity), Bhaskara did correctly state other properties of zero, such as square of zero is zero and square root of zero is also zero.

The Islamic and Arabic mathematicians took the ideas of the Indian mathematicians to further west. (FOC)

# The quarrel and its solution

**Pt. Ram Krishan Sharma**

Once upon a time there was a lawyer who had a seven years old son. This son was in the habit of attending Satsangs (religious lectures). One day he heard a piece of song whose chorus was ‘Speak on harsh words’ (Kadve bol na bol). The child picked up the line and kept singing it at home. One day the lawyer and his wife had a quarrel and sharp words were exchanged from both sides and this quarrel was followed by a gloomy silence. Both did not speak to each other for several days; but after all, being husband and wife, how long could they refrain from speaking? Both were desirous of breaking the ice. Returning from the office, the lawyer would proceed to his room. His wife would cook the meal and to avoid encounter would sent it to him through the servant.

One day, as the lawyer returned from office and proceeded to his room, he heard his son singing, ‘Speak no harsh words’. The lawyer had a brain wave and directed the child, “Sunny, go and sing this to your mother in her room.” The child obeyed and entered his mother’s bedroom and started singing, ‘Speak no harsh words’. On hearing him, the mother promptly enjoined “No, why are you singing it here; go to your father’s room and sing it there.” The child returned to his father’s room, whereupon the father said, “Hadn’t I asked you to sing it in your mother’s room? Go back and sing it there.” He was shuttled back to his mother’s room. Once again she admonished him, “Did I not bid you to sing it in your father’s room and not here? Now go back.” The child was bewildered and stood in between the two rooms and said, “Neither of you let me sing my song. Now I will stand in between your rooms and sing.” He began to sing “Speak no harsh words” (Kadve bol na bol).

On hearing his innocent voice, both the parents broke into laughter. They could not control themselves any longer. The rooms rang with music in which the barriers of vanity and anger melted away. Both of them came near the son and looking into each other’s eyes laughed and sang the same song, “Speak no harsh words” (Kadve bol na bol). As smiles burst forth, the clouds of anger flew away.