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Zero as a Number

The idea to use zero as a number was probably not recognized until the early centuries A.D. The Hindus and Arabs appear to have led the way. Two things helped bring this about. The first was attempts at solving the quadratic equation ax2 - bx = 0, where one root is zero and the other is a rational number different than zero. The second was a more organized systematic study of the properties of operations on numbers.

Considering the first, most mathematicians at the time accepted zero as a possible solution, but did not think much of it since it was not very useful as a solution to practical problems.

With regard to the second idea, some quotes are listed here. In about 850 A.D. Mahavir, a Hindu mathematician, wrote in his book, Ganita-Sara-Sahgraha ("The Compendium of Calculation"):

A number multiplied by zero is zero, and that number remains unchanged which is divided by, added to, or diminished by zero.

Today we agree on most of what he says. We can add or subtract zero to a number and just get back that number. When we multiply a number by zero we get zero. The division by zero is where we disagree. Today division by zero is undefined.

About three hundred years after Mahavir, another Hindu mathematician, Bhaskara, states, "a definite number divided by cipher (zero) is a submultiple of nought" and goes on to write, "These fractions of which the denominator is cipher are termed infinite quantities." Not exactly the definition we use today, but a concept that is useful in many areas of mathematics. It is important to note that Bhaskara did recognize dividing by zero. Others of his time either avoided it or declared the result of it to be meaningless.