The oldest known text to use a decimal place-value system, including a zero, is the Jain text from India entitled the Lokavibhâga, dated 458 CE. This text uses Sanskrit numeral words for the digits, with words such as the Sanskrit word for void for zero.[20] The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876 CE.[21][22] There are many documents on copper plates, with the same small o in them, dated back as far as the sixth century CE, but their authenticity may be doubted.[8]
Rules of Brahmagupta
The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe),[24] written in 628. Here Brahmagupta considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Here are the rules of Brahmagupta:[24]
* The sum of zero and a negative number is negative.
* The sum of zero and a positive number is positive.
* The sum of zero and zero is zero.
* The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
* A positive or negative number when divided by zero is a fraction with the zero as denominator.
* Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
* Zero divided by zero is zero.
In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value to this, whereas computers and calculators sometimes assign NaN, which means "not a number." Moreover, non-zero positive or negative numbers when divided by zero are either assigned no value, or a value of unsigned infinity, positive infinity, or negative infinity. Once again, these assignments are not numbers, and are associated more with computer science than pure mathematics, where in most contexts no assignment is done.