Brahmagupta, an Indian mathematician and astronomer, made revolutionary contributions to mathematics, particularly in defining and operating with zero. His most influential work, Brahmasphutasiddhanta (628 CE), introduced rules for calculations involving zero and negative numbers.
Key Contributions of Brahmagupta:
- Defining Zero and Its Operations:
He was the first to treat zero as an actual number and provided clear rules for its arithmetic.
0+x=x,0−x=−x,x−0=x,0×x=00 + x = x, \quad 0 – x = -x, \quad x – 0 = x, \quad 0 \times x = 0
However, he struggled with division by zero, stating:
00=0\frac{0}{0} = 0
- Negative Numbers and Their Arithmetic:
He extended arithmetic to negative numbers, establishing rules such as:
A positive number minus a larger positive number gives a negative number.\text{A positive number minus a larger positive number gives a negative number.}
He also discussed multiplication of negative numbers.
- Solutions to Quadratic Equations:
He provided the first known solution formula for quadratic equations:
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
- Brahmagupta’s Theorem in Geometry:
He described a formula for the area of a cyclic quadrilateral:
A=(s−a)(s−b)(s−c)(s−d)A = \sqrt{(s-a)(s-b)(s-c)(s-d)}
where s is the semi-perimeter.
Bhaskara I (c. 600 – c. 680 CE): Early Insights on Zero and Infinity
Bhaskara I was another Indian mathematician and astronomer who made important contributions to the understanding of zero and infinity.
- Properties of Zero and Infinity:
He recognized that dividing a number by zero leads to infinity:
x0=∞\frac{x}{0} = \infty
This was an intuitive but incomplete understanding, as modern mathematics defines division by zero as undefined.
- Approximation of Sine Functions:
He provided an accurate approximation of sinx\sin x for small angles.
- Commentary on Aryabhata’s Work:
He wrote an important commentary on Aryabhata’s mathematics, refining earlier ideas.
Brahmagupta’s work on zero and negative numbers later influenced Islamic and European mathematics. His ideas reached Europe through Arabic translations.
Bhaskara I’s insights on zero and infinity were far ahead of his time, prefiguring modern concepts in calculus.
