Short answers, pulled from the story.
The decimal place-value system in use today was first recorded in India. The earliest surviving evidence appears in a copper plate from Gujarat dated 595 CE and in stone inscriptions from Indonesia and Cambodia recording the year 683 CE. The system later passed through the Islamic world before reaching Europe.
The classical period, roughly 400 to 1300 CE, included Aryabhata (476-550), who calculated pi to four decimal places and produced the earliest sine and cosine tables; Brahmagupta, whose 628 CE work gave the first systematic treatment of zero and negative numbers; and Bhaskara II (1114-1185), who stated an early form of Rolle's theorem and computed pi to five decimal places.
The Kerala school, founded by Madhava of Sangamagrama, developed infinite series expansions for sine, cosine, and arc tangent two centuries before calculus was developed in Europe. They also derived a rational approximation of 104348/33215 for pi, accurate to nine decimal places. However, they did not develop a formal theory of differentiation, integration, or the fundamental theorem of calculus.
The Bakhshali Manuscript is the oldest extant mathematical manuscript from India, discovered in 1881 by a farmer in the village of Bakhshali near Peshawar. Written on birch bark in Buddhist hybrid Sanskrit, it covers arithmetic, algebra, and arithmetic progressions, and employs a decimal place-value system with a dot for zero. Radiocarbon dating in 2017 showed its fragments come from three different centuries: 224-383 CE, 680-779 CE, and 885-993 CE.
Mathematical works were transmitted orally until approximately 500 BCE using rigorous memorization techniques. These included up to eleven forms of recitation of the same text, which were then compared to catch errors. The most complex method, dense recitation, wove words in intricate forward and backward sequences. Mathematical content was encoded in sutras, highly compressed verse formulas, and was paired with oral commentary passed from teacher to student.
The Sanskrit word shunya, meaning void, was used by Jain mathematicians as the earliest recorded term for zero. Shunya was calqued into Arabic as sifr, then borrowed into Medieval Latin as zephirum, passed through Romance languages including French and Italian, and eventually entered English as zero.