Short answers, pulled from the story.
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, 5, and so on, and over the centuries the notion was extended to include zero, negative numbers, rational numbers, real numbers, and complex numbers.
Brahmagupta is usually considered the first to formulate the mathematical concept of zero. The first known recorded use of zero as an integer dates to AD 628, in his work the Brahmasphutasiddhanta, where he treated 0 as a number and discussed operations involving it, including division by zero.
According to legend, Pythagoras believed in the absoluteness of numbers and could neither disprove nor accept the existence of irrational numbers. When the Pythagorean Hippasus proved that the square root of 2 is irrational, Pythagoras is said to have sentenced him to death by drowning to stop the spread of the news.
The abstract concept of negative numbers was recognized as early as 100 to 50 BC in China, where the Nine Chapters on the Mathematical Art used red rods for positive coefficients and black rods for negative ones. They were used in India during the 600s to represent debts, but European mathematicians mostly resisted them until the 17th century.
A number is the mathematical object, while a numeral is the symbol used to represent it. For example, eleven is a number word and 11 is the corresponding numeral, though in common usage a numeral is not clearly distinguished from the number it represents.
A transcendental number is a value that is not the root of any polynomial with integer coefficients, which makes it not algebraic and excludes all rational numbers. Hermite proved in 1873 that e is transcendental, and Lindemann proved in 1882 that pi is transcendental.
The irrationality of pi was proved in 1761 by Johann Lambert. Later, Lindemann proved in 1882 that pi is also transcendental, meaning it is not the root of any polynomial with integer coefficients.