Short answers, pulled from the story.
Arithmetic is an elementary branch of mathematics that deals with numerical operations such as addition, subtraction, multiplication, and division. In a wider sense it also includes exponentiation, the extraction of roots, and taking logarithms. It forms the basis of branches like algebra, calculus, and statistics.
The ancient Indians were the first to develop zero as a number to be used in calculations. Brahmagupta wrote down the exact rules for its operation around 628 CE, and he also discussed calculations with negative numbers and their application to credit and debt.
A rational number can be written as the ratio of two integers, such as one half, and corresponds to a finite or repeating decimal. An irrational number, such as pi or the hypotenuse of a right triangle with legs of length 1, cannot be expressed as a ratio of two integers, and its decimal representation is infinite without repeating.
The first positional numeral system was developed by the ancient Babylonians starting around 1800 BCE, and it used a radix of 60. A positional system gives a digit a value based on its place, which made representing large numbers and calculating with them far more efficient.
The fundamental theorem of arithmetic states that every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, 18 can be factored into primes, while 19 is a prime number with no other prime factorization.
Computers commonly approximate real numbers using floating-point arithmetic, representing each number through a significand, a base, and an exponent. Because the number of bits is limited, results are rounded to the closest representable number, which causes rounding errors. The most common technical standard for this is IEEE 754.
Integer arithmetic is not closed under division because dividing one integer by another does not always produce an integer. For example, 7 divided by 2 is 3.5, not a whole number. This can be handled by rounding, by keeping a remainder, or by using rational number arithmetic, which is closed under division as long as the divisor is not 0.