In mathematics and computer science, the binary numeral system, or base-2 numeral system, represents numeric values using two symbols: typically 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Numbers represented in this system are commonly called binary numbers. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices such as mobile phones.
A numeral system (or system of numeration) is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.
Ideally, a numeral system will:
A repeating or recurring decimal is a way of representing rational numbers in arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated digit is not zero. The decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single-digit sequence "3" forever. A more complicated example is 3227/555, whose decimal becomes periodic after the second digit following the decimal point and then repeats the sequence "144" forever. At present, there is no universally-accepted notation or phrasing for repeating decimals.
If the repeated digit is a zero, the rational number is called a terminating decimal, since the number is said to "terminate" before these zeros. Instead of taking any note of the repeated zeroes, they are simply omitted. All terminating decimals can be written as a decimal fraction whose divisor is a power of 10 (1.585 = 1585/1000); they may also be written as a ratio of the form k/2n5m (1.585 = 317/2352). However, every terminating decimal also has a second representation as a repeating decimal. This is obtained by decreasing the final non-zero digit by one and appending an infinitely-repeating sequence of nines, a non-obvious phenomenon that many find puzzling. 1 = 0.999… and 1.585 = 1.584999… are two examples of this. (This type of repeating decimal can be obtained by long division if one uses a modified form of the usual division algorithm.)
Elementary arithmetic is the simplified portion of arithmetic which includes the operations of addition, subtraction, multiplication, and division.
Elementary arithmetic starts with the natural numbers and the written symbols (digits) which represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division.