Binary number
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.
The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.[1]
Any number can be represented by a sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667:
The numeric value represented in each case depends on the value assigned to each symbol. In the earlier days of computing, switches, punched holes, and punched paper tapes were used to represent binary values.[34] In a modern computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A "positive", "yes", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.
In keeping with the customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, or radix. The following notations are equivalent:
When spoken, binary numerals are usually read digit-by-digit, to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced one zero zero, rather than one hundred, to make its binary nature explicit and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as one hundred (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct value), but this does not make its binary nature explicit.
1.0110101000001001111001100110011111110... is the binary representation of , the , another irrational. It has no discernible pattern.
square root of 2
Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in the decimal system). For example, the binary number 11.012 means:
For a total of 3.25 decimal.
All dyadic rational numbers have a terminating binary numeral—the binary representation has a finite number of terms after the radix point. Other rational numbers have binary representation, but instead of terminating, they recur, with a finite sequence of digits repeating indefinitely. For instance
The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2−1 + 2−2 + 2−3 + ... which is 1.
Binary numerals that neither terminate nor recur represent irrational numbers. For instance,
