Browse Definitions :

irrational number

An irrational number is a real number that cannot be reduced to any ratio between an integer p and a natural number q . The union of the set of irrational numbers and the set of rational numbers forms the set of real numbers. In mathematical expressions, unknown or unspecified irrationals are usually represented by u through z. Irrational numbers are primarily of interest to theoreticians. Abstract mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.

Examples of irrational numbers are 2 1/2 (the square root of 2), 3 1/3 (the cube root of 3), the circular ratio pi, and the natural logarithm base e . The quantities 2 1/2 and 3 1/3 are examples of algebraic numbers. Pi and e are examples of special irrationals known as a transcendental numbers. The decimal expansion of an irrational number is always nonterminating (it never ends) and nonrepeating (the digits display no repetitive pattern).

If x and z are irrationals such that x < z, then there always exists an irrational y such that x < y < z. The set of irrationals is "dense" like the set Q of rationals. But theoretically, the set of irrationals is "more dense." Unlike Q , the set of irrationals is nondenumerable. There are more nonterminating, nonrepeating decimals than is possible to list, even by implication. To prove this, suppose there is an implied list of all the nonterminating, nonrepeating decimal numbers between 0 and 1. Every such number consists of a zero followed by a decimal point, followed by an infinite sequence of digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Suppose the elements of the list are denoted x 1, x 2 , x 3, ... and the digits in the numbers are denoted a ii. The list can be written like this:

x 1 = 0. a 11 a 12 a 13 a 14 a 15 a 16 ...
x 2 = 0. a 21 a 22 a 23 a 24 a 25 a 26 ...
x 3 = 0. a 31 a 32 a 33 a 34 a 35 a 36 ...
x 4 = 0. a 41 a 42 a 43 a 44 a 45 a 46 ...
x 5 = 0. a 51 a 52 a 53 a 54 a 55 a 56 ...
x 6 = 0. a 61 a 62 a 63 a 64 a 65 a 66 ...

Even though we don't know the actual values of any of the digits, it is easy to imagine a number between 0 and 1 that can't be in this list. Think of a number y of the following form:

y = 0. b 11 b 22 b 33 b 44 b 55 b 66 ...

such that no b ii in y is equal to the corresponding a ii in the list. The resulting number y is nonterminating and nonrepeating, is between 0 and 1, but is not equal to any x i in the list, because there is always at least one digit that does not match.

The non-denumerability of the set of irrational numbers has far-reaching implications. Perhaps most bizarre is the notion that "not all infinities are created equal." Although the set of rationals and the set of irrationals are both infinite, the set of irrationals is larger in a demonstrable way.

This was last updated in October 2016

Continue Reading About irrational number

  • OPSEC (operations security)

    OPSEC (operations security) is a security and risk management process and strategy that classifies information, then determines ...

  • smart contract

    A smart contract is a decentralized application that executes business logic in response to events.

  • compliance risk

    Compliance risk is an organization's potential exposure to legal penalties, financial forfeiture and material loss, resulting ...

  • security token

    A security token is a physical or digital device that provides two-factor authentication for a user to prove their identity in a ...

  • hardware security module (HSM)

    A hardware security module (HSM) is a physical device that provides extra security for sensitive data.

  • buffer overflow

    A buffer overflow occurs when a program or process attempts to write more data to a fixed-length block of memory, or buffer, than...

  • What is risk mitigation?

    Risk mitigation is a strategy to prepare for and lessen the effects of threats faced by a business.

  • change control

    Change control is a systematic approach to managing all changes made to a product or system.

  • disaster recovery (DR)

    Disaster recovery (DR) is an organization's ability to respond to and recover from an event that affects business operations.

  • What is RAID 6?

    RAID 6, also known as double-parity RAID, uses two parity stripes on each disk. It allows for two disk failures within the RAID ...

  • PCIe SSD (PCIe solid-state drive)

    A PCIe SSD (PCIe solid-state drive) is a high-speed expansion card that attaches a computer to its peripherals.

  • VRAM (video RAM)

    VRAM (video RAM) refers to any type of random access memory (RAM) specifically used to store image data for a computer display.