(repeats the pattern from blue area)

(repeats the pattern from blue area)

In mathematics, an imaginary number (or purely imaginary number) is a complex number whose squared value is a real number not greater than zero. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Algebra, the square of a number is that number multiplied by itself In Mathematics, the real numbers may be described informally in several different ways The imaginary unit, denoted by i or j, is an example of an imaginary number. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation If y is a real number, then i·y is an imaginary number, because:

Imaginary numbers were defined in 1572 by Rafael Bombelli. Rafael Bombelli (1526–1572 was an Italian Mathematician. Born in Bologna, he is the author of a treatise on Algebra and is a central figure At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by some as fictitious or useless. Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La Géométrie, where the term was meant to be derogatory. La Géométrie was published in 1637 as an appendix to Discours de la méthode ( Discourse on Method) written ^[1]

Although Descartes originally used the term imaginary number to mean what is currently meant by the term complex number, the term imaginary number today usually means a complex number with a real part equal to 0, that is, a number of the form i·y. Zero (0) is the only number that is both real and imaginary.

An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Geometric interpretation

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented orthogonal to the real axis. One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. In mathematics a number line is a picture of a straight line in which the Integers are shown as specially-marked points evenly spaced on the line At 0 on this x-axis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then "increase" in magnitude upwards, and "negative" imaginary numbers "decrease" in magnitude downwards. This vertical axis is often called the "imaginary axis" and is denoted , or simply Im.

In this representation, multiplication by −1 corresponds to a rotation of 180 degrees about the origin. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation Multiplication by i corresponds to a 90-degree rotation in the "positive" direction (i. e. counter-clockwise), and the equation i² = − 1 is interpreted as saying that if we apply 2 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" direction (i. e. clockwise) also satisfies this interpretation. This reflects the fact that −i also solves the equation x² = − 1 — see imaginary unit. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

Applications of imaginary numbers

For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as ⅔ and ⅛ are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as −3 and −5 are meaningless when weighing the mass of an object, but essential when keeping track of monetary debits and credits^[1]. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Debit and credit are formal Bookkeeping and Accounting terms They are the most fundamental concepts in accounting representing the two records that one Similarly, imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, and many others. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Control theory is an interdisciplinary branch of Engineering and Mathematics, that deals with the behavior of Dynamical systems The desired output Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

In electrical engineering, for example, the voltage produced by a battery is characterized by one real number (called amplitude), such as +12 volts or −12 volts. But the "AC" voltage in a home requires two parameters. An alternating current ( AC) is an Electric current whose direction reverses cyclically as opposed to Direct current, whose direction remains constant One is an amplitude, such as 120 volts, and the other is an angle (called phase). The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 The voltage is said to have two dimensions. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it A 2-dimensional quantity can be represented mathematically as either a vector or as a complex number (known in the engineering context as phasor). In Physics and Engineering, a phase vector ("phasor" is a representation of a Sine wave whose amplitude ( A) phase ( θ) In the vector representation, the rectangular coordinates are typically referred to simply as X and Y. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane But in the complex number representation, the same components are referred to as real and imaginary. When the complex number is purely imaginary, such as a real part of 0 and an imaginary part of 120, it means the voltage has an amplitude of 120 volts and a phase of 90°, which is physically very real.

Some programming languages have built-in support for imaginary numbers. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. For example, in the Python interpreter, one may use them by appending a lowercase or uppercase J to the number^[2]:

>>> (5+2j) * (8+5j)
(30+41j)

Matlab examples:

   >> (5+2j) * (8+5j)
   ans =
     30. Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability J is the tenth letter in the modern Latin alphabet; it was the last of the 26 letters to be added 0000 +41. 0000i
   >> (5+i*2) * (8+5j)
   ans =
     30. 0000 +41. 0000i
   >>

History

Descartes was the first to use the term “imaginary” number in 1637. However, imaginary numbers were discovered much earlier by Gerolamo Cardano in the 1500s but they were not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German

Powers of i

The powers of i repeat in a cycle:

This can be expressed with the following pattern where n is any integer:

This leads to the conclusion that

See also

• Quaternion
• Octonion
• Imaginary correlation

Notes

1. ^ ^{a b} Alberto A. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent (Princeton University Press, 2005), discusses ambiguities of meaning in imaginary expressions in historical context.
2. ^ The leading angle brackets in the first line are part of the interpreter's syntax and are not part of the equation. Brackets are Punctuation marks used in pairs to set apart or interject text within other text

References

• Paul Nahin, An Imaginary Tale: the Story of the Square Root of -1 (Princeton University Press, 1998), explains many applications of imaginary expressions.

External links

• Why imaginary numbers really do exist