... (repeats the pattern from blue area) |

i^{−3} = i |

i^{−2} = −1 |

i^{−1} = −i |

i^{0} = 1 |

i^{1} = i |

i^{2} = −1 |

i^{3} = −i |

i^{4} = 1 |

i^{5} = i |

i^{6} = −1 |

i^{n} = i^{m} where m ≡ n mod 4 |

An **imaginary number** is a complex number that can be written as a real number multiplied by the imaginary unit i,^{ [note 1] } which is defined by its property *i*^{2} = −1.^{ [1] }^{ [2] } The square of an imaginary number bi is −*b*^{2}. For example, 5*i* is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary.^{ [3] }

- History
- Geometric interpretation
- Square roots of negative numbers
- See also
- Notes
- References
- Bibliography
- External links

Originally coined in the 17th century by René Descartes ^{ [4] } as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).

An imaginary number *bi* can be added to a real number a to form a complex number of the form *a* + *bi*, where the real numbers a and b are called, respectively, the *real part* and the *imaginary part* of the complex number.^{ [5] }^{ [note 2] }

Although the Greek mathematician and engineer Hero of Alexandria is noted as the first to have conceived imaginary numbers,^{ [6] }^{ [7] } it was Rafael Bombelli who first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, such as in work by Gerolamo Cardano. At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his * La Géométrie * in which the term *imaginary* was used and meant to be derogatory.^{ [8] }^{ [9] } The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).^{ [10] }

In 1843, William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries in which three of the dimensions are analogous to the imaginary numbers in the complex field.

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, which allows them to be presented perpendicular 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. At 0 on the 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 increase in magnitude downwards. This vertical axis is often called the "imaginary axis" and is denoted or ℑ.

In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin. Multiplication by i corresponds to a 90-degree rotation in the "positive" counterclockwise direction, and the equation *i*^{2} = −1 is interpreted as saying that, if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" (clockwise) direction also satisfies that interpretation, which reflects the fact that −*i* also solves the equation *x*^{2} = −1. In general, multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude.

Care must be used when working with imaginary numbers that are expressed as the principal values of the square roots of negative numbers:^{ [11] }

That is sometimes written as:

The fallacy occurs as the equality fails when the variables are not suitably constrained. In that case, the equality fails to hold as the numbers are both negative, which can be demonstrated by:

where both x and y are non-negative real numbers.

In mathematics, the **absolute value** or **modulus** of a real number x, denoted |*x*|, is the non-negative value of x without regard to its sign. Namely, |*x*| = *x* if x is positive, and |*x*| = −*x* if x is negative, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

In classical mathematics, **analytic geometry**, also known as **coordinate geometry** or **Cartesian geometry**, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

In mathematics, a **complex number** is a number that can be expressed in the form *a* + *bi*, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation *i*^{2} = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number *a* + *bi*, a is called the **real part** and b is called the **imaginary part**. The set of complex numbers is denoted by either of the symbols or **C**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

A **Cartesian coordinate system** in a plane is a coordinate system that specifies each point uniquely by a pair of numerical **coordinates**, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a *coordinate axis* or just *axis* of the system, and the point where they meet is its *origin*, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

**Euler's formula**, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

In mathematics, a **polynomial** is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate *x* is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

In algebra, a **quadratic equation** is any equation that can be rearranged in standard form as

In mathematics, a **square root** of a number *x* is a number *y* such that *y*^{2} = *x*; in other words, a number *y* whose *square* (the result of multiplying the number by itself, or *y* ⋅ *y*) is *x*. For example, 4 and −4 are square roots of 16, because 4^{2} = (−4)^{2} = 16. Every nonnegative real number *x* has a unique nonnegative square root, called the *principal square root*, which is denoted by where the symbol is called the *radical sign* or *radix*. For example, the principal square root of 9 is 3, which is denoted by because 3^{2} = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the *radicand*. The radicand is the number or expression underneath the radical sign, in this case 9.

The **imaginary unit** or **unit imaginary number** is a solution to the quadratic equation *x*^{2} + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3*i*.

In mathematics, the **complex plane** or ** z-plane** is the plane associated with complex coordinate system, formed or established by the

In mathematics, an ** nth root** of a number

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called **mathematical fallacy**. There is a distinction between a simple *mistake* and a *mathematical fallacy* in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof.

In mathematics, an **algebraic equation** or **polynomial equation** is an equation of the form

In mathematics, **tetration** is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

In linear algebra, a **rotation matrix** is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

**Mohr's circle** is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.

In algebra, a **nested radical** is a radical expression that contains (nests) another radical expression. Examples include

In abstract algebra, a **bicomplex number** is a pair (*w*, *z*) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate , and the product of two bicomplex numbers as

In mathematics, the concept of **sign** originates from the property that every real number is either positive, negative or zero. Depending on local conventions, zero is either considered as being neither a positive number, nor a negative number, or as belonging to both negative and positive numbers. Whenever not specifically mentioned, this article adheres to the first convention.

In mathematics, the **argument** of a complex number *z*, denoted arg(*z*), is the angle between the positive real axis and the line joining the origin and *z*, represented as a point in the complex plane, shown as in Figure 1. It is a multi-valued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument is used. It is often chosen to be the unique value of the argument that lies within the interval (–*π*, *π*].

- ↑ Uno Ingard, K. (1988). "Chapter 2".
*Fundamentals of Waves and Oscillations*. Cambridge University Press. p. 38. ISBN 0-521-33957-X. - ↑ Weisstein, Eric W. "Imaginary Number".
*mathworld.wolfram.com*. Retrieved 2020-08-10. - ↑ Sinha, K.C. (2008).
*A Text Book of Mathematics Class XI*(Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9. - ↑ Giaquinta, Mariano; Modica, Giuseppe (2004).
*Mathematical Analysis: Approximation and Discrete Processes*(illustrated ed.). Springer Science & Business Media. p. 121. ISBN 978-0-8176-4337-9. Extract of page 121 - ↑ Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009).
*College Algebra: Enhanced Edition*(6th ed.). Cengage Learning. p. 66. ISBN 1-4390-4379-5. - ↑ Hargittai, István (1992).
*Fivefold symmetry*(2nd ed.). World Scientific. p. 153. ISBN 981-02-0600-3. - ↑ Roy, Stephen Campbell (2007).
*Complex numbers: lattice simulation and zeta function applications*. Horwood. p. 1. ISBN 1-904275-25-7. - ↑ Descartes, René,
*Discourse de la Méthode*... (Leiden, (Netherlands): Jan Maire, 1637), appended book:*La Géométrie*, book three, p. 380. From page 380:*"Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x*(Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x^{3}– 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires."^{3}– 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].) - ↑ Martinez, Albert A. (2006),
*Negative Math: How Mathematical Rules Can Be Positively Bent*, Princeton: Princeton University Press, ISBN 0-691-12309-8 , discusses ambiguities of meaning in imaginary expressions in historical context. - ↑ Rozenfeld, Boris Abramovich (1988). "Chapter 10".
*A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space*. Springer. p. 382. ISBN 0-387-96458-4. - ↑ Nahin, Paul J. (2010).
*An Imaginary Tale: The Story of "i" [the square root of minus one]*. Princeton University Press. p. 12. ISBN 978-1-4008-3029-9. Extract of page 12

- Nahin, Paul (1998).
*An Imaginary Tale: the Story of the Square Root of −1*. Princeton: Princeton University Press. ISBN 0-691-02795-1., explains many applications of imaginary expressions.

Look up in Wiktionary, the free dictionary. imaginary number |

- How can one show that imaginary numbers really do exist? – an article that discusses the existence of imaginary numbers.
- 5Numbers programme 4 BBC Radio 4 programme
- Why Use Imaginary Numbers? Basic Explanation and Uses of Imaginary Numbers

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