Imaginary number

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 i2 = −1.[1][2] The square of an imaginary number bi is b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary.[3]

... Exponents repeat the pattern
from blue area
i−3 = i
i−2 = −1
i−1 = −i
i0 = 1
i1 = i
i2 = −1
i3 = −i
i4 = 1
i5 = i
i6 = −1
in = im where m ≡ n mod 4

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]

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