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The imaginary unit i in the complex plane: Real numbers are conventionally drawn on the horizontal axis, and imaginary numbers on the vertical axis.. The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 1 = 0.
A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a + 0i = a, 0 + bi = bi, and a + (−b)i = a − bi; for example, 3 + (−4)i = 3 − 4i.
In complex numbers, the cube of a purely imaginary number is also purely imaginary. For example, i 3 = −i. The derivative of x 3 equals 3x 2. Cubes occasionally have the surjective property in other fields, such as in F p for such prime p that p ≠ 1 (mod 3), [11] but not necessarily: see the counterexample with rationals above.
Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. Well, one of those three possibilities for odd numbers causes an issue.
An imaginary number is the product of a real number and 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, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]
Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers. [13] For instance, the complex number i {\displaystyle i} has the formulas − 1 {\displaystyle {\sqrt {-1}}} or 0 − 1 {\displaystyle {\sqrt {0-1}}} , and its real and imaginary parts are ...
Here is an angle in the unit circle; taking 1 / 3 of that angle corresponds to taking a cube root of a complex number; adding −k 2 π / 3 for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by corrects for scale.
Equivalently, an elementary cube is any translate of a unit cube [,] embedded in Euclidean space (for some , {} with ). [3] A set X ⊆ R d {\displaystyle X\subseteq \mathbf {R} ^{d}} is a cubical complex (or cubical set ) if it can be written as a union of elementary cubes (or possibly, is homeomorphic to such a set).