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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]
All rational numbers are real, but the converse is not true. Irrational numbers (): Real numbers that are not rational. Imaginary numbers: Numbers that equal the product of a real number and the imaginary unit , where =. The number 0 is both real and imaginary.
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.
A simple example of the use of i in a complex number is 2 + 3i. Imaginary numbers are an important mathematical concept; they extend the real number system to the complex number system , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra).
Imaginary time is a mathematical representation of time that appears in some approaches to special relativity and quantum mechanics. It finds uses in certain cosmological theories. Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary unit i.
If the real part of a complex number is 0, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers.
The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist physically in nature. Proponents of intuitionism, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. Consequently, they reconstruct the foundations of ...
The imaginary unit or unit imaginary number, denoted as i, is a mathematical concept which extends the real number system to the complex number system . The imaginary unit's core property is that i 2 = −1. The term "imaginary" was coined because there is no number having a negative square.