: Mathematics: real part

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real part

A complex number is a number of the form $a + bi$ , where $a$ and $b$ are real numbers, and $i$ is an indeterminate satisfying $i 2 = -1$ . For example, $2 + 3 i$ is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate $i$ , for which the relation $i 2 + 1 = 0$ is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation $i 2 + 1 = 0$ induces the equalities $i 4 k = 1, i 4 k +1 = i, i 4 k +2 = -1,$ and $i 4 k +3 = - i,$ which hold for all integers $k$ ; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in $i$ , again of the form $a + bi$ with real coefficients $a, b.$
The real number $a$ is called the real part of the complex number $a + bi$ ; the real number $b$ is called its imaginary part. To emphasize, the imaginary part does not include a factor $i$ ; that is, the imaginary part is $b$ , not $bi$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Complex_number#Definition)

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