Complex number formula definition and theorem
i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.
Complex number extend the concept of the one-dimensional number line to the two-dimensional
l complex plane using the horizontal axis for the real part and the vertical axis for the imaginary 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 i2 = −1. For example, 2 + 3i 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 i2 + 1 = 0 is imposed.
Complex number
Based on this definition, complex numbers can, using the addition and multiplication for polynomials. The relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, and i4k+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.
Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate the ideal generated .
