complex number (redirected from Complex number field)

complex number: see number.

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complex number

Any number consisting of both real numbers and imaginary numbers. It has the form a + bi, where a and b are real numbers and i = √−1; a is called the real part and bi the imaginary part. Because a or b can equal 0, any real or imaginary number is also a complex number. Invented as an extension of the real numbers so that certain algebraic equations such as x² + 1 = 0 would have solutions, the complex numbers form an algebraic field, meaning that they obey the commutative law and the associative law (with respect to addition and multiplication), as well as certain other rules in much the same way real numbers do (see field theory).

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(mathematics)

complex number - A number of the form x+iy where i is the square root of -1, and x and y are real numbers, known as the "real" and "imaginary" part. Complex numbers can be plotted as points on a two-dimensional plane, known as an Argand diagram, where x and y are the Cartesian coordinates. An alternative, polar notation, expresses a complex number as (r e^it) where e is the base of natural logarithms, and r and t are real numbers, known as the magnitude and phase. The two forms are related: r e^it = r cos(t) + i r sin(t) = x + i y where x = r cos(t) y = r sin(t) All solutions of any polynomial equation can be expressed as complex numbers. This is the so-called Fundamental Theorem of Algebra, first proved by Cauchy. Complex numbers are useful in many fields of physics, such as electromagnetism because they are a useful way of representing a magnitude and phase as a single quantity.