A complex number is a real number plus a real number times the square root of -1; in other words, a number of the form z = a + ib, where a and b are real and i = √(-1). The term ib is known as an imaginary number or the imaginary part of the complex number a + ib; a is called the real part. The names "complex," "real," and "imaginary," which came about historically, are totally misleading because complex numbers are not particularly complex and imaginary numbers are no less real than real numbers! Another way to represent a complex number is as an ordered pair of real numbers (a, b) together with the operations: (a, b) + (c, d) = (a + c, b + d) and (a, b) × (c, d) = (ac - bd, bc + ad). Alternatively, complex numbers can be shown as points on an Argand diagram (a representation of the complex plane) in which the horizontal axis is the real number line and the vertical axis represents all possible purely imaginary numbers. Any point that appears on the complex plane off-axis has both real and imaginary parts. On an Argand diagram a complex number can also be shown as a vector, or directed line segment (a line of a certain length with an arrow), extending from the origin (0 + 0i) to the number (a + bi). The absolute value or magnitude of a complex number z, thought of as a point on a plane, is its Euclidean distance from the origin, and is denoted |z|; this is always a non-negative real number. Algebraically, if z = a + ib, we can define |z| = √(a2 + b2). If the complex number z is written in polar coordinates z = r eiφ, then |z| = r.
Complex numbers are a natural extension of the real numbers and form what is called an algebraically closed field. Because of this, mathematicians sometimes consider the complex numbers to be more "natural" than the real numbers: all polynomial equations have solutions among the complex numbers, which is not true for the real numbers. Complex numbers are used in electrical engineering and other branches of physics as a convenient description for periodically varying signals. In an expression z = r eiφ one may think of r as the amplitude and phi as the phase of a sine wave of given frequency. In special and general relativity theory, some formulas for the metric on spacetime become simpler if the time variable is taken to be imaginary.