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Numerical no.2 based on complex numbers in Part 4 | Ekeeda.com

134 views · Published 29 September 2015 · 13:27 · Indexed 22 September 2026

Channel: Ekeeda · 2015 · Education

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Complex Number:

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone.

As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them ""fictitious"" during his attempts to find solutions to cubic equations in the 16th century.

A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-Robert Argand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical. These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.

A position vector may also be defined in terms of its magnitude and direction relative to the origin. These are emphasized in a complex number's polar form. Using the polar form of the complex number in calculations may lead to a more intuitive interpretation of mathematical results. Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin.

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