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Complex Number Complex Number A complex number is a number that can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where i2 = −1.[1] In this expression, a is called the real part and b the imaginary part of the complex number. Complex numbers extend the idea 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). 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 contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with only real numbers. Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, and applied mathematics, such as chaos theory. 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.Complex numbers allow for solutions to certain equations that have no real solution: the equation, has no real solution, since the square of a real number is either 0 or positive. Complex numbers provide a solution to this problem. Know More About :- Associated Property of Addition

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Definition :- An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y. 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 i2 = −1. For example, −3.5 + 2i is a complex number. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi with b > 0 instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i. The set of all complex numbers is denoted by ℂ, or . The real number a of the complex number z = a + bi is called the real part of z, and the real number b is often called the imaginary part. By this convention the imaginary part is a real number – not including the imaginary unit: hence b, not bi, is the imaginary part.[3][4] The real part is denoted by Re(z) or ℜ(z), and the imaginary part b is denoted by Im(z) or ℑ(z). Some authors write a+ib instead of a+bi (scalar multiplication between b and i is commutative). In some disciplines, in particular electromagnetism and electrical engineering, j is used instead of i, since i is frequently used for electric current. In these cases complex numbers are written as a + bj or a + jb. A real number a can usually be regarded as a complex number with an imaginary part of zero, that is to say, a + 0i. However the sets are defined differently and have slightly different operations defined, for instance comparison operations are not defined for complex numbers. A pure imaginary number is a complex number whose real part is zero, that is to say, of the form 0 + bi. Complex plane :- A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; is the rectangular expression of the point. 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 (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form. The defining characteristic of a position vector is that it has magnitude and direction. These are emphasised in a complex number's polar form and it turns out notably that the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors. Read More About :- Round Whole Numbers

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Complex Number