Category: Complex Number

All the algebraic identities apply equally for complex numbers. The addition and subtraction of complex numbers and with exponents of 2 or 3 can be easily solved using algebraic identities of complex numbers. (z1+z2)2=z21+2z1z2+z22(z1+z2)2=z12+2z1z2+z22 (z1−z2)2=z21−2z1z2+z22(z1−z2)2=z12−2z1z2+z22 (z1+z2)3=z31+3z21z2+3z1z22+z32(z1+z2)3=z13+3z12z2+3z1z22+z23 (z1−z2)3=z31−3z21z2+3z1z22−z32(z1−z2)3=z13−3z12z2+3z1z22−z23 (z1+z2)(z1−z2)=z21−z22(z1+z2)(z1−z2)=z12−z22 (z1+z2+z3)2=z21+z22+z23+2z1z2+2z2z3+2z3z1

The various operations of addition, subtraction, multiplication, division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows. Addition of Complex Numbers Th addition of complex numbers is similar to the addition of natural numbers. Here in complex numbers, the real part is added… Continue reading Operations on Complex Numbers

The ordering of complex numbers is not possible. Real numbers and other related number systems can be ordered, but complex numbers cannot be ordered. The complex numbers do not have the structure of an ordered field, and there is no ordering of the complex numbers that are compatible with addition and multiplication. Also, the non-trivial… Continue reading Ordering of Complex Numbers

The equality of complex numbers is similar to the equality of real numbers. Two complex numbers z1=a1+ib1z1=a1+ib1 and z2=a2+ib2z2=a2+ib2 are said to be equal if the rel part of both the complex numbers are equal a1=a2a1=a2,  and the imaginary parts of both the complex numbers are equal b1=b2b1=b2. Also, the two complex numbers in the polar form are equal, if and… Continue reading Equality of Complex Numbers

The reciprocal of complex numbers is helpful in the process of dividing one complex number with another complex number. The process of division of complex numbers is equal to the product of one complex number with the reciprocal of another complex number.. The reciprocal of the complex number z = a + ib is z−1=1a+ib=a−iba2+b2=aa2+b2+i(−b)a2+b2z−1=1a+ib=a−iba2+b2=aa2+b2+i(−b)a2+b2. This… Continue reading Reciprocal of a Complex Number

The following properties of complex numbers are helpful to better understand complex numbers and also to perform the various arithmetic operations on complex numbers. Conjugate of a Complex Number The conjugate of the complex number is formed by taking the same real part of the complex number and changing the imaginary part of the complex number to… Continue reading Properties of a Complex Number

With the modulus and argument of a complex number and the representation of the complex number in the argand plane, we have a new form of representation of the complex number, called the polar form of a complex number. The complex number z = a + ib, can be represented in polar form as z… Continue reading Polar Representation of a Complex Number

The angle made by the line joining the geometric representation of the complex number and the origin, with the positive x-axis, in the anticlockwise direction is called the argument of the complex number. The argument of the complex number is the inverse of the tan of the imaginary part divided by the real part of… Continue reading Argument of the Complex Number

The complex number consists of a real part and an imaginary part, which can be considered as an ordered pair (Re(z), Im(z)) and can be represented as coordinates points in the euclidean plane. The euclidean plane with reference to complex numbers is called the complex plane or the Argand Plane, named after Jean-Robert Argand. The complex number… Continue reading Graphing of Complex Numbers

The alphabet i is referred to as the iota and is helpful to represent the imaginary part of the complex number. Further the iota(i) is very helpful to find the square root of negative numbers. We have the value of i2 = -1, and this is used to find the value of √-4 = √i24 = +2i  The value… Continue reading Power of i