Complex numbers come into existence when the square of a number is negative because we know it very well that the square of a number will always be positive doesn’t matter whether the number is positive or negative.
In cases like or here, if we solve, we find the square of x=-1. We say that x is not real here. Generally, these types of cases are considered as Complex numbers. Complex numbers were first observed by mathematician Girolamo Cardano (1501-1575). In his book Ars Magna, he discussed the mechanics of complex numbers in details and thus he started Complex Algebra.
Standard form of Complex Numbers-
Complex numbers are defined as expressions of the form a + ib where a,b R & i =
It is denoted by Z i.e. z= a + ib.
‘a’ is called as real part of z= (Re z)
and ‘b’ is called as imaginary part of z =(Im z).
i or IOTA- iota is a unique symbol. it’s the ninth letter of Latin alphabet. It’s used to denote imaginary numbers whose square root is -1.
Click here to download the book “An Imaginary Tale The Story of i” a very interesting book on iota by Paul J. Nahin.
► Zero is both purely real as well as purely imaginary but not imaginary.
► i = is called the imaginary unit. Also i² =-l, = -i, = 1 etc.
► only if at least one of either a or b is non-negative.
► Conjugate Complex- If z=a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z ¯ or z*. i.e. z* = a – ib.
►z + z* = 2 Re(z) v
► z – z* = 2i Im(z)
► zz* = a² + b² which is real If z lies in the 1st quadrant then lies z* in the 4th quadrant and -z* lies in the 2nd quadrant.
Algebraic Operations: The algebraic operations on complex numbers are similar to those on real numbers treating i as a polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative.
e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless .
However in real numbers, if then a = 0 = b but in complex numbers,
does not imply
Equality In Complex Number: Two complex numbers
are equal if and only if their real & imaginary parts coincide.
Representation Of A Complex Number In Various Forms:
(a) Cartesian Form (Geometric Representation): Every complex number z = x + i y can be represented by a point on the cartesian plane known as a complex plane (Argand diagram) by the ordered pair (x, y).
length OM is called modulus of the complex number denoted by & is called the argument or amplitude.
► is always non-negative. Unlike real numbers
► Argument of a complex number is a many-valued function. If is the argument of a complex number then 2n+ where n I will also be the argument of that complex number. Any two arguments of a complex number differ by 2n
►The unique value of such that is called the principal value of the argument. Unless otherwise stated, amp z implies the principal value of the argument.
►By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0.i the argument is not defined and this is the only complex number which is given by its modulus.
► There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.
Few Basic Questions on complex numbers–
Example-1 Compute real and imaginary part of
so clearly Re(z)=14/13 and Im(z)=5/13
In the next article on complex numbers, we will learn about Polar fom of complex numbers and some other properties of complex numbers