Complex plane

Complex plane is the geometrical representation of complex numbers drawn between real axis and imaginary axis. The other name for this is 'z-plane'. In Cartesian plane the real part is expressed by the x-axis and the imaginary part of a complex number is expressed by the y-axis.

Complex plane is sometimes called as 'Argand plane'.  It is called as Argand plane because it is used in Argand diagrams, which are used to plot the position of the poles and zeroes of position in the z-plane.

Complex plane representation

Examples :

Represent the following complex numbers in a complex plane.

1. 3+7i
2. 2-i
3. -4+5i
4. -2-2i

The above complex numbers can be represented in Argand diagram by taking the real part of the complex number in the x-axis and the imaginary part in the y-axis.

1.  3+7i

In this the real part of the complex number is 3 and the imaginary part is 7.  So let us plot the number 3 in the real axis which we are taking as the x-axis and 7 in the imaginary axis that is y-axis.

Similarly we can plot other numbers also.

We represent all 4 complex numbers in the same Argand diagram.

History of complex number:

Till now we know that we can not take the square root for a negative number. How to take a square root of a negative number? For that a new form of number was introduced by the Italian mathematician Girolamo Cardano. Even though the first mention of imaginary number started in the 1st century. But for a long time no one tried to give the correct explanation of imaginary numbers. In 1545 Girolamo Cardano he solved an equation x(10-x)=40 and got one real number solution and another one square root of a negative number. That gave him the idea to use the imaginary number (whose square is a negative number). In 1833  William Rowan Hamilton expressed this number as a pair of real numbers, for example a+ib is expressed as (a,b) making this form of numbers more believable.

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