Simulation · Maths · Class 11

Points of the Argand plane

From the lesson The Argand plane in Complex Numbers and Quadratic Equations. Change the values and watch what happens.

The idea behind it

NCERT §4.5

The complex number x + iy corresponds to the ordered pair (x, y), so it can be drawn as the point P(x, y) of the coordinate plane, and each point stands for exactly one complex number.
A plane in which every point is read as a complex number is called the complex plane or the Argand plane. For example 3 + 4i, −2 + 3i, 5 and 2i sit at (3, 4), (−2, 3), (5, 0) and (0, 2).
The x-axis carries the numbers a + i0 and is called the real axis; the y-axis carries the numbers 0 + ib and is called the imaginary axis.
|x + iy| = √(x² + y²) is the distance of P from the origin O. So all the numbers with |z| = 5, such as 5, 3 + 4i, −4 + 3i and −5i, lie on the circle of radius 5 about O.
The conjugate x − iy is the point Q(x, −y), the mirror image of P in the real axis. The negative −z = −x − iy is the point opposite P through O.
The distance between the points z₁ and z₂ is |z₁ − z₂|. The distance from 1 + i to 4 + 5i is |3 + 4i| = 5.
The signs of the parts fix the quadrant: −2 + 3i is in the second quadrant, −5 − 2i in the third and 1 − 2i in the fourth.

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