Modulus of a Complex Number

The geometry of complex numbers plays a vital role in dealing with problems. Sometimes a purely geometric approach minimises the solution of the problem. Leonard Euler: Solving equations such as x² + n² = 0, n belongs to Z were thought to be impossible. However, mathematicians managed to come up with the idea to create such a number to solve these equations.

Indian mathematician Shri Dharacharya gave us the  formula for the solution of ax2+bx+c = 0 as

{-b ± √( b2 – 4ac)} / 2a

This formula is named after the scientist as the Shri Dharacharya method/formula.

Discovery of Complex Numbers

The first step in developing the quadratic equation was recognising that it was linked to a highly practical situation that required a “fast and dirty” solution. In this context, it is important to emphasise that Egyptian mathematics did not use equations and numbers in the same way that we do today; rather, it was descriptive, rhetorical, and at times difficult to understand.

The Egyptian scholars were well aware of this shortcoming, but they devised a solution: rather than learning an operation or a formula for calculating the sides from the required area, they calculated the area for all possible sizes and shapes of squares and rectangles and created a look-up table. This method worked much like we learn the multiplication tables by heart in school instead of doing the operation correctly.

In 700 AD, the general solution for the quadratic equation came up this time using numbers, which was devised by an Indian mathematician called Brahmagupta. He used irrational numbers to solve this problem.

The final, appropriate solution as we know it today came in 1100AD by Baskhara. Baskhara was the first mathematician to recognise that any positive number has two square roots. Indian mathematician Shari Dharacharya gave a formula for the solution of ax2+bx+c = 0 as 

(-b±b2 – 4ac  ) / 2a

Complex Numbers

A number in the form x+iy, where i=-1 and x, y may be real numbers or the numbers containing i are called a complex number. For example 2 + 3i, 2-3i, 3, 0, 2 + i(2+i) all are complex numbers.

Real and Imaginary Part

If z = x + iy, where x, y e R and i=-1, is a complex number, then x is called the real part of z and y’s called the imaginary part of z, which we can write as

Re(z) = x and Im(z) = y

If Re(z) = 0, then z is purely imaginary complex numbers.

If I’m(z) = 0, then z is a purely real complex number.

z = 0 is purely real as well as purely imaginary.

Modulus of Complex Numbers

The modulus of a complex number is calculated as the distance from (0, 0). If z = x + iy, then its modulus is represented by |z| = x² + y² = (Re(z))² + (Im(z))².

In other words, the modulus of a complex number is the square root of the sum of the square of the real part of the complex number and the square of the imaginary part of the complex number.

If we assume z, z1 and z2  to be complex numbers, then

1. |z1z2|=|z1| |z2|
2. |-z| = |z|
3. |z| =0 if and only if z=0
4. |z1| / |z2| =|z1 / z2|

Modulus of a Complex Number: Examples

Example 1

Find the modulus of z=3 + i.

Solution

Here imaginary part of the equation is i, and the real part of the equation is 3

Therefore we can write x=3 and y=1

On applying formula,

|z| = x² + y² = (Re(z))² + (I’m(z))²

We get the modulus of z as

|z| =3+1=2

Example 2

Find the modulus of z=x+i1-2x, x belongs to R and x ≤ ½.

Solution

Here imaginary part of the equation is 1-2x and the real part of the equation is x.

On applying formula,

|z| = x² + y² = (Re(z))² + (Im(z))²

We get

|z| =x² + (1-2x)=x² + 1-2x 

=(x-1)(x-1)

=|x-1|

The Logarithm of a Complex Number

To express loge(x + iy) in the form A + iB , we have

z=x+iy = r(cosӨ + isinӨ)

Where

r=|z|=x² + y² ,

Ө = arg(z) = tan-1(y/x)

=reiө.1

=reiө. ei2k

z=rei(2k + ө); k is an integer

Therefore on taking log both the side we get:

Logez = log(x+iy) = ½log(x2+y2) + i(2k + tan-1y/x), which is the required form.

Conclusion

For every x € R,  x2 >= 0, implying that the square of a real integer (positive, negative, or zero) is non-negative. As a result, the equations x2 = – 1, x2 = – 5,  x2+ 7 = 0 and others cannot be solved in the real number system. As a result, we need to expand the real number system to a bigger system in order to solve such equations. We’ve extended the real number system to a larger system called the complex number system, allowing us to solve quadratic equations like ax2 +bx+c=0, where a, b, and c are all real values.