Complex numbers extend the concept of real numbers by introducing an imaginary unit, denoted as "i." Real numbers can be considered a subset of complex numbers with the imaginary part equal to zero. Complex numbers include both a real and imaginary component, allowing for operations like addition, subtraction, multiplication, and division.

The first step when dividing complex numbers is to find the conjugate of the denominator, which is the same expression but with the sign of the imaginary part changed. This is done to eliminate the imaginary part in the denominator.

In the complex plane, each complex number is represented by a point, with the real part as the x-coordinate and the imaginary part as the y-coordinate. The mapping of complex numbers in the complex plane allows us to visualize operations like addition, subtraction, multiplication, and division geometrically. It also enables us to study properties such as modulus, argument, and conjugate of complex numbers.

A: "bicentillion" doesn't exist, I think you mean: centillion, or here in Britan a vigintillion, a vigintillion has 120 zeros after it and a centillion has 303 zeros after it.

Gang-nam is a city in south Korea where Psy was raised it translates into English as city south of river it is a business city but Psy claims the style there is dress classy act cheesy

so when Psy says Gang-nam style he means in the style of the people of Gang-nam

This is a rather loose application. The related link shows the fluid flow field of two

water sources near each other.

Refer to related link down below.

typedef struct complex {

double real, imag;

} complex;

...

complex x, y, z;

...

/* add */

z.real = x.real + y.real;

z.imag = x.imag + y.imag;

/* sub */

z.real = x.real - y.real;

z.imag = x.imag - y.imag;

/* mul */

z.real = x.real*y.real - x.imag*y.imag;

z.imag =x.imag*y.real + x.real*y.imag;

/* div */

double d = y.real*y.real + y.imag*y.imag;

z.real = (x.real*y.real + x.imag*y.imag)/d;

z.imag = (x.imag*y.real - x.real*y.imag)/d;

There is an infinite number of answers, but anything that is divisible by 3, 6, 8, 9, 11, 12, 13 and so on

eg 3, 6, 9, 12...

eg 11, 22, 33, 44...

eg 13, 26, 39, 52,

The conjugate of a complex number can be found by multiplying the imaginary part by -1, then adding the "real" part back. (-2i) * -1 = 2i, so the conjugation is 7+2i

this is a very good question. lets solve (2+3i)/(4-2i).

we want to make 4-2i real by multiplying it by the conjugate, or 4+2i

(4-2i)(4+2i)=16-8i+8i+4=20, now we have

(2+3i)/20 0r 1/10 + 3i/20

notice that -2i times 2i = -4i^2 =-4 times -1 = 4

In (x+iy), x=any value in the real number line.

1. i1 = i, i2= -1, i3= -i, i4= 1, i5= i, i6= -1 and so on.

2. Euler's constant e, when raised to the power of a complex number, gives a point in the complex plane according to Euler's Identity, e^(iθ

) = cos(θ

) + i*sin(θ

).

Whenθ

is replacedby pi, we get:

e^(iπ)

= cos(π

) + i*sin(π

)

= -1 + 0

= -1

In other words, e^(iπ)

+ 1 = 0.

3. Many intriguing fractals have been discovered in the complex plane, the most famous of which is the Mandelbrot Set. This is formed by an iterative process of a point in the plane. It produces a beautiful image, nicknamed by many as the "Thumbprint of God". (Google Images is your friend here - check it out.)

4. De Moivre's Theorem holds true in the realm of the imaginary, stating that

(r cisθ

)n= rncis nθ

.

This is a fascinating theorm in itself, but it also has hidden implications.

For example, if r = 1 (the point was 1 unit away from the origin), and you were to plot this point with a small angle above the positive real axis, then raising this number to an increasing power would result in that point making a full revolution about the origin and coming right back to where it started.

This is because rnstays the same (1^(anything) = 1), but the angle is slowly increasing as the power does. Therefore, as the angle increases, a circle is plotted around the origin with radius 1.

There are many other fascinations around complex numbers, but this covers the basics.