The Complex numbers in real life


In this article, I will show the utility of complex numbers, and how physicists describe physical phenomena using this kind of numbers.

What is the complex number?

A complex number can be written in this form:

z=x+iy

Where x and y is the real number, and

i=√(-1)

In complex number x is called real part and y is called the imaginary part.

For example:

z=3+5i
z=6+4i

The imaginary part is not something doesn’t exist, but it is only the part of the complex number.

Real Numbers:

Real numbers are the all positive and negative integers numbers (… ,-3, -2, -1, 0, 1, 2, 3, 4, …) , rational numbers that can written in the fraction or ratio (e.g. 1/2 or 0.5) and irrational numbers that can’t be written in the fraction or ratio because it has infinite numbers after decimal point.

e.g.

√2=1.41421356237… ,π=3.14159265359…

Dimensions:

Dimension is the way to describe the point (position or direction of motion) by writing it down.

If you familiar with paly chess and use chess notation to write down all moves in chess, you can understand the dimensions in a very easy way. But don’t worry, I will explain it in details.

chessboard

In chess, you can notice some numbers and letters on the corner of the chessboard. I can describe any position of chess piece by using these numbers and letters.

For example, if I want to describe the position of the left knight, I can write “b1”. And if I want to describe the motion of this knight, I can write the final position “c3”.

In this very basic example, you can consider these numbers and letters as coordinates (two-dimensional coordinates). In addition, you use these coordinates to describe the position of any chess pieces.

But What Does N Dimensions Mean?

While you read a book, you will notice some topics deal with 4 dimensions or even 12 dimensions. But as we know, the real world is three dimensions only.

One dimension can describe any point (or thing) on the straight line (e.g. street).

Two dimensions can describe any point on the plane (e.g. GPS on your phone)

Three dimensions can describe any point in space.

From our understanding, the dimension is the way to write down the position of any point or object.

If you want to describe the position of a point in space, you need three dimensions. But if the point moves over time, you will need to add one more dimension that contains time. In this case, we have four dimensions.

Ok, what about five dimensions.

In complex motion like CNC machine, you need to give the machine specific information about the position that works in.

The arm of CNC may have many motors (motor units) and each motor can move in three dimensions.

If you have two motors can each one move in three dimensions, you have now six dimensions. And you must write down the six dimensions in the program software of that machine.

CNC arm

In this picture, this CNC arm has motor moves in one dimension (1D) (up and down) and the other two motors move in two dimensions (2D)(up and down, and rotate). Therefore, this is five dimensions (5D) CNC arm.

Phase difference:

If you want to make an experiment and try to count how many rotations of your fan per minute. Of course, you should have a very slow fan to make this experiment easy for you.

While you count the rotation, you need a fixed point (like something in your room). Every time the fan blade reach to this fixed point, you increase the count.

If you begin this experiment and the fan blade not at the fixed point, the distance (rotational angle) between the fixed point and the fan blade is the phase difference.

phase difference

Any delay between two physical phenomena can describe as phase difference.

For example, if you apply AC voltage to the conductor coil, the current across this coil will be a delay with respect to voltage. In this case, we have a phase difference between voltage and current.

The importance of complex number in real life:

In real numbers, we can represent this number as a straight line.

straight line

A real number can store the information about the value of the number and if this number is positive or negative.

But in complex number, we can represent this number (z = a + ib) as a plane.

Complex plane

If you notice, this number has one more information. This new information is the angle (θ).  

To extract this information from the complex number

z=a+ib

by using these relations.

r=√(a^2+b^2 )
θ=tan^(-1)⁡(b/a)

Where

 r is the absolute value of the complex number, or the distance between the origin point (0,0) and (a,b) point.

In this example, z = 2 + 3i

Complex plane - Example
r=√(2^2+3^2 )
θ=tan^(-1)⁡(3/2)

In most cases, this angle (θ) is used as a phase difference.

From trigonometry and right-angled triangle:

cos⁡θ=Opposite/Hypotenuse
sin⁡θ=Adjacent/Hypotenuse
tan⁡θ=Opposite/Adjacent

From these

cos⁡θ=a/r
a=r cos⁡θ
sin⁡θ=b/r
b=r sin⁡θ
z=a+ib
z=r cos⁡θ+i (r sin⁡θ)
z=r(cos⁡θ+i sin⁡θ)

From Euler’s formula:

e^iθ=cos⁡θ+i sin⁡θ
z=r e^iθ

There are many applications that use complex numbers instead of real numbers to represent the value of physical phenomena in real life because the importance to store the phase shift inside these numbers.

The equation of wave and the phase angle:

Moving an object in a circle

If an object moves in a uniform circle, the equation of the projection of this object in x-axis is

x=x_0 cos⁡θ

And the angle velocity (ω) is equal to

ω=θ/t

Where t is the time and x0 is the amplitude (the largest displacement from equilibrium)

θ=ωt
x=x_0 cos⁡(ωt)

But if we choose t=0 at anywhere else.

Moving an object with the phase difference
x=x_0 cos⁡(θ+ϕ)

Where φ is the phase angle.

x=x_0 cos⁡(ωt+ϕ)

The importance of complex number in travelling waves

In the travelling wave, the complex number can be used to simplify the calculations by convert trigonometric functions (sin(x) and cos(x)) to exponential functions (ex) and store the phase angle into a complex amplitude.

The angle velocity (ω) unit is radians per second. If you measure an angle using radian, you can simply calculate the length of the arc.

The circumference (C) of the circle is

C=2πr

And the circumference (C) of the unit circle (r=1) is

C=2π

The angle of the complete circle is (2π) in radian. Therefore, in the unit circle, you can deal with an angle in radian as the length of the arc.

cosine wave

The equation of non-travelling wave

The equation of non-travelling wave is

y(θ)=A cos⁡θ
θ=2π (x/λ)

Because (2π) is the angle of full-wave

In the full-wave:

x=λ
θ=2π(λ/λ)=2π

And in the half-wave:

x=λ/2=(1/2)λ
θ=2π (x/λ)=2π((1/2) λ/λ)=2π(1/2)=π

For the quarter-wave:

x=1/4 λ
θ=2π (x/λ)=2π((1/4) λ/λ)=2π(1/4)=π/2

From that

θ=2π (x/λ)

Then

y(x)=A cos⁡(2π (x/λ))

The equation of travelling wave

If you want to make any wave to travel with velocity (v), you should replace any x with (x – vt) if the motion in the positive direction, and (x + vt) if the motion in the negative direction.

But why.

Consider a point(x) moves in a line.

Motion on the straight line

This point (x) moves from x1 to x2 with velocity (v) in (t) time.

v=Distance/Time=(x_2-x_1)/t
vt=x_2-x_1
x_1=x_2-vt

Any positive change of (x) will subtract with the same amount of change (vt). Therefore, we can convert any wave to travelling wave by replacing (x) with (x-vt) in positive and (x+vt) in a negative direction.

Then, the travelling wave equation is

y(x,t)=A cos⁡((2π/λ) (x-vt))

We can use the wavenumber k to simplify this equation, where the number is 

k=2π/λ

And

v=λ/T

Where v is the velocity of the travelling wave.

1/T=v/λ

And T is the periodic time.

And also

ω=2π/T

Substitute in equation

ω=2π (v/λ)=(2π/λ) v=kv
v=ω/k

Then, the equation of the travelling wave is

y(x,t)=A cos⁡(k(x-(ω/k) t))
y(x,t)=A cos⁡(kx-ωt)

If you have phase difference angle (φ)

y(x,t)=A cos⁡(kx-ωt+ϕ)

We can simply this equation by converting it from trigonometric functions (sin(x) and cos(x)) to exponential functions (ex) using complex wave function.

y(x,t)=A e^i(kx-ωt+ϕ)

But this equation equal to

y(x,t)=A cos⁡(kx-ωt+ϕ)+i A sin⁡(kx-ωt+ϕ)

We will neglect the imaginary part and deal with the real part only. This is because we want to simplify the original equation for calculations.

Now we want to store the phase different angle inside the complex amplitude.

The real amplitude (A) should be converted to a complex amplitude (y0).

y(x,t)=Ae^i(kx-ωt) e^iϕ
y(x,t)=(Ae^iϕ ) e^i(kx-ωt)

The complex amplitude (y0) is

y_0=Ae^iϕ

From that, the complex form of the one-dimensional travelling wave equation is

y(x,t)=y_0 e^i(kx-ωt)

In this equation, we used the complex function to simplify the calculations and used the complex amplitude to store the phase different angle.

Conclusion

In real numbers you can store the information of magnitude (scale value) for example “5” and if that value is positive or negative. But in the complex numbers you can store one more information “angle”, and this angle in most cases used as a phase difference angle.

The complex numbers can be used also to simplify the calculations by convert trigonometric functions (sin(x) and cos(x)) in equations to exponential functions (ex) by adding an imaginary part to these equations and use Euler’s formula:

e^iθ=cos⁡θ+i sin⁡θ

I hope this article will help you to understand the importance of complex numbers to describe the physical phenomena.


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