Imaginary Numbers

  • Invented to solve problems where an equation has no real roots, e.g., x2+16=0. The idea of declaring the existence of a quantity i such that i2=1 allows us to express the solution as:
x=16=16i2=±4i

The set represented by I defines an imaginary number as:

i2=1

Powers of i

If i2=1, then i4=i2i2=11=1.

Therefore, we have the sequence:

ii2i3i4i5i1i1i

Complex Numbers

A complex number is just the sum of a real and an imaginary number:

z=a+bi,a,bR,i2=1

Operations on Complex Numbers

Given two complex numbers:

z1=a1+b1iz2=a2+b2i

Addition and Subtraction

z1±z2=a1±a2+(b1±b2)i

Product

z1z2=a1a2+a1b2i+a2b1i+b1b2i2given that i^2 = -1=(a1a2b1b2)+(a1b2+b1a2)i

Given the complex number:

z=a+bi

Norm (Modulus or Absolute Value)

|z|=a2+b2

Complex Conjugate

The product of two complex numbers where the only difference between them is the sign of the imaginary part is:

(a+bi)(abi)=a2abi+abib2i2=a2+b2

This quantity abi is called the complex conjugate of z (denoted as z). It implies that:

zz=|z|2

Inverse

z1=1z

Multiplying the numerator and denominator with the conjugate of z (so that we have a real part on the denominator):

z1=1zzz=zzz=z|z|2

Square Root of i

We’re trying to find a complex number z such that:

i=zi=z2

Assuming that z is the complex number z=a+bi:

i=(a+bi)2=(a+bi)(a+bi)(1)=a2b2+2abi

Therefore:

(a2b2)+(2ab)i=0+1i

Equating real and imaginary parts:

a2b2=02ab=1

Therefore, a=±b. Replacing a=b in the second equation, we obtain 2b2=1, which is not satisfied by any real number b. Therefore, the case a=b is impossible. Replacing a=b in the second equation, we obtain 2a2=1, so:

2a2=1a2=12a=b=±12=±12

Finally, the value of i is:

i=(a+bi)=±12(1+i)

The value of i is found in the same way (by replacing b=a in the equation 2ab=1 found from multiplying (1) by 1):

i=(a+bi)=±12(1i)

Matrix Representation of a Complex Number

The matrix C for a complex number is the sum of two other matrices representing the real R and imaginary I parts:

C=R+I

which can be written as:

C=aR^+bI^,a,bR

Where R=1 and I=i.

The matrix representation of R=1 in 2D is the identity matrix:

[1001]

To find the matrix representation of i, we have to analyze the definition of i, which is a quantity which squares to 1. Given that we already know the value of 1 in matrix form:

i2=1[1001]=[1001]

Squaring the following matrix gives the matrix above. Then the value of i expressed in matrix form is:

I=[0110]

Finally, the value of C is:

C=a[1001]+b[0110]=[abba]

The Complex Plane

The powers of i give rise to the sequence (1,i,1,i,1,), which is quite similar to the pattern (x,y,x,y,x,). The resemblance is no coincidence, as complex numbers belong to a 2-dimensional plane. This complex plane allows us to visualize complex numbers using the horizontal axis for the real part and the vertical axis for the imaginary part.

−6−5−4−3−2−10123456−3 i−2 i−1 i0 i1 i2 i3 i
1,i,1,i

We can see that the positions of i0=1,i1=i,i2=1,i3=i, suggest that the multiplication of a complex number by i is equivalent to rotating through 90 degrees.

e.g.,

z1=2+iz2=(2+i)(i)=1+2iz3=(1+2i)(i)=2iz4=(2i)(i)=12iz5=(12i)(i)=2+i=z1
−6−5−4−3−2−10123456−3 i−2 i−1 i0 i1 i2 i3 i

A complex number is rotated ±90 by multiplying it by ±i.

Let’s graph the roots of i=±12(1+i):

−6−5−4−3−2−10123456−3 i−2 i−1 i0 i1 i2 i3 i

We can see that 12(1+i) is exactly at 45 and 12(1+i) is exactly at 225.

Let’s multiply the complex number 2+i by i (it should rotate it by 45):

z1=2+iz2=(2+i)(12+12i)=12+312i
−6−5−4−3−2−10123456−3 i−2 i−1 i0 i1 i2 i3 i

Multiplying z2 by i again should be equal to multiplying z1 by i (because z2 is already rotated by 45):

z2=12+312iz3=(12+312i)(12+12i)=(1232)+(12+32)i=1+2i

Which is exactly what we find if we multiply z1 by i. These observations suggest that we can build a complex number which can rotate another complex number by any angle.

A complex number is rotated 45 by multiplying it by 12+12i.

A complex number is rotated 225 by multiplying it by 1212i.

Polar Representation

Instead of using coordinates in the complex plane, we can represent a polar number with the length of the vector from the origin to the complex coordinate and the angle between the complex vector and the positive real axis:

r=|z|=a2+b2θ=arctan(ba)

The horizontal component of z is then rcos(θ), and the vertical component is rsin(θ). Expressing the complex number using these quantities:

z=a+bi=rcosθ+risinθ=r(cosθ+isinθ)

Euler provided the identity:

(2)eiθ=cosθ+isinθ

Which allows us to represent any complex number as:

z=reiθ

Given two polar numbers:

z=reiθw=seiϕ

Their product is:

zw=rsei(θ+ϕ)=rs[cos(θ+ϕ)+isin(θ+ϕ)]

Which effectively rotated the complex number z by ϕ angles! However, the quantity zw was scaled by s units. To avoid scaling, we can normalize w (i.e., making r=1, which is equal to (2)).

A rotor is a complex number that rotates another complex number by an angle θ (through multiplication) and has the form:

eiθ=cosθ+isinθ

Rotating a complex number x+yi by an angle θ:

x+yi=(x+yi)(cosθ+isinθ)=(xcosθysinθ)+(xsinθ+ycosθ)i

Which in matrix form is:

[xyyx]=[xyyx][cosθsinθsinθcosθ]

Note that because of the way the complex product is defined, the multiplication between two complex numbers commutes:

x+yi=(cosθ+isinθ)(x+yi)=(xcosθysinθ)+(xsinθ+ycosθ)i