Physics Lournal

Mathematical Interlude - Complex Numbers

Complex numbers are numbers that can be expressed in the form $a + bi$ , or $x + iy$ , where a/x & b/y are Real Numbers and i is the Imaginary Unit, where $\large i^{2} = 1$ .

While $b$ is a real number, $bi$ constitutes an imaginary value.

They can also be written in Polar Form:

$\Large re^{i \theta} = r(cos\theta + i\cdot sin\theta)$ .

$\theta$ is the angle measured counterclockwise from the x-axis.

We visualize complex numbers using a Cartesian Coordinate System, where the y-axis is the imaginary line, and the x-axis is the real line. This is often referred to as the Complex Plane.

Complex numbers can be manipulated arithmetically, as with real numbers.

Particularly, multiplying complex numbers is much easier using the Polar form:

$\Large r_1(e^{i \cdot \theta_1}) \cdot r_2(e^{i \cdot \theta_2}) = (r_1 \cdot r_2)e^{i(\theta_1 + \theta_2)}$

Every complex number $z$ , has a conjugate $z^{\star}$ , which can be found by flipping the sign of the imaginary component.

If $\Large z = x + iy = re^{i\cdot \theta}$ ,

$\Large z^{\star} = x + (-i)y = re^{-i\cdot \theta}$ .

Worked Example Under-Construction

{{TODO}} Return after studying some

{{TODO}} Return after studying some Trigonometry {{∆:1+0}}

{{r/moved}}* {{r/moved}}

Given some complex number like $7 + 4i$ , we take the pure form of a complex number, $z = a + bi$ , and observe the pure polar form:

$z = r(cos \theta + i (sin \theta))$

We take the value of $r$ to be the absolute value of $z$ , which is equivalent to the sum of the squares of $a\; and\; b$ :

$r = |z| = \sqrt{a^2 + b^2}$

Focusing on the sum of squares, we input our numbers from before:

$\sqrt{7^2 + 4^2}$

Simplifying these with some basic arithmetic we get:

$\sqrt{49 + 16}, -> \sqrt{65}$

$\sqrt{65} \approx 8.06$

Going back to our original formula:

$r(cos \theta + i(sin \theta))$ , we have a value for r:

$8.06(cos \theta + i(sin \theta))$

Because $a > 0$ , we use the following formula to calculate the correct value for $\theta$ (theta):

$tan^{-1}(\frac{b}{a})$

The value for $tan^{-1}$ was incorrect, and seems to be tricky to find, however the end result should give us 0.51.

Or we could be a physicist and say $0.44 \pm{0.7}$ .

So what we want to do is the following calculation:

$tan^{-1}(\frac{4}{7})$ , which gives us back: $0.51$ , for our value of $\theta$

So we have a value for $r$ , and $\theta$ , allowing us to fill out the polar form equation entirely:

$8.06(cos(0.44) + i(sin(0.44)))$

Multiplying a number by its complex conjugate, always yields a non-negative real number.

Operations

Magnitude: $|z| = r = \sqrt{x^{2} + y^{2}}$

Phase: $\varphi = tan^{-1}(x/y)$

Addition: $z_1 + z_2 = (a +bi) \times (c + di) = (ac -bd) + (ad + bc)i$

Referenced in

1. Systems of Linear Equations and Matrices

Thoughts: One of the great things about Mathematics is how things get built upon each other, into wildly new fields, yet retaining fundamental concepts, and being relatable to other field: $c$ and $d$ are functions of $ax + by$ and $ax + by + cz$ respectively, and this also relates to the function focused aspect of the fundamentals of calculus, in that we can say $y = f(x) = ax+by+cz$ . Also, we know that we are dealing with real numbers, due to the nature of the coordinate system not being complex, so we can assume that $a,b,c,x,y,z \in \mathcal{R}$ , which is the set of real numbers for which they have membership, so there are even reminisces of set theory hidden within this. This makes sense though, because the foundations of mathematics have been set theoretic since the early 1900's.

The Theoretical Minimum: Quantum Mechanics

Mathematical Interlude - Complex Numbers

(Game) Formalism in the Philosophy of Mathematics

One common presentation of the formalism is that it holds that mathematics is not representative of an abstract sector of reality, but rather that is somewhat of a game, being no more wed to an ontology than chess. There is some intuitive support for this, when one considers the novice working on calculations or the application of some method or technique for solving problems. It also refers to the nature of the work done by advanced mathematicians, such as the handling of complex numbers after their introduction, or the opinion of mathematicians on higher levels of set theory. Lastly, it is the position to which many will gesture, when questioned about the nature of mathematics.

Chapter 1: Vector Spaces

Within this study, it is beneficial not to study just real numbers, but complex numbers as well.

Mathematical Interlude - Complex Numbers

Mathematical Interlude - Complex Numbers

Complex numbers are numbers that can be expressed in the form a+bia + bia+bi, or x+iyx + iyx+iy, where a/x & b/y are Real Numbers and i is the Imaginary Unit, where i2=1\large i^{2} = 1i2=1.

While bbb is a real number, bibibi constitutes an imaginary value.

θ\thetaθ is the angle measured counterclockwise from the x-axis.

We visualize complex numbers using a Cartesian Coordinate System, where the y-axis is the imaginary line, and the x-axis is the real line. This is often referred to as the Complex Plane.

Complex numbers can be manipulated arithmetically, as with real numbers.

Particularly, multiplying complex numbers is much easier using the Polar form:

Every complex number zzz, has a conjugate z⋆z^{\star}z⋆, which can be found by flipping the sign of the imaginary component.

If z=x+iy=rei⋅θ\Large z = x + iy = re^{i\cdot \theta}z=x+iy=rei⋅θ,

z⋆=x+(−i)y=re−i⋅θ\Large z^{\star} = x + (-i)y = re^{-i\cdot \theta}z⋆=x+(−i)y=re−i⋅θ.

Worked Example Under-Construction

{{TODO}} Return after studying some Trigonometry {{∆:1+0}}

Given some complex number like 7+4i7 + 4i7+4i, we take the pure form of a complex number, z=a+biz = a + biz=a+bi, and observe the pure polar form:

z=r(cosθ+i(sinθ))z = r(cos \theta + i (sin \theta))z=r(cosθ+i(sinθ))

We take the value of rrr to be the absolute value of zzz, which is equivalent to the sum of the squares of a and ba\; and\; baandb:

r=∣z∣=a2+b2r = |z| = \sqrt{a^2 + b^2}r=∣z∣=a2+b2​

Focusing on the sum of squares, we input our numbers from before:

72+42\sqrt{7^2 + 4^2}72+42​

Simplifying these with some basic arithmetic we get:

49+16,−>65\sqrt{49 + 16}, -> \sqrt{65}49+16​,−>65​

65≈8.06\sqrt{65} \approx 8.0665​≈8.06

Going back to our original formula:

r(cosθ+i(sinθ))r(cos \theta + i(sin \theta))r(cosθ+i(sinθ)), we have a value for r:

8.06(cosθ+i(sinθ))8.06(cos \theta + i(sin \theta))8.06(cosθ+i(sinθ))

Because a>0a > 0a>0, we use the following formula to calculate the correct value for θ\thetaθ (theta):

tan−1(ba)tan^{-1}(\frac{b}{a})tan−1(ab​)

The value for tan−1tan^{-1}tan−1 was incorrect, and seems to be tricky to find, however the end result should give us 0.51.

So what we want to do is the following calculation:

tan−1(47)tan^{-1}(\frac{4}{7})tan−1(74​), which gives us back: 0.510.510.51, for our value of θ\thetaθ

So we have a value for rrr, and θ\thetaθ, allowing us to fill out the polar form equation entirely:

8.06(cos(0.44)+i(sin(0.44)))8.06(cos(0.44) + i(sin(0.44)))8.06(cos(0.44)+i(sin(0.44)))

Multiplying a number by its complex conjugate, always yields a non-negative real number.

Operations

Referenced in

1. Systems of Linear Equations and Matrices

The Theoretical Minimum: Quantum Mechanics

Mathematical Interlude - Complex Numbers

(Game) Formalism in the Philosophy of Mathematics

Chapter 1: Vector Spaces

Within this study, it is beneficial not to study just real numbers, but complex numbers as well.

Complex numbers are numbers that can be expressed in the form $a + bi$ , or $x + iy$ , where a/x & b/y are Real Numbers and i is the Imaginary Unit, where $\large i^{2} = 1$ .

While $b$ is a real number, $bi$ constitutes an imaginary value.

$\theta$ is the angle measured counterclockwise from the x-axis.

Every complex number $z$ , has a conjugate $z^{\star}$ , which can be found by flipping the sign of the imaginary component.

If $\Large z = x + iy = re^{i\cdot \theta}$ ,

$\Large z^{\star} = x + (-i)y = re^{-i\cdot \theta}$ .

Given some complex number like $7 + 4i$ , we take the pure form of a complex number, $z = a + bi$ , and observe the pure polar form:

$z = r(cos \theta + i (sin \theta))$

We take the value of $r$ to be the absolute value of $z$ , which is equivalent to the sum of the squares of $a\; and\; b$ :

$r = |z| = \sqrt{a^2 + b^2}$

$\sqrt{7^2 + 4^2}$

$\sqrt{49 + 16}, -> \sqrt{65}$

$\sqrt{65} \approx 8.06$

$r(cos \theta + i(sin \theta))$ , we have a value for r:

$8.06(cos \theta + i(sin \theta))$

Because $a > 0$ , we use the following formula to calculate the correct value for $\theta$ (theta):

$tan^{-1}(\frac{b}{a})$

The value for $tan^{-1}$ was incorrect, and seems to be tricky to find, however the end result should give us 0.51.

$tan^{-1}(\frac{4}{7})$ , which gives us back: $0.51$ , for our value of $\theta$

So we have a value for $r$ , and $\theta$ , allowing us to fill out the polar form equation entirely:

$8.06(cos(0.44) + i(sin(0.44)))$