Demystifying Complex Numbers: A Practical Guide
At first glance, complex numbers might seem mystifying. Yet, they’re undeniably pivotal in science, engineering, and, naturally, mathematics. Dive in, and the puzzle pieces start to fit!
Why are these weird number useful to us you may ask? The most logical answer is that we would simply not be able to solve equations such as x2 + 1 = 0. Some may say, well it’s simple:
x² + 1 = 0
x² = -1
x = ± √-1, but we know that whatever is under square root is always positive or equal to 0. Not only seen in mathematics, but in electrical engineering, computer science and even in quantum mechanics.
The history of imaginary numbers is really a fascinating journey. Our greatest mathematicians have fought with these “impossible” equations like the one from above for more than five centuries. The first mention of imaginary number was made by an Italian mathematician, Rafael Bombelli in the 1500s.
Fast forward almost two centuries, Leonhard Euler introduced the notation i for the square root of -1. What is the use of this i ? A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which is defined as
i² = -1. For example, 5+6i is a complex number with “5” being called the real part and “3” being called the imaginary part of the complex number.
Carl Friedrich Gauss further developed the idea of representing complex number graphically, leading to the concept of complex plane.
Ever heard of the complex plane, also known as the Argand plane? It’s a visual playground for complex numbers. While it might sound intricate, it’s simpler than you’d expect, mirroring the familiar Euclidean plane.
Just as we plot real numbers on a line, complex numbers have their stage: the horizontal axis showcases the real part, while the vertical axis displays the imaginary. So, every point tells a unique story of a complex number, blending the real and the imaginary. Dive in, and watch these numbers come to life!
A complex number a+bi can be represented as a point in the complex plane where:
- The x-coordinate (horizontal position) is the real part, a.
- The y-coordinate (vertical position) is the imaginary part, b.
For example, the complex number 3+4i would be plotted as the point (3,4) in the complex plane. Let’s add two more complex numbers, 1+ 2i and 4i. For the first complex number we already have it’s point, let’s call that A(3,4). For the others, we will have B(1,2) and C(0, 4).
This graphical representation provides insights into the nature of complex numbers and their relationships, especially when performing operations like addition or multiplication.
Let’s talk about the operation of complex number, our first stop being additions. To add two complex numbers, you simply add their real parts together and their imaginary parts together.
In general, the formula looks something like this:
(a + bi) + (c + di) = (a + c) + (b + d)i ;
Let’s add 3+4i and 1+2i:
3 + 4i + 1 + 2i = (3+1) + (4+2)i = 4 + 6i
In order to subtract one complex number from another, you subtract the real part of the second number from the real part of the first number, and the imaginary part of the second number from the imaginary part of the first number. It is very similar to addition.
In general, the formula looks something like this:
(a + bi)−(c + di)=(a − c)+(b − d)i
Great job! Now that you’ve mastered adding and subtracting complex numbers, let’s dive into the intriguing world of multiplication and division. Multiplying complex numbers? It’s all about using the distributive property, often known as the FOIL method for binomials. Just remember to navigate the imaginary unit with care, and you’re set!
The formula goes like this:
(a + bi)×(c + di)= ac + adi + bci+ bdi², but it is crucial that we already know that i2 = -1 when multiplying complex numbers. Let’s multiply 2+3i and 4+i.
(2 + 3i)×(4+) = 2×4 + 2×i + 3i×4 + 3i×i = 8 + 2i + 12i — 3 =5+14i
(Note: 3i×i=3i²=3×(−1)=−3)
Now that you know how to add, subtract and multiply, let’s learn division. Dividing complex numbers involves a few more steps than other arithmetic operations, primarily due to the need to eliminate the imaginary unit from the denominator. This is where the concept of the conjugate comes into play.
The conjugate, or noted as
where a+bi represents the complex number, is essentially changing the sign of the imaginary part to get the conjugate.
When you multiply complex number by its conjugate, the result is a real number. Specifically:
(a + bi)(a — bi) = a² + b²
To divide two complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. This process is often referred to as “rationalizing the denominator”, eliminating the imaginary part from the denominator, but the result will still remain a complex number.
Enough of theory; let’s dive into the real-world applications! As hinted earlier, complex numbers aren’t just mathematical marvels. They’re powerhouses in engineering, physics, and computer science.
In engineering, one of the standout applications of complex numbers is in Alternating Circuits. Here’s why:
1. AC signals, which vary sinusoidally, are represented as phasors — rotating vectors or complex numbers. This simplifies time-varying signals into a static representation, capturing magnitude and phase angle.
2. Using phasors, Ohm’s law is V = IZ, where V and I are voltage and current phasors, respectively. This makes circuit analysis more algebraic and less differential.
3. Power in AC circuits is complex, comprising real power (actual work) and reactive power (energy oscillating in the circuit).
To sum it up, in AC circuit analysis, the dynamic nature of signals is elegantly transformed into static phasors, making the mathematical exploration more intuitive. This approach is pivotal for grasping how circuits respond across diverse frequencies.
Shifting gears to physics, brace yourself for a captivating journey. While the terrain might seem unconventional, your progress so far assures you’re up for the challenge. Two standout concepts where imaginary numbers shine are Superposition and the Schrödinger equation. These aren’t just fancy terms; they’re central to quantum mechanics, offering a profound understanding of the universe’s intricate dance. Curious to delve deeper?
In quantum mechanics, particles don’t have definite states but exist in a simultaneous combination of multiple states. This phenomenon is called superposition. The coefficients representing these combined states are complex numbers. When a measurement is made, the quantum system collapses to one of the possible states. The probability of collapsing to a specific state is determined by the square magnitude of its complex coefficient. Superposition is behind many quantum phenomena, such as quantum interference and entanglement.
The Schrödinger equation is the fundamental equation of motion in non-relativistic quantum mechanics. It describes how the wave function (or the quantum state) of a system changes over time. This equation is inherently complex, with its solutions often being complex-valued wave functions. The equation can be time-dependent, showing the evolution of a system, or time-independent, revealing the stationary states of a system. The Schrödinger equation provides a bridge between the abstract world of quantum mechanics and observable phenomena, guiding predictions and experiments in the quantum realm.
Complex numbers are pivotal in various algorithms and transformations. Here’s a breakdown of their role:
Fourier Transform:
1. Basics: The Fourier Transform is a mathematical operation that decomposes a signal into its constituent frequencies. It’s a fundamental tool in signal processing.
2. Complex Exponentials: The Fourier Transform uses complex exponentials as its basis functions. These complex sinusoids are essential for representing both the amplitude and phase of each frequency component in a signal.
3. Applications: It’s used in many applications, from audio processing to image analysis. For instance, in audio equalization, the Fourier Transform can isolate specific frequency bands for amplification or attenuation.
Despite their integral role in many areas of science, complex numbers are frequently misunderstood. One of the most common misconceptions stems from the term “imaginary” used to describe them. This label might suggest that these numbers don’t have a basis in reality. However, in mathematics, “real” and “imaginary” are merely terminologies. Both the real and imaginary parts of a complex number have tangible applications in various fields. For instance, in electrical engineering, while the “real” part might denote resistance, the “imaginary” part can represent reactance.
A related misconception is that a number must possess both a real and an imaginary component to be labeled “complex.”
In reality, a complex number can have just a real part, just an imaginary part, or both. If its imaginary component is zero, the complex number is essentially a real number.
Lastly, there’s a notion that introducing complex numbers complicates problems. In many cases, the opposite holds true. Complex numbers can often simplify issues, especially in fields like signal processing or fluid dynamics. They have the power to transform differential equations into algebraic ones or highlight certain mathematical properties more clearly.
In conclusion, complex numbers, despite their name, are neither impractical nor overly intricate. They’re a natural progression of the number system and are indispensable in a wide range of scientific and engineering contexts. The misunderstandings surrounding them often arise from unfamiliarity or a lack of appreciation for their extensive applications.
I hope that after all this information that you acquired in this article you don’t have a headache because in that field complex numbers can’t really help. For those who don’t have a headache and have been intrigued by the multifaceted world of complex numbers and eager to delve deeper, I gathered some resources where you can learn more about complex numbers.
