What Are Imaginary Numbers? Why Are They So Important?

Imaginary numbers? As if the numbers we already have weren’t enough. The commentary on mathematics’ difficulty has become a platitude. We’re all aware that some proportion of all high schoolers are terrified by the unintelligible language their math textbooks are scribbled with, like Victorian readers encountering Ulysses for the very first time.

However, rather than experiencing relief, their terror is only exacerbated when the turn of a thin page marking the end of another arduous chapter introduced them to an entirely new dimension of numbers – imaginary numbers. Eventually, the introduction of imaginary numbers opened our eyes to an entirely novel branch of mathematics, another of nature’s absurd languages – complex mathematics. Why are they so significant if they’re not even real?

“Impossible” Numbers: What are imaginary numbers?

If the number 1 is the unit or identity of real numbers, such that each number can be written as that number multiplied by 1, then imaginary numbers are real numbers multiplied with the imaginary identity or unit ‘, What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC‘.

, What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABCThe imaginary unit represents a clever way around a mathematical roadblock. Consider the simple quadratic equation , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC. The value of , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC for which the sum becomes zero must be a number whose square is -1. But that is impossible! Squares of all real numbers, either positive or negative, can only be positive numbers. In other words, the problem has no real number solution.

Even so, let’s solve for , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC anyway. The solution turns out to be , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC. There is the roadblock – negative numbers don’t have a legitimate square root. This indivisible unit , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC is denoted by , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC, such that the solution to the given equation becomes , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC. All numbers multiplied by this unit have come to be known as imaginary numbers. The unit can be used to derive the solutions for other similar quadratic equations. Consider , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC. Here, , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC turns out to be , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC, which can be written as , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC.

Witnessing this, impossible is the first word that would pop into anyone’s mind. The square root of negative numbers is highly counterintuitive, but so were negative numbers when they were first introduced. Even Euler was confounded by them. So, don’t worry if you can’t wrap your head around imaginary numbers; initially, even the most brilliant of mathematicians couldn’t.

, What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC

The incredulity results from, of course, the denial of the comfort of certainty that we find in rituals to which we are indelibly accustomed. Imagine the astonishment of the Romans when they were introduced to Base-10 numbers. Or, the infinite numbers between them. If zero, or a symbol, a “something” to describe “nothing” wasn’t enough, the concept of negative numbers was introduced – numbers that represent a quantity less than nothing. This was a Roman’s worst nightmare. Despite their horror, we needed these new systems… all of them.

The existence of equations that are unsolvable in a single system is quite common. With only natural numbers, , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC would be unsolvable, unless we introduce integers. Now, despite the knowledge of both natural numbers and integers, the equation , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC would be unsolvable without introducing rational numbers. The equation , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC would be unsolvable without irrational numbers. And finally, , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC would be unsolvable without imaginary numbers.

Prime number art Pure-mathematics-formulæ-blackboard

Ahhhh, good ol’ math. (Photo Credit: Wallpaper / Wikimedia Commons)

One would assume that imaginary numbers arose from our need to solve quadratic equations, but their existence was actually hinted at by a cubic equation. One of the oldest and most popular problems that involved them was “To divide 10 into two parts, the product of which is 40”. This was solved by Girolamo Cardano, who divided 10 into two equal parts of 5. He squared them (25) and subtracted 40 from it, which left him with -15. He concluded that the square root of -15 added to or subtracted from 5 gives two parts, the product of which is 40.

Doesn’t make any sense? This would be much easier to understand if the operations were written this way: , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC. A quick calculation would transmute this product into a simple subtraction: , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC

Now, because , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC (for simplicity, let’s ignore the other root: , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC) the operation becomes a sum:

, What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC

, What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC

, What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC

What Are Complex Numbers?

Subsequently, in 1572, the mathematician Rafael Bombelli published his treatise on algebra, which expounded the nature of imaginary numbers in splendid detail. This made him a central figure in the understanding of complex numbers. Complex numbers are a combination of both real and imaginary numbers. A complex number Z is the sum or subtraction of a real number A and an imaginary number Bi, such that , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC., What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC

Despite this work of genius, Bombelli’s book was frowned upon. The numbers were dubbed fictitious – or even useless – by his peers. Descartes referred to them as imaginary in a tone that verged on ridicule. The numbers accrued acceptance when Euler and Gauss began to recruit them in their work. Other mathematicians gradually caught up and realized how immensely important they are, that these numbers fill an empty void in the subject. However, Descartes’ slur stuck, and we continued to refer to them as imaginary to this day.

In a way, every number seems fictitious. This is the central argument in the debate of whether math was discovered or invented. The Romans invented their system to count whole, countable things, such as to keep count of the number of elephants they lost in a war. The process can be roughly translated to: “Let us represent this exclusive set of elephants with this symbol. The introduction of a new elephant will transform this set into a new, different, exclusive set which can be represented by another symbol and so on.”

Child learning a numbers

Is math a property of the Universe, or something we devised to make sense of it? Was it invented or discovered? (Photo Credit: Flickr)

In that case, what do negative numbers entail? One can say that a negative number is a highly convenient way to limn debt. Without negative numbers, quantifying debt would have been extremely difficult. Similarly, complex numbers represent rotation. This renders them ubiquitous in branches of physics that involve vectors and waves, such as the laws of electromagnetism and Fourier Transform, an indispensable tool that we use to analyze music.

The new dimension I mentioned in the prelude is real. There is a plane solely dedicated to plotting complex numbers known as the complex plane. To understand what I meant by rotation, let’s step into this new dimension. A complex plane is a 2D plane on which real numbers are plotted on the horizontal axis, known as the real axis, and complex numbers are plotted on the vertical axis, known as the imaginary axis.

Axis Complex Plane

The complex plane

A peculiar pattern can be observed each time we multiply a real number with , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC. Graphically, the product , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABCrepresents a counterclockwise rotation of a vector of size , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC by 90º each time an , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC is multiplied to it. If we assume the value of , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC to be 1, only 4 vectors repeatedly emerge: , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC and , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC; a quarter of a circle, a semi-circle, a three-quarter circle and a full circle.

However, if we were to rotate this vector by, say, only 45º, the resulting vector would lie in the middle of the two axes; the area it sweeps represents a small part – in this case, exactly one-eighth of a circle. A 45º line implies the equivalence of coordinates or, as it is popularly known, the , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC line. If one such coordinate is (A, B): (1,1), then one can conclude that the drawn vector represents the complex number , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC.

Complex number triangle

Rotation is build into complex numbers

The same vector can also be represented as a vector sum of its horizontal and vertical components — , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC. Here, , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC is the hypotenuse of a triangle formed by the vector and its perpendicular components and , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC is the angle the vector makes with the horizontal axis. The modulus , What Are Imaginary Numbers? Why Are They So Important?, Science ABC, Science ABC’s value is the Pythagorean sum of the two components. This property binds trigonometry and complex numbers in a conjugal relationship, rendering them utterly inseparable.

No wonder they are so ubiquitous in electronics and wave theory. This characteristic illustrates how deeply rotation is ingrained in complex numbers. This makes for efficient problem solving, problems that concern rotation, problems that would have been troublesome or tedious when tackled with conventional math – doable but inefficient. However, personally, their most commendable achievement remains their direct involvement in the formation of one of the most beautiful structures in abstract math, one of the most beautiful sentences written in nature’s languages.

Mandel zoom 07 satellite

The Mandelbrot set (Photo Credit : Creative Commons / Wikimedia Commons)

The curve drawn on a complex plane, known as the Mandelbrot set, looks like an enormous garden that comprises a long nonlinear trail of infinite coniferous trees and spiraling flowers. It is absurd how the menial act of simply plotting coordinates can lead to such an aesthetic marvel. It is a visual treat for Mandala artists, a true testimony to splendor and detail. And remember, all of this came from imaginary numbers. Math never ceases to amaze me.

References

  1. KhanAcademy.org
  2. University of Rhode Island
  3. Lamar University
  4. University of Toronto
The short URL of the present article is: http://sciabc.us/yFobI
Help us make this article better

Akash Peshin is an Electronic Engineer from the University of Mumbai, India and a science writer at ScienceABC. Enamored with science ever since discovering a picture book about Saturn at the age of 7, he believes that what fundamentally fuels this passion is his curiosity and appetite for wonder.

.
Science ABC YouTube Videos

  1. What's the Mysterious & Super Awesome Thing That Occupies 90% of Your Brain?What's the Mysterious & Super Awesome Thing That Occupies 90% of Your Brain?
  2. Why Don't They Have Parachutes For Passengers In Commercial Planes?Why Don't They Have Parachutes For Passengers In Commercial Planes?
  3. Methusaleh: The oldest tree in the world | What's the mystery of trees' immortality?Methusaleh: The oldest tree in the world | What's the mystery of trees' immortality?
  4. 7 Scientifically Inaccurate Things They Show in Movies: Most Common Movie Mistakes and Myths7 Scientifically Inaccurate Things They Show in Movies: Most Common Movie Mistakes and Myths
  5. Why Venus and Mercury have no Moons?Why Venus and Mercury have no Moons?
  6. What Does It Take To Make Vaccines?What Does It Take To Make Vaccines?
  7. Are Zebras Black with White Stripes or White with Black Stripes?Are Zebras Black with White Stripes or White with Black Stripes?
  8. What Are Asteroids And Where Do They Come From?What Are Asteroids And Where Do They Come From?

Tags: