BySanjenbam Jugeshwor Singh
Updated 9 Jan 2022, 6:24 am
In the 16th century, Venice formulae for solving equations were closely guarded intellectual property. Of particular interest to ballistics and fortifications, expert Niccolo Tartaglia pointed out quadratic and cubic equations which model the behavior of projectiles in flight among other things. These may well ring a bell with you to form the school math-quadratic equations having an x2 term in them and Cubic’s an x3 term. Tartaglia and other mathematicians noticed that some solutions required the square root of negative numbers and herein lies a problem.
Negative numbers do not have square roots- there is no number that when multiplied by itself gives a negative number. This is because negative numbers when multiplied together yield a positive result like (-2) x (-2) =4 not -4. Tartaglia and his rival, Girolamo Cardano, observed that if they allowed negative square root in their calculations, they could still give valid numerical answers (real number as mathematicians call them).
Tartaglia learned this the hard way when he was beaten by one of Cardano’s student in a month –long equation –solving duel in 1530. Mathematicians use i to represent the square root of minus one. This is called imaginary unit-it is not a real number, does not exist in real life. We can use it to find the square root of negative numbers though. This means that the square root of -4 is the square root of 4 multiplied by the square root of -1. In symbol √ =√ √ . The square root of 4 is 2 and square root of -1 is i, giving square root of -4 as 2i and -2i. The arithmetic of i itself initially posed an obstacle for mathematicians.
With the imaginary unit, this seems to break down, with two positives multiplying to give a negative number i.e. ixi=i2 =-1. Equally, here two negatives multiply to give a negative i.e. -ix-i =i2 =-1. This was a problem for some time and made some people feel that using them in formal mathematics was not rigorous. The work of mathematicians on imaginary numbers allowed the development of what is now called the fundamental Theorem of algebra.
In basic terms, the number of solutions to an equation is always equal to the highest power of the unknown in the equation. According to the University of Toronto, there are a variety of uses for imaginary numbers in the real world, most notably in the field of electrical engineering and measuring natural phenomena. An electromagnetic field, for example, requires imaginary numbers to measure because the strength of the field is determined by both electrical and magnetic components that must be combined into a single complex imaginary number to get an accurate measurement.
According to Drexel University, imaginary numbers are further used when measuring phenomena that occurs in nature such as disruption created when water flows around an object. Imaginary numbers are quite useful in many situations where more than one force is acting simultaneously and the combined output of these forces needs to be measured. These forces can be measured using imaginary number makes getting an accurate measurement much easier. Imaginary numbers have long been used in the most important equation of quantum mechanics. A field of physics that describes a very small world.
When you add an imaginary number to a real number, the two together form complex number allowing physics to write out quantum equations in simple terms. But whether quantum theory requires these mathematical chimeras or simply uses them as a convenient shortcut has long been controversial. In fact, even the founders of quantum mechanics thought the implications of including complex numbers in equations were disturbing. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrodinger was the first person to introduce complex number into quantum theory using the quantum wave function.
Schrodinger found a way to represent an equation with real numbers only, along with a set of additional rules about how to use equation. Later physicists did the same in other parts of quantum field theory. However the question remains if there is no solid experimental evidences governing the prediction of these ‘’all real’’ equations.
Is imaginary a simplification of option or does trying to work without imaginary losing the quantum theory of the ability to describe reality? Another popular idea of imaginary number is the Fourier Transform, which states that any function can be made up of a lot of sine and cosine functions added up. This is particularly used in messy signals since they could be broken up into the frequencies that they are composed of, which can then be analyzed and manipulated. In the real world, this is helpful for audio processing, speech recognition, radar etc.
The equation for Fourier Transform involved calculus but it is based on imaginary numbers. Imaginary numbers have prevalence in other areas as well. One of the ideas is in quantum mechanics as discussed above. When studying this alongside, other ideas you can see the prevalence of imaginary number in math. Additionally when dealing with control theory in engineering, graphs are used to portray the stability of system and how certain parameters change with respect to individual components of the control system. These types of systems are found in rockets, fighter jets, robots, autonomous vehicle and more. It is important to understand that you can not necessarily respect imaginary number in the real world. For instance you cannot just have5i pounds of cheese or 3i grams of sugar.
Overall imaginary number helps simply the math for dealing with real world and has a variety of useful applications. The result of study by researchers on imaginary number suggest that the possible ways we can explain the universe in mathematics are actually much more controversial than we thought. By just observing what comes out of some experiments, we can rule out many potential explanations without making any assumptions on the reliability of the physical device used in the experiment.
In the future, physicists may need a small number of experiments built from first principles to reach complete quantum theory. In addition to this, researchers also said that their experimental set up, a rudimentary quantum network, may help outline the principles at which future quantum internets may work. Thank goodness that mathematicians from 500 years ago to the present day, decided that imaginary numbers were worth investigating after all.
(The views expressed is personal)
First published:9 Jan 2022, 6:21 am
sciencemathematicsImaginary number iota (i)math
Sanjenbam Jugeshwor Singh
Assistant Professor, JCRE Global College, Babupara, Imphal. The writer can be reached at firstname.lastname@example.org