What is the significance of imaginary numbers

This sporting life. Stage and screen. Birds and the bees. Do they have purpose or is it just mathematicians having some fun? Bob Jones , Aberdeen Scotland All numbers are imaginary even "zero" was contentious once. Introducing the square root s of minus one is convenient because i all n-degree polynomials with real coefficients then have n roots, making algebra "complete"; ii it saves using matrix representations for objects that square to -1 such objects representing an important part of the structure of linear equations which appear in quantum mechanics, heat diffusion, optics, etc.

The hottest contenders for numbers without purpose are probably the p-adic numbers an extension of the rationals , and perhaps the expiry dates on army ration packs. Michael Hall, Canberra Australia Don't forget that maths is an invention, if you like the rules of a game by which we play. It is designed for convenience. Since the roots of maths predate the complexity that led in much later centuries to imaginary numbers, it is no surprise that later elaborations for functions originally unforseen proved problematic.

Further: do numbers have a use in the "real" world? Well, yes, but apparently crucial and well-established! Same with imaginary numbers. The discomfort you feel is the awkwardness not between reality and the i series but between the deceptively named "real" series and its i counterpart. And if by "use" you mean application, I am sure plenty of mathematicians and physicists and engineers will swear to their importance.

M B Drennan, Oxford UK On the contrary, purely real numbers only describe a perfect, simplified world in physics while imaginary numbers must be used to include the myriad complicating factors found in the "real" world. For example, the school text-book pendulum was a doddle, with nice real numbers and a swing which carried on forever. However, friction with the air causes damping of the motion so that it decreases over time, and the only way to deal with damping is using imaginary numbers.

Perhaps it would've been better to label the types of number the other way around, or simply "plus-root" and "minus-root" numbers or something.

A complex number is the fancy name for numbers with both real and imaginary parts. Not too bad. In the case of negatives this is:. Which is another way to find the absolute value. But for complex numbers, how do we measure two components at 90 degree angles? Geez, his theorem shows up everywhere , even in numbers invented years after his time. Yes, we are making a triangle of sorts, and the hypotenuse is the distance from zero:.

I want to change my heading 45 degrees counter-clockwise. Sorry, did I break your calculator? Care to answer that question again? So our new orientation is 1 unit West -1 East , and 7 units North, which you could draw out and follow.

But yowza! We found that out in 10 seconds, without touching sine or cosine. There were no vectors, matrices, or keeping track what quadrant we are in. It was just arithmetic with a touch of algebra to cross-multiply. Imaginary numbers have the rotation rules baked in: it just works.

Even better, the result is useful. How, exactly, were you planning on drawing and following that angle? With the protractor you keep around? Complex numbers beat you to it, instantly, accurately, and without a calculator.

This is just a preview; later articles will give you the full meal. Ok, look at your right hand. Good luck figuring that out on your own. That was a whirlwind tour of my basic insights. Take a look at the first chart — it should make sense now.

My goals were simple:. Imaginary numbers have been a bee in my bonnet for years — the lack of an intuitive insight frustrated me. Happy math. I try to put myself in the mind of the first person to discover zero.

But both zero and complex numbers make math much easier. If you want more nitty-gritty, check out wikipedia , the Dr. Complex numbers are also utilised in calculations of current, voltage or resistance in AC circuits AC stands for Alternating Current, which is a current that changes magnitude and direction over time.

On the right is an example of such a calculation. To find the combined potential difference, simply adding VA and VB together will not work. However, we can express both voltages as the Real Part x-coordinate on the Argand Diagram of a complex number. Furthermore, complex numbers are also used to express the magnitude and phase of impedance in an AC circuit. Impedance is very similar to resistance - it slows down the electrons in the circuit. The distinction is that impedance causes a phase shift on the electrical current, while resistance does not.

Impedance takes place in common electrical components such as inductors and capacitors, and so having a complex number representation is crucial. In general, complex numbers serve as a representation of phase, which is essential to analysing AC circuits. Quantum Mechanics is a field of Physics that deals with the motions and interactions between subatomic particles - mainly Bosons e. It provides a mathematical description of their behaviour in terms of probabilities.

In fact, complex numbers form the fundamental basis of Quantum Mechanics. The system of complex numbers is essential to the field because it is a convenient language for expressing wave functions without breaking the rules.

Furthermore, a direct application of Quantum Mechanics is that it accelerated the expansion of Chemistry. In , Walter Heitler not Hitler! A wave function, simply put, is a complex probability distribution indicating the possible positions of the particle on a specific time.