Can imaginary numbers have real world applications?

[Stratosphere]

Why do we have an imaginary number? I don't see it's usefulness. Why dos it matter if we can make up a number that satisfies this equation

($$\ x^{2}+1=0$$ )? It must have real world applications that I'm unaware of.

 

[karkas]

Well, I guess a mathematician would be of much more use in such a subject, but I'll try and give an answer.

The imaginary unit, leading later to complex numbers and the corresponding set, were first introduced, as mine and every other school book says, to solve several third-grade equations, where the determinant is negative, that would previously mean that there are no solutions in the real number system, but on the contrary there were real obvious solutions of the equation. Therefore mathematicians had to accept that several real numbers could be written as expressions that included roots of negative numbers. I say these things in risk of you knowing them already, but I want to point out that there are a lot of equations that probably are encountered in nature and require "higher" sets than the real set to be explained and solved. In other words, it's useful as a notation and has helped to simplify several procedures. That's what I think.

 

[DyslexicHobo]

I can't give you an answer as to why they were first defined, but they simplify many analysis of real-world applications. I'm only a third-year in college studying engineering, but I've already run into several topics where using the imaginary plane is very useful. In the study of vibrations, Euler's identity is used to simplify an equation that might be 10 times its size without the use of imaginary numbers.

Imaginary numbers are also used widely in the study of electronics. It's not my area of expertise, but from what I've learned it's used for calculating the impedance of capacitors and inductors.

 

[LCKurtz]

Why do we have an imaginary number? I don't see it's usefulness. Why dos it matter if we can make up a number that satisfies this equation

($$\ x^{2}+1=0$$ )? It must have real world applications that I'm unaware of.

Although it doesn't talk about applications, you might find this interesting:

http://math.asu.edu/~kurtz/complex.html

 

[Bacle]

i allows you to extend the ring R[x] into the complexes.

R[x]/[(x²+1)] ~ C

where R[x] is the ring of polys. over the reals (as a ring of polys. in one variable) ,
and C is the complexes, and (x²+1) is the maximal ideal generated by the irreducible (over R[x]) poly. x²+1 .
Maybe someone who has their abstract algebra fresher/better than mine can give you
a proof or good argument for why an ideal generated by an irreducible poly. in F[x]
is maximal in F[x].

 

[Mute]

There are always applications. For the imaginary unit, the biggest one is probably quantum mechanics. Unlike classical mechanics, which can be written entirely in real numbers, quantum mechanics necessarily requires the use of complex numbers. If we didn't have $i$, we wouldn't have a theory of quantum mechanics.

 

[Stratosphere]

What's the difference between $$i$$ and complex numbers?

 

[jasonRF]

The complex numbers are used all the time in applications - certainly in many areas of engineering. The primary reason the imaginary number is useful is because of Euler's identity
$$e^{i x} = cos(x) + i \, sin(x)$$
So systems that deal with sinusoidal-like signals use complex numbers just to make analysis easier. Examples include such every-day things as dealing with the 60 Hz (US) power distribution and communications systems. Analog circuits (eg the tuner or amplifier in your stereo) are also designed and analyzed using complex numbers. I promise you that the power company, the cell-phone companies, etc., use complex numbers in their design and analysis work. There are many, many other examples as well. These could be done without complex numbers in principle, but it is sooo much easier with them that no-one in their right minds would do it without them.

If you know calculus, I recommend the book "an imaginary tale" by Paul Nahin, as it has some great history as well as showing how functions of a complex variable can be used to do things like evaluate integrals in a way that is sometimes much easier than if you stuck to real variables.

jason

 

[Integral]

What's the difference between $$i$$ and complex numbers?

i is defined to be $$\sqrt {-1}$$

A complex number is a number which takes the form a + bi. Where a and b are both real; a is called the real part, b is the imaginary part.

 

[Redbelly98]

A similar discussion took place last fall.

... we often encounter differential equations that describe oscillatory phenomena. Familiar examples of this are the Schrodinger Equation, and the voltage & current in capacitors and inductors.

What makes complex numbers convenient is the fact that the derivative of exp(iωt) is proportional to exp(iωt), which greatly simplifies the solving of linear differential equations. Since sin(ωt) and cos(ωt) do not have this property, it is advantageous to use exp(iωt) instead.

In the end, we obtain real-valued answers by either taking the real part of the answer (in the case of voltages and currents) or multiplying by a complex conjugate (in the case of quantum mechanics). The use of complex numbers is just an intermediate mathematical tool towards finding real-valued answers.

 

[turbo]

It might be helpful to audit some of the popular talks given by Penrose. He elucidates quite clearly the power of imaginary and complex numbers, and gives the viewer an insight into the power of numbers with multiple (tiered) exponents.