Sometimes you may need to square the root(s) of an equation, but the roots of a quadratic equation are not always real. Then, without imaginary roots, your ability to solve a problem stops there...
However, complex number applications are a lot more extensive.
In mathematics there are operators and functions which can reduce a complex operation to a simple one. For example, logarithms can easy the multiplication by reducing it to addition: using logarithmic tables, one may transform large factors to logarithm terms, add them and reverse again obtaining the multiplication result. (Not in use anym
Sometimes you may need to square the root(s) of an equation, but the roots of a quadratic equation are not always real. Then, without imaginary roots, your ability to solve a problem stops there...
However, complex number applications are a lot more extensive.
In mathematics there are operators and functions which can reduce a complex operation to a simple one. For example, logarithms can easy the multiplication by reducing it to addition: using logarithmic tables, one may transform large factors to logarithm terms, add them and reverse again obtaining the multiplication result. (Not in use anymore since hand-held computers...)
With complex numbers the same is possible for transforming differential equations in algebraic ones. Algebraic equation are a lot easier to solve then differential, while differential calculus governs a lot of dynamic processes as electricity, mechanic, acoustic, aero and hydro- dynamics etc.
Even more, simple conventions may reduce the dynamics of variable process (as alternate currents, or even signals of any sort) to static-like behaviors.
For example, in an electric circuit there are resistors, capacitors and inductances. It is easy to solve a circuit with only resistors, but adding capacitors, inductances and AC makes the life a lot more difficult for the electric engineer.
Yet, as an application of complex numbers theories, the capacitor may be represented as a resistance of -j/wC, and the inductance as jwL (where w stands for omega - related to AC, j is sqrt(-1) conveniently noted as such for not confusing with i - intensity of the current.)
With these conventions, now the circuit can be solved as in DC (algebraic, linear equations.) The roots (currents and/or node potentials in the circuit) are functions of w, so giving values to w for the frequency of interest we get frequency-determined roots whose real parts represent the amplitude of AC, while the factor of the exponent in the polar representation of the complex roots represent the phase of the signal.
But wait - there is no need to convert or solve the algebraic equation!
The complex theory of electric circuits tells us (and we should relay on the interpretation of those theories' theorems) that the equation of a circuit (passive or not!) reduces itself to a fraction of polynomials, and the polynomials can be reduced to some products of 1 or two order polynomials. We call the roots of the upper poly - zeros, and the other one (denominator) - poles. Knowing the zeros and poles of the equation, the electric engineer can depict directly the behavior of the circuit on logarithmic graphs with straight lines which would be the asymptotic behavior (amplitude/phase) on frequency in dB (decibels - units for physiologic response of human senses to signals' intensity (as in acoustic sounds.))
And this is not all: by combining in a Cartesian graph the real and imaginary parts of the circuit's function (by eliminating w in the process,) we obtain a curve which tells easily the stability of an active circuit: if this curve goes behind the point (-1, 0) - the circuit is not stable! (do you know that high-pitch sound produced sometimes by microphones?)
This example was about electronics, but in mechanics, mass (inertia) is like inductance, elasticity is like capacitance and friction - like resistance, so same theory applies in evaluating the response of a mechanical system to vibrations (like in to a car or airplane that should not vibrate (too much) because of its engine to any speed.) And also in aerodynamics (an airplane should not vibrate or go unstable because of air flow around it.)
To make the story interesting, almost half century ago, I was student in CS University.
One day, in the afternoon the teacher-assistant came in class with a magazine he wanted to read. He has put a circuit schema on the blackboard asking us for the next 2 hours to figure out what it was.
By the time he went back to his desk, I said - "I know what it is!" I gave the functionality of the circuit (active tone corrector) and picture the graphs.on the board.
He was amazed - how could I have done so fast! - I have had learned in the morning the theory from the head professor of Automation ans System Theory!
"Yes - can be done this way, but I was not aware you have assimilated that theory yet! This is how I wanted you to do! (differential calculus)"
And he has spent the next two hours solving on the board the circuit using the old-fashion method and integrating partial differential equations...
Complex numbers make 2D analytic geometry significantly simpler.
The discovery of analytic geometry dates back to the 17th century, when René Descartes came up with the genial idea of assigning coordinates to points in the plane. Now it seems almost trivial, but this was a huge leap for mathematics: it connected two previously separate areas. Suddenly, you could do geometry by doing calculations with numbers!
Getting a new point of view like this one is huge and it usually leads to lots of new interesting results: because now you can use a new, better language that allows you to think about new
Complex numbers make 2D analytic geometry significantly simpler.
The discovery of analytic geometry dates back to the 17th century, when René Descartes came up with the genial idea of assigning coordinates to points in the plane. Now it seems almost trivial, but this was a huge leap for mathematics: it connected two previously separate areas. Suddenly, you could do geometry by doing calculations with numbers!
Getting a new point of view like this one is huge and it usually leads to lots of new interesting results: because now you can use a new, better language that allows you to think about new concepts in an easier way.
Example: There were many open problems in ancient Greek geometry. Among them, Angle trisection, Squaring the circle, and Doubling the cube. All of these are impossible when using just a compass and a straightedge. These problems were open for centuries because there is basically no way you can prove that they cannot be solved, just by thinking in terms of geometry. We needed algebra. Once we started studying the algebraic properties of geometric constructions, we discovered, for example, that all lengths constructible using a compass and a straigthedge are algebraic numbers such that the degree of their minimal polynomial is a power of 2. This, among other things, rules out the constructibility of [math]\sqrt[3]{2}[/math].
Now, analytic geometry gave us a nice new tool that was easy to work with -- as long as you dealt with points and linear objects only. Checking whether two lines are parallel? Easy. Finding the intersection of two line segments? Easy. Rotating an object around a point? Possible, but painful. Once you start dealing with angles and rotations, the notation starts to be really clumsy. Part of the reason is that you have to work with each coordinate separately, and you don't really see the connections between the coordinates and the angles.
This all changed once we realized that the Complex plane is isomorphic to the standard Cartesian plane. Thus, when doing analytic geometry in 2D, instead of representing a point by a pair of reals, we can represent it by a single complex number. Instant profit!
Why? Because for complex numbers we have the polar form (see Complex number) and we have a very good idea how they relate to angles: namely, when you multiply two complex numbers, you multiply their sizes (absolute values) and add their polar angles (arguments). When doing 2D analytic geometry using complex numbers, operations that involve angles and rotations become as simple as translations and resizing.
Here's one nice example. Go ahead and try solving it without complex numbers, before reading the solution. The question is simple: what is the sum of the three angles shown in the picture?
Here's the answer: The three angles correspond to the complex numbers [math]1+i[/math], [math]2+i[/math], and [math]3+i[/math]. To add those three angles together, we simply multiply those three numbers. We get:
[math](1+i)(2+i)(3+i) = 10i[/math].
Hence, the sum of those three angles is precisely the right angle. Neat, right?
It is really surprising rather amusing to see that a branch of Mathematics that was once neglected for being weird and meaningless is now probably one of the most powerful weapon for a physicist or for that matter for the entire science community in general.
There are already many great answers to this question and to sum it up they all basically hint at one common point that is complex number simp
It is really surprising rather amusing to see that a branch of Mathematics that was once neglected for being weird and meaningless is now probably one of the most powerful weapon for a physicist or for that matter for the entire science community in general.
There are already many great answers to this question and to sum it up they all basically hint at one common point that is complex number simplifies mathematical analysis of many physical problem to a great extent. Had we not had this powerful tool, calculations of many problem would definitely have been a lot more difficult.
In my knowledge, what I have seen or read up till now I think one of it's heaviest uses is in the area of electrical circuits.
Apart from that another very important area where it is used in the analysis of the motion of a particle executing forced vibration.
If you want to know how, I will tell you.
First of all let me tell you that in case of S.H.M, we take into consideration basically two forces. One is the restoring forc proportional to displacement and a damping force proportional to its instantaneous velocity. Both these forces oppose the motion of the particle and thus forms the basis of our mathematical calculation.
Now in case of forced vibration, where does the change in the scenario occur? Nothing else but just an external period force is supplied so as to overcome the damping and to keep the body in motion.
So the standard equation of forced vibration is given by
[math]m\frac{d^{2}x}{dt^{2}}=-sx-R_{m}\frac{dx}{dt}+F_{0}cos\omega t\tag{1}[/math]
where [math]x[/math] is the displacement from the equilibrium position at any instant of time [math]t[/math], [math]sx[/math] is the restoring force and [math]s[/math] is called stiffness or force constant,[math]R _{m}\frac{dx}{dt}[/math] is the damping force.
The equation may be rewritten as
[math]\frac{d^{2}x}{dt^{2}}+2b\frac{dx}{dt}+\omega_{0}^{2}x=f_{0}cos\omega t[/math]
[math]or,\frac{d^{2}x}{dt^{2}}+2b\frac{dx}{dt}+\omega_{0}^{2}x=Re[f_{0}e^{i\omega t}]\tag{2}[/math]
Where [math]2b=\frac{R_{m}}{m}[/math],[math]\omega_{0}^{2}=\frac{s}{m}[/math] and [math]f_{0}=\frac{F_{0}}{m} [/math]and [math]Re[/math] stands for the real part of the complex quantity on its right hand side .
Here we have expressed the actual driving force as the real part of a complex driving force,it is expected that the actual displacement will be given by the real part of a complex displacement will be given by the real part of a complex displacement.
To solve equation [math](2)[/math] we note that for small damping the complementary function( C.F.) i.e. (the solution with R.H.S zero) represents a damped oscillatory motion which does out after some time. The particular integral represents the motion of forced vibration in calculation of which we use concept of complex number.
Since ultimately in the steady state the system vibrates with the frequency of the driving force we may try a solution of the form
[math]x=Re\left[X_{0}e^{i\omega t}\right]\tag{3}[/math]
Here [math]x[/math] is simply [math]X_{0}\cos\omega t[/math]
Differentiating once we get,
[math]\frac{\text{d}x}{\text{d}t}=-X_{0}\omega \sin\omega t\tag{4}[/math]
Now, [math]i\omega e^{i\omega t}=i\omega [\cos\omega t+i\sin\omega t][/math]
[math]\therefore i\omega e^{i\omega t}=i\omega\cos\omega t-\omega\sin\omega t\tag{5}[/math]
So, comparing [math](4)[/math] and [math](5)[/math] we get,
[math]\frac{dx}{dt}=Re\left[X_{0}(i\omega e^{i\omega t})\right]\tag{6}[/math]
Again differentiating [math](4)[/math] we get,
[math]\frac{d^{2}x}{dt^{2}}=-X_{0}\omega^{2}\cos\omega t=Re\left[-X_{0}\omega^{2}e^{i\omega t}\right]\tag{7}[/math]
Substituting theses values of [math]x,\frac{dx}{dt} and \frac{d^{2}}{dt^{2}}[/math] in equation [math](2)[/math] we get,
[math]Re\left[(-\omega^{2}+i2b\omega+\omega_{0}^{2})X_{0}e^{i\omega t}\right]=Re\left[f_{0}e^{i\omega t}\right][/math]
This is true for all [math]t[/math] and hence [math]X_{0}[/math] will be given by
[math]\large\boxed{X_{0}=\frac{f_{0}}{(\omega_{0}^{2}-\omega^{2})+i2b\omega}=Ae^{-i\phi}(let)}[/math]
Let us represent [math]X_{0}[/math] in the form of [math]a+ib[/math]
[math]X_{0}=\frac{f_{0}\lef...
Complex numbers have a myriad of uses. Let’s use them here to understand Euclid’s recipe for Pythagorean Triples.
A Pythagorean Triple is a triple of natural numbers [math](a,b,c)[/math] such that [math]a^2+b^2=c^2[/math]. It represents a right triangle whose legs and hypotenuse lengths are all natural numbers. Typically we count [math](b,a,c)[/math] as the same triple as [math](a,b,c)[/math].
Euclid said to create a Primitive Pythagorean Triple (one with coprime sides) we choose coprime natural numbers [math]m[/math] and [math]n[/math], [math]m-n[/math] positive and odd. Then
[math](m^2 - n^2, 2mn, m^2+n^2)[/math]
is a Primitive Pythagorean Triple, and furthermore, all primitive triples are of this
Complex numbers have a myriad of uses. Let’s use them here to understand Euclid’s recipe for Pythagorean Triples.
A Pythagorean Triple is a triple of natural numbers [math](a,b,c)[/math] such that [math]a^2+b^2=c^2[/math]. It represents a right triangle whose legs and hypotenuse lengths are all natural numbers. Typically we count [math](b,a,c)[/math] as the same triple as [math](a,b,c)[/math].
Euclid said to create a Primitive Pythagorean Triple (one with coprime sides) we choose coprime natural numbers [math]m[/math] and [math]n[/math], [math]m-n[/math] positive and odd. Then
[math](m^2 - n^2, 2mn, m^2+n^2)[/math]
is a Primitive Pythagorean Triple, and furthermore, all primitive triples are of this form.
It’s easy to verify that these make a Pythagorean Triple,
[math](m^2-n^2)^2+(2mn)^2 = m^4 - 2m^2n^2+n^2+4m^2n^2=m^4+2m^2n^2+n^4 \\ \qquad =(m^2+n^2)^2 \quad\checkmark[/math]
But what do [math]m[/math] and [math]n[/math] mean? What do they have to do with our triple triangle?
We can answer this by noting what Euclid is essentially doing is squaring a complex number:
[math](m+ni)^2 = (m^2-n^2) + 2mni, \quad |(m+ni)^2|=m^2+n^2[/math]
That makes it clear. We know complex multiplication adds angles. So squaring doubles angles.
We have a right triangle with natural legs [math]m[/math] and [math]n.[/math] When we square [math]m+ni[/math] we double the angle opposite [math]n.[/math] If we start from a Gaussian Integer (a complex number with integer parts, here integer legs), squaring will still give us integer legs. (I wrote integer instead of natural number here because unless we’re careful we’ll sometimes get negative legs, which we can just negate.) But it also gives us an integer hypotenuse, because [math]|(m+ni)^2|=|m+ni|^2=m^2+n^2[/math].
We end up with an interesting theorem: Twice the small acute angle of any right triangle with natural sides is an acute angle in a Primitive Pythagorean Triple. Another way to say the same thing is, if [math]\tan \theta[/math] is rational, so are [math]\cos 2 \theta[/math] and [math]\sin 2 \theta. [/math] We can also say: every angle in a Pythagorean Triple right triangle is twice the small acute angle of a right triangle with natural sides.
So complex numbers help us understand things, even things that are very old.
Complex numbers, which consist of a real part and an imaginary part (expressed as [math]a + bi[/math], where [math]a[/math] and [math]b[/math] are real numbers and [math]i[/math] is the imaginary unit), have a wide range of applications across various fields. Here are some key applications:
1. Electrical Engineering
- AC Circuit Analysis: Complex numbers are used to analyze alternating current (AC) circuits. Impedance, which combines resistance and reactance, is often represented as a complex number.
- Phasors: Complex numbers simplify calculations involving sinusoidal functions by representing them as phasors.
2. Signal Processing
- Fourier Transform: Co
Complex numbers, which consist of a real part and an imaginary part (expressed as [math]a + bi[/math], where [math]a[/math] and [math]b[/math] are real numbers and [math]i[/math] is the imaginary unit), have a wide range of applications across various fields. Here are some key applications:
1. Electrical Engineering
- AC Circuit Analysis: Complex numbers are used to analyze alternating current (AC) circuits. Impedance, which combines resistance and reactance, is often represented as a complex number.
- Phasors: Complex numbers simplify calculations involving sinusoidal functions by representing them as phasors.
2. Signal Processing
- Fourier Transform: Complex numbers are fundamental in the Fourier transform, which decomposes signals into their frequency components. This is crucial for audio processing, image compression, and telecommunications.
- Filtering: Complex numbers allow for the design of filters in signal processing, enabling the manipulation of different frequency components of signals.
3. Fluid Dynamics
- Potential Flow Theory: In fluid dynamics, complex analysis is used to model fluid flow around objects, particularly in potential flow theory, which simplifies the analysis of two-dimensional flows.
4. Control Systems
- Stability Analysis: The behavior of control systems is often analyzed using complex numbers, particularly through the use of the Laplace transform, which helps determine system stability and response.
5. Quantum Mechanics
- Wave Functions: In quantum mechanics, the state of a quantum system is described by wave functions, which are often complex-valued. The probabilities of different outcomes are derived from the modulus squared of these complex numbers.
6. Computer Graphics
- Fractals: Complex numbers are used to generate fractals like the Mandelbrot set, where the iteration of complex functions creates intricate and beautiful patterns.
- Transformations: Complex numbers facilitate transformations such as rotations and scaling in 2D graphics.
7. Mathematics
- Complex Analysis: This branch of mathematics studies functions of complex variables and has implications in various theories, including contour integration and residue theorem.
- Roots of Polynomials: The Fundamental Theorem of Algebra states that every polynomial has roots in the complex number system, which is essential in many areas of mathematics.
8. Economics and Finance
- Modeling Oscillations: Complex numbers can model cyclical behaviors in economic data, such as seasonal fluctuations in sales or interest rates.
9. Robotics and Control Theory
- Kinematics: Complex numbers can represent rotations and translations in robotic motion, simplifying calculations for movement and orientation.
Summary
Complex numbers are a powerful mathematical tool with applications ranging from engineering and physics to finance and computer science. Their ability to represent oscillatory behavior and simplify calculations makes them invaluable in both theoretical and practical contexts.
Quantum mechanics is a pretty big one.
There's a lot to say about this, but I'll keep it relatively simple. Let's have a look at the equation that governs most all of non-relativistic quantum mechanics, the Schödinger equation (for one particle in a potential V(x) in one dimension):
[math] i \hbar \frac{\partial}{\partial t} \Psi = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi + V(x) \Psi [/math]
Notice that the left hand side has an i and the right hand side has only real terms. This means that, for the equation to be true, [math] \Psi [/math] must be a complex number! Otherwise we'd have a real number equalin
Quantum mechanics is a pretty big one.
There's a lot to say about this, but I'll keep it relatively simple. Let's have a look at the equation that governs most all of non-relativistic quantum mechanics, the Schödinger equation (for one particle in a potential V(x) in one dimension):
[math] i \hbar \frac{\partial}{\partial t} \Psi = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi + V(x) \Psi [/math]
Notice that the left hand side has an i and the right hand side has only real terms. This means that, for the equation to be true, [math] \Psi [/math] must be a complex number! Otherwise we'd have a real number equaling an imaginary number, which is nonsense. Wavefunctions in quantum mechanics have to be complex numbers. The theory just doesn't work if they're not.
Why? Well, in quantum mechanics the probability is given by the length squared of the wavefunction. But you get more information in the complex-valued wavefunction than you do when you get a real-valued probability distribution. You have phases as well as lengths when you have complex numbers. The simplest expression of this is in the interference between two wavefunctions, say in a two-slit experiment. You add the wavefunctions, not the probability distributions. It looks something like this:
Say [math] \Psi_L [/math] is the wavefunction for a particle coming from the left slit and [math] \Psi_R [/math] is the wavefunciton for a particle coming from the right slit. Then the probability distribution when both slits open is given by
[math] P(x) = | \Psi_L + \Psi_R |^2 = | \Psi_L |^2 + | \Psi_R |^2 + 2 Re(\Psi_L \bar{\Psi_R}) [/math]
One of the most common applications of complex numbers are in electrical engineering. They are used whenever Alternating Currents (AC) are being modeled -- everything from the AC power delivered to your home, to RF signals in your cellular phone and wi-fi. This is because complex numbers succinctly incorporate the modeling of sinusoidal waves.
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Whenever a force/ influence divides itself into 2 or more components, complex numbers come into pictures.
Usually this happens due to inclination. An inclined surface or a force divides its own influence into two or more components.
One component parallel to a reference plane (for eg ground) is called as real part.
Other component perpendicular to this reference point is called as imaginary part.
Whenever a force/ influence divides itself into 2 or more components, complex numbers come into pictures.
Usually this happens due to inclination. An inclined surface or a force divides its own influence into two or more components.
One component parallel to a reference plane (for eg ground) is called as real part.
Other component perpendicular to this reference point is called as imaginary part.
What a great question, and in the responses I see a host of practical answers, all good. So perhaps the application of complex numbers to philosophy might be of some interest. Descartes hated negative numbers, though they solved many equations of interest, and called them "false roots". You can imagine his disdain, then, when the "n" roots of some "nth order" polynomials--the Fundamental Theorem of Algebra--occasionally contained some roots that were the square roots of negative numbers. In his anger he gave them the derogatory label "imaginary roots", a label persisting to this day that gives
What a great question, and in the responses I see a host of practical answers, all good. So perhaps the application of complex numbers to philosophy might be of some interest. Descartes hated negative numbers, though they solved many equations of interest, and called them "false roots". You can imagine his disdain, then, when the "n" roots of some "nth order" polynomials--the Fundamental Theorem of Algebra--occasionally contained some roots that were the square roots of negative numbers. In his anger he gave them the derogatory label "imaginary roots", a label persisting to this day that gives these "complex roots" the symbol "i".
Leibniz saw the work of God here, a mathematical middle-place between the concepts of the void--represented by zero--and infinity in his binary scheme, comparing "i" to properties of the Holy Spirit that are ethereal and imaginative, yet barely substantial.
The concept remained largely in the realm of mathematics until the self-taught Oliver Heaviside (1850-1925) used complex numbers to solve differential equations---and thus predict and solve the difficulties being encountered with the recent introduction of electrical transmission lines. Though mathematicians spurned him and he died a recluse who signed his name W.O.R.M., he remains one of my heroes. Check him out on Wikipedia...!
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One real life application is in circuit analysis! And of course circuit analysis has plenty of applications (think about cellphones, tablets, anything electrical)!
They come up when studying circuits with frequency dependent components, namely capacitors and inductors. These frequency dependent components involve derivatives when circuit analysis is performed and hence this results in the need to solve differential equations (can get ugly!).
A method of solving for the steady-state solution of these circuits (AKA the one we desire, with respect to voltages & currents in the circuit) involves tra
One real life application is in circuit analysis! And of course circuit analysis has plenty of applications (think about cellphones, tablets, anything electrical)!
They come up when studying circuits with frequency dependent components, namely capacitors and inductors. These frequency dependent components involve derivatives when circuit analysis is performed and hence this results in the need to solve differential equations (can get ugly!).
A method of solving for the steady-state solution of these circuits (AKA the one we desire, with respect to voltages & currents in the circuit) involves transforming a circuit into what is called the phasor domain. To be brief, this transforms the circuit from being real-valued to complex valued and allows for a succinct method of finding solutions using good ol' Ohm's Law in this complex domain (no more differential equations to fight off!).
Another interesting application of complex numbers is seen through the man Joseph Fourier . He developed the Fourier transform. This mathematical transform on a function of time (let's say a voltage signal!) allows us to gain a measure of what frequency information is in the signal. Is the signal composed of very high frequencies (think high pitch of your voice) or low frequencies (think low pitch of your voice)? By gaining this type of information, systems can be created to extract certain frequency information. A prime application (of the many) of this is your radio and how it tunes in to specific frequencies!
Hope this was interesting!
You can have complex variables instead in any equation and the reason is in the role complex variables can play sometimes. There was no answer and people convinced themselves that we are seeing a new numbering system called the irrationals that can't be formed from the then existing integers and their ratios- the fractions.
This idea of creating a new number system helped when people asked what is the square root of a negative number. Many thought of kind of a solution but Descartes finally solved it by putting an i for the answer and called it an imaginary number. A new number was born despite
You can have complex variables instead in any equation and the reason is in the role complex variables can play sometimes. There was no answer and people convinced themselves that we are seeing a new numbering system called the irrationals that can't be formed from the then existing integers and their ratios- the fractions.
This idea of creating a new number system helped when people asked what is the square root of a negative number. Many thought of kind of a solution but Descartes finally solved it by putting an i for the answer and called it an imaginary number. A new number was born despite Descartes himself stressing that this i is truly imaginary- see Wikepedia/Complex number.
Then the complex number plane was discovered and this unveiled the vector like properties of complex numbers, which connected math and geometry in an unexpected way. This prompted people to look for complex numbers in 3D too, and Hamilton found the quaternions. The reason why people loved the new complex number representation of vectors is that these can be treated like numbers in calculations— not needing many components like vectors with special and more difficult formulae to handle. This then found the immediate application in harmonic analysis wherein functions in 0-infinite time can be broken into the sum of components of variable frequencies/circles instead of variable time. Like fxx+fyy=0, fxx+ftt=0, representing Laplace equation in 2D with t behaving like a finite spatial dimension (not infinite like time). Complex numbers are now looked at as an umbrella of all the previously discovered numbers. The vectors, tensors etc types are only a collection of numbers and don't add a new number system.
For my dissertation, I used the complex plane to integrate around a singularity which existed on the real number line. Since the function went to zero at infinity, the integral along the line in the complex plane was equal to the one on the real line, by Cauchy's integral theorem. Otherwise the singularity would have prevented the calculation. Here is the illustration from the dissertation:
For my dissertation, I used the complex plane to integrate around a singularity which existed on the real number line. Since the function went to zero at infinity, the integral along the line in the complex plane was equal to the one on the real line, by Cauchy's integral theorem. Otherwise the singularity would have prevented the calculation. Here is the illustration from the dissertation:
For those of who have taken a course on complex analysis I think Joachim actually hits it right on the head when he brilliantly summarizes a large body of mathematics: Simplify computations involving real numbers.
But there probably needs to be some explanation for everyone else who does not have the privilege of having a grandfather [Einar] write a book on this topic, take a course or two or actually apply this technique to an engineering problem... It has been a while since I have used this so probably Joachim can elaborate more and correct any of my miss statements.
Particularly in the analy
For those of who have taken a course on complex analysis I think Joachim actually hits it right on the head when he brilliantly summarizes a large body of mathematics: Simplify computations involving real numbers.
But there probably needs to be some explanation for everyone else who does not have the privilege of having a grandfather [Einar] write a book on this topic, take a course or two or actually apply this technique to an engineering problem... It has been a while since I have used this so probably Joachim can elaborate more and correct any of my miss statements.
Particularly in the analysis of turbulence or other tough problems in engineering there arises the need to solve some mathematical and calculus problems that are particularly hard to solve with the conventional tool set of real numbers. One typical such problem is integrating through a point where the function being integrated goes to infinity, such as integrating f(x) = 1/(x^2 ) across x = 0, say from x =-1 to x = 1 [Joachim may be able to provide a better example and one where he knows the solution]. The trick here is that this problem can be solved not by integrating through x=0 but by integrating from x = -1 around but not through x = 0 to the point x =1. What is really nice is that by integrating around the singularity (point of infinity) you actually capture the correct solution to the problem and the answer is a real number for a solution to your real world problem and is not a complex number. This hits back to Joachim's answer: to simplify computations involving real numbers.
The skill is in picking a path around x = 0 where the function f(x) is not infinity and where f(x) is some convenient well defined number. Here the trick to integrating around x=0 is to chose some path through the complex plane where defining x to be x = y + z i where y and z are real number, i = square root of -1, and thus z i is imaginary--complex component of x. Often this is done by integrating in a circle with a constant radius r around the point x = 0, and then the integration is usually done by representing the complex number in polar coordinates as a radius and angle.
Because this is such a useful technique for engineers to solve some of their problem, there is a whole field of mathematics that works on this type of problem. Much of this work in mathematics is to directly solve the engineers problems but sometimes the mathematicians go off on their own path to ask and solve related problems that may or may not one day help solve engineering problems or problems in other fields.
I am sure there are other strong applications of complex analysis and an engineer or mathematician may be better qualified to elaborate on these applications. There are also some really neat graphical pasterns and pictures that are drawn when representing the outcome of complex analysis graphically such as the Mandelbrot set. And I am sure that there are some set of mathematical or applied math problems that this and similar sets help solve or illustrate; but most of you will just be familiar with beautiful ornately complex pleasing pasterns that the graphical representation of the Mandelbrot set produce.
You can use complex numbers whenever you like. There is no rule or law to prevent it, and you don’t need a license.
I use complex numbers at least several times a week, but this is unusual. I have applied math modeling problems where the problem and the solution are expressed in terms of real numbers, but using complex numbers makes the intermediate calculations simpler.
Some beautiful results from Complex numbers and complex Analysis:
1. Evaluate Fourier Transforms of a real-valued function f(x), with the power Residue Theorem in an instant.
2. The integrals from -infinity to infinity you do in Calculus 2 can be done without tedious techniques like Change of Variables, Substitution, By Parts etc.
3. Complex Analysis has marvelous applications from extremely pure math topics (Functional Analysis, Cstar Algebras, Analytic Number Theory) to very applied topics (Statistics, Fluid Dynamics, Electrical Engineering)
Just in case you really did mean applications in mathematics, the basic use is as an algebraically closed field, in which all sorts of things with an embedded polynomial structure can be resolved into products of linear factors. A great deal of applicable mathematics depends on this, whether indirectly through transform methods, or with raw obviousness like linear differential equations. A vast amount of stuff which is really about real variables can be made easier by embedding it in the complex numbers.
Straddling both pure and applied mathematics, one extraordinary feature of the complex numb
Just in case you really did mean applications in mathematics, the basic use is as an algebraically closed field, in which all sorts of things with an embedded polynomial structure can be resolved into products of linear factors. A great deal of applicable mathematics depends on this, whether indirectly through transform methods, or with raw obviousness like linear differential equations. A vast amount of stuff which is really about real variables can be made easier by embedding it in the complex numbers.
Straddling both pure and applied mathematics, one extraordinary feature of the complex numbers is the way in which calculus works. When integration is possible, so is differentiation; and it is possible to extend functions differentiable over the reals to be differentiable essentially everywhere. This provides ways of transforming problems into path integrals which can be very unexpected.
In your daily, role of complex number depends on your profession
- Students :- to do homework
- Teachers :- to give homework
- Scientist/research :- calculation for their research work
- Everyone else :- absolutely 0 utilization
Complex numbers are specially important in electric and electronic engineering. They are used to model electric currents, voltages and impedance.
In most of our electricity use, like the sockets in your home, the current is alternating (called AC - alternating current), which means it alternates in magnitude and direction. To represent this 2-dimension quantity, we use complex numbers. A current, and also a voltage and a impedance, is represented as a function of time as:
I(t) = A*exp(iwt+o), where:
A is the amplitude
w=2*pi*f is the angular velocity (f being the frequency)
o is the init
Complex numbers are specially important in electric and electronic engineering. They are used to model electric currents, voltages and impedance.
In most of our electricity use, like the sockets in your home, the current is alternating (called AC - alternating current), which means it alternates in magnitude and direction. To represent this 2-dimension quantity, we use complex numbers. A current, and also a voltage and a impedance, is represented as a function of time as:
I(t) = A*exp(iwt+o), where:
A is the amplitude
w=2*pi*f is the angular velocity (f being the frequency)
o is the initial phase
Electric and electronic engineering are entirely based in complex numbers.
Pretend that the only type of numbers you know about are the Whole Numbers: W = {0, 1, 2, 3 …….}
If all problems you did had solutions in W then you would have no need to extend your understanding.
Eg If x + 3 = 17
then x = 14
——————-
If 5x – 7 = 13
then 5x = 20
and x = 4
But suppose you meet x + 5 = 2. There is no answer to this in W but we do not say that “There is no solution”. We have to extend our id
Pretend that the only type of numbers you know about are the Whole Numbers: W = {0, 1, 2, 3 …….}
If all problems you did had solutions in W then you would have no need to extend your understanding.
Eg If x + 3 = 17
then x = 14
——————-
If 5x – 7 = 13
then 5x = 20
and x = 4
But suppose you meet x + 5 = 2. There is no answer to this in W but we do not say that “There is no solution”. We have to extend our idea of “what numbers are” to a bigger set called the Integers Z
Z = {….–3, –2, –1, 0, 1, 2, 3, ….}
Now we can solve problems such as: x + 5 = 2 so x = –3
Now consider 5x = 6. There is no answer in Z.
We cannot find an integer to be the solution, so we again extend our idea of “what numbers are” to the Rational Numbers, Q
Contrary to popular belief, there is no fraction (ie no decimal) which when squared equals 2.
The solution is x = +√2 or –√2.
Most people get out their calculator and say √2 = 1.414 but this is simply not true because 1.414 is rational! These people are just trying to make the value of √2 into a rational answer.
√2 cannot be represented by any number of decimal places. We can only give an approximation to any number of decimal places we like..
This is exactly like saying that the solution of 5x = 16 (when we only knew about whole numbers) is x = 3 because it is the nearest whole number!
√2 is a new type of number called Irrational. In fact they seem absurd, which is probably where the word “surd” comes from!
Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction. Instead he proved you could not write the square root of 2 as a fraction, which seemed to be quite “irrational”.
Pythagoras could not accept the existence of irrational numbers. He believed that all numbers were rational but he could not disprove the existence of these "irrational numbers" and so Hippasus was thrown overboard and drowned! (so the story goes!)
The following equation can be solved by a method called “completing the square”.
(Calculators can only deal with rational numbers. The exact answers are not 4.414, 1.586)
These new Irrational numbers are of the form a + b√c and they simply do not exist as decimal fractions.
Incidentally, after some operations, irrational numbers can become rational.
Eg 3 + 2√5 + 7 – 2√5 = 10
and (3 + √7)(3 – √7)
= 9 + 3√7 – 3√7 – 7
= 2
The Rationals and the Irrationals together are called the REAL NUMBERS and they can be represented by either points on a number line or vectors on a number line.
Now we are prepared to ACCEPT i = √(–1)
Things like √(–1), √(–9), √(–7) are not in the set of Rationals neither are they in the set of Irrationals.
It is no use looking for a decimal approximation to √(–1) , just as it is no use looking for an integer equal to 7/ 3
The number √(–1) is not found anywhere in the set of REAL Numbers so we could say it...
An eigenvalue of a matrix is a complex number in general. It is a good thing the complex numbers are algebraically closed, or else you'd have some huge extension field to worry about.
In electrical engineering, solutions to the differential equations governing capacitance and inductance, in many cases must be expressed in complex valued functions. This is because of Euler's Rule, and the complex definitions of sine and cosine. If you dig deep enough, it's actually as a result of our approximation of electromagnetic oscillations as simple harmonic oscillators. Any physical application which uses this approximation - such as the movement of astronomical bodies, satellites and planets - will require the use of the complex plane if the mathematics is expressed in Cartesian c
In electrical engineering, solutions to the differential equations governing capacitance and inductance, in many cases must be expressed in complex valued functions. This is because of Euler's Rule, and the complex definitions of sine and cosine. If you dig deep enough, it's actually as a result of our approximation of electromagnetic oscillations as simple harmonic oscillators. Any physical application which uses this approximation - such as the movement of astronomical bodies, satellites and planets - will require the use of the complex plane if the mathematics is expressed in Cartesian coordinates. In many cases, complex terms cancel out or simplify to real valued functions if these problems are expressed in cylindrical or spherical coordinate systems. These coordinate systems build trigonometric identities into the orthogonality of their axes. This means that the coordinates are specifically designed to represent sinusoidal mathematics very simply. Any physical system that will have terms in sin and cosine, is an application where complex numbers will appear in the solutions in non-trivial (realistic, not out of a textbook) applications. Hope this is helpful.
One usage (I would say need) for complex numbers that I found in a book from R. Penrose is to solve very simple higher order equations that at first sight could seem anything but requiring complex numbers, while keeping the rule: degree of the equation = number of solutions.
Let's take this very simple equation: [math]x^3=1[/math]
The solution seems to be x = 1. But it is a cubic equation. Where are the other two solutions?
Introducing complex numbers the other two solutions are
[math]-\frac{1}{2}+i\frac{\sqrt{3}}{2}[/math]
[math]\frac{1}{2}-i\frac{\sqrt{3}}{2}[/math]
So: cubic equation -> three solutions
3D animation.
Before I get into why complex numbers are important for 3D animation, I will first briefly describe how 3D animation works.
The way that 3D animation works is, briefly speaking, poses are specified at different points in times with different positions and rotations. The position and rotation of each joint is interpolated at each point in time and from this vertex positions are calculated and displayed on a screen.
It is when interpolating rotations that complex numbers come in. Interpolating Euler rotations doesn’t work very well. To remedy this, quaternions are used to represent th
3D animation.
Before I get into why complex numbers are important for 3D animation, I will first briefly describe how 3D animation works.
The way that 3D animation works is, briefly speaking, poses are specified at different points in times with different positions and rotations. The position and rotation of each joint is interpolated at each point in time and from this vertex positions are calculated and displayed on a screen.
It is when interpolating rotations that complex numbers come in. Interpolating Euler rotations doesn’t work very well. To remedy this, quaternions are used to represent the joint transformation.
A quaternion is pretty much a 4-dimensional complex number.
Complex numbers are also used in quaternions
Quaternions and spatial rotation
These are used heavily in computer games and simulations.
As an Electrical Engineer I can assure you that my profession would be almost impossible to practice without their aid. A very simple example: take a capacitor, an inductor and a resistor. Connect them in series and apply a AC voltage to the whole contraption. Now try to compute the intensity of the current flowing thru this circuit or the voltage difference in every one of the components. Let me assure you that using complex numbers it is a piece of cake making those computations. Without hem you will need to be a full fledge mathematicians to accomplish the task.
Complex numbers come up all the time in engineering, when dealing with a sinusoidal, sine wave, you can think of them as resulting from the sum of the sine and cosine functions, by changing the magnatude of the sine and cosine you can change the phase, offset, of the resulting sinusoidal, these sine and cosine functions can be understood as being orthogonal to each other, so we can plot their magnitudes at right angles to each other on the complex plane, if you work out the magnatude of this complex number you will get the magnatude of your sinusoidal, and if you work out the angle your comple
Complex numbers come up all the time in engineering, when dealing with a sinusoidal, sine wave, you can think of them as resulting from the sum of the sine and cosine functions, by changing the magnatude of the sine and cosine you can change the phase, offset, of the resulting sinusoidal, these sine and cosine functions can be understood as being orthogonal to each other, so we can plot their magnitudes at right angles to each other on the complex plane, if you work out the magnatude of this complex number you will get the magnatude of your sinusoidal, and if you work out the angle your complex number makes, you will get the phase of your sinusoidal. We can take this further and represent any signal as being the sum of a whole lot of sinusoidals of different frequencies, each with a complex number to represent their amplitude and phase. We could probable do a lot of the same stuff using 2D vectors, but complex numbers are neater way to represent this as they have all the properties of numbers except ordering, you can not do that with vectors of arbitrary number of dimensions.
Dear Bro David Chanda,
I have this same question, (both with Complex Numbers as well as Imaginary Numbers).
I think, there is an Application for them in Quantum Mechanics, where Matrices having Imaginary and / or Complex number values are used, but I am Not an expert on Quantum.
Best thing maybe, if we start a discussion about Complex Numbers, so that questions maybe answered to an extensive level.
For high school level, it’s enough to imagine and accept a number i, such that i^2 = -1, it is called an imaginary number. Then, set C = {a+bi, where a and b are real numbers} is the set of a type of new numbers, called complex numbers. Since b can be equal to 0, set C includes set R of the real numbers. So a complex number z can be written in the form z = a + bi, a is called the real part of number z and b is called the imaginary part of z.
Then, it is said that two numbers that are complex, z = a + bi and w = c + di are equal if and only if a = c and b = d, so they must be identical.
Two opera
For high school level, it’s enough to imagine and accept a number i, such that i^2 = -1, it is called an imaginary number. Then, set C = {a+bi, where a and b are real numbers} is the set of a type of new numbers, called complex numbers. Since b can be equal to 0, set C includes set R of the real numbers. So a complex number z can be written in the form z = a + bi, a is called the real part of number z and b is called the imaginary part of z.
Then, it is said that two numbers that are complex, z = a + bi and w = c + di are equal if and only if a = c and b = d, so they must be identical.
Two operations are defined, the sum : z + w =(a + c) + (b + d)i and the multiplication, (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
Due to the definition of the operations and stating that i^2 = -1, complex numbers behave like the algebraic expressions when operated.
Example: if z = 3 — 2i and w = 4 + i, then z + w = (3 + 4) + (-2 + 1)i = 7 -i and
zw = (3 - 2i)(4 + i) = 12 + 3i - 8i -2i^2 = 12 -5i -2(-1) = 12 - 5i +2 = 14 - 5i.
And there is more to learn, like the fact that complex numbers have a module and also a geometrical representation, or that any quadratic expression can be factorized in set C, or that now there are three famous numbers, e, π and i.
Many and varied.
The easiest is perhaps it's use in electronics.
School physics tells us that V=IR for resistances but this is easily generalised to V=IZ where Z is impedance and V voltage and I current as before.
By working with complex Z inductance and capacitance just become complex resistances and real components are more realistically modeled using this.
It is possible to represent square root of any number but,is it possible to represent square root of -1?,the square root of -1 is taken as i or j.and,to rotate a image 90 degree in counter clockwise direction,just multiply it with the imaginary number.
Eg:x axis × i = y axis,
Again y axis × i = -x axis and so on.
Just another one to the growing list:
In fluid dynamics you will sometimes find it easier to solve the problem in complex space - your actual answer is just the real part of your solution (imaginary velocities have no bearing on what you can actually observe).
When you are working with equations and suddenly appears a square root of a negative number, you do not have to worry about it neither to find a way to avoid it. You just continue working with the complex number. If you are lucky perhaps the added i disappears in the process.
Here is a short list of Applications
- Discrete Fourier Transform which is used in signal analysis
- Algebra of Quaternions used in 3D Animation
- Conformal Mapping which is used to extend solutions of differential equations
- Modelling electrical circuits
- Quantum Mechanics and Quantum Computation
- Control Engineering
Complex Numbers are used in the field of Physics to great extent.In the circuit of resistance,inductance and capacitance ,the addition and multiplication of the complex numbers are used.Complex numbers are also used in electrical engineering with the help of Fourier transforms which help in understanding oscillations occurring both in alternating current and in signals modulated by electromagnetic waves.
Complex numbers are one of the most conceptual part of mathematics and mathematically they are the numbers which can be expressed in the format of a+ib, here a and b are real numbers and ‘i’ is regarded as an imaginary number and typically called as ‘iota’. The ‘i’ has the value of (√-1). For example, 3+7i is a complex number, where 3 being a real number (Re) and 7i is imaginary number (Im).
In mathematics both i or j is used in representing the imaginary number whose value is equal to √-1. Thus, on squaring the imaginary number we get a negative -1 value. Complex number is mainly used in repre
Complex numbers are one of the most conceptual part of mathematics and mathematically they are the numbers which can be expressed in the format of a+ib, here a and b are real numbers and ‘i’ is regarded as an imaginary number and typically called as ‘iota’. The ‘i’ has the value of (√-1). For example, 3+7i is a complex number, where 3 being a real number (Re) and 7i is imaginary number (Im).
In mathematics both i or j is used in representing the imaginary number whose value is equal to √-1. Thus, on squaring the imaginary number we get a negative -1 value. Complex number is mainly used in representation of motions which are periodic in nature for example waves, current in alternating form that is sine wave, light waves, etc. These waves are relied on sine function or cosine function.
Since real numbers are those numbers which exist in number system such that they are positive, negative, zero, integer, rational, irrational, fractions, etc. The representation of real numbers is with Re (). Some of the examples of real numbers are: 12, -45, 0, 1/7, 2.8, √5, etc. Thus, the imaginary numbers are just those numbers which are not real numbers. We have alreadyseen that on squaring the imaginary number we get a negative number. Some of the examples of imaginary numbers are: √-2, √-7, √-11 are all imaginary numbers.
The introduction of complex numbers was introduced when we needed to solve the equation x^2+1 = 0. On finding out the roots of the above equation we got the form of x = ±√-1 with no real roots existing. Therefore, the complex number was introduced where we had Imaginary roots. The symbol ‘i’ which is called as iota is being used to denote √-1 as an imaginary number.
The notation of complex number is of the equation form z= a+ib, where a and b are real numbers. Re z = a is the real part of the complex number and Im z = ib is the imaginary part of the complex number
The term absolute value is frequently used in complex numbers which is nothing but the positive value of the number or magnitude. For example, the real number has its absolute value of the number itself. The absolute value is represented by modulus, i.e. |x|. Therefore, the modulus of any value always provides us with a positive value, such that;
|5| = 5
|-5| = 5
But in complex number a different method is used in determining the modulus of complex number.
Assume, z = x+iy is a complex number. Then modulus of the complex number z, will be given by:
|z| = √(x^2+y^2)
Above expression was obtained on applying the Pythagorean theorem in a complex plane. Thus, it can be said that the mod of the complex number, z lies from 0 to z and modulus of the real numbers x and y is from 0 to x and 0 to y. Since these values take a formation of a right triangle, here 0 is the vertex of the angle less than 90 degrees that is acute angle. On applying Pythagoras theorem,
|z|^2 = |x|^2+|y|^2
|z|^2 = x^2 + y^2
|z| = √(x^2+y^2)
We can perform various algebraic operations on complex numbers like addition, subtraction, multiplication and division.
For solving a quadratic equation in the form of ax^2 +bx + c = 0, The determination of the roots of the equations can be done in three forms as either two different Real Roots, equal roots or no real roots which are basically complex roots.
It performs the arithmetic operations on complex numbers which could be addition and subtraction. It simply states that combination of real with real number and imaginary with imaginary number.
Addition operation
(p + iq) + (x + iy) = (p + x) + i(q + y)
Subtraction operation
(p + iq) – (x + iy) = (p – x) + i(q – y)
On multiplication operation on complex number with each other there is a method called distributive multiplication process.
(x + iy). (p + iq) = (xp – yq) + i(xq + yp)
Division operation on complex numbers
(x + iy) / (p + iq) = (xp+yq)/ (p2 + q2) + i(yp – xq) / (p^2 + q^2)
Some of the identities of complex numbers are as follows:
a) (z1 + z2)^2 = (z1)^2 + (z2)^2 + 2 z1 × z2
b) (z1 – z2)^2 = (z1)^2 + (z2)^2 – 2 z1 × z2
c) (z1)^2 – (z2)^2 = (z1 + z2) (z1 – z2)
d) (z1 + z2)^3 = (z1)^3 + 3(z1)^2 z2 +3(z2)^2 z1 + (z2)^3
e) (z1 – z2)^3 = (z1)^3 – 3(z1)^2 z2 +3(z2)^2 z1 – (z2)^3
Some of the important properties associated with the complex numbers are as below:
a) On adding two conjugate complex numbers will result in a real number
b) On multiplying two conjugate complex number will also result in a real number
c) If x and y are the real numbers and x+yi =0, then x =0 and y =0
d) If p, q, r, and s are the real numbers and p+qi = r+si, then p = r, and q=s
e) The complex number obeys the commutative law of addition and multiplication.
f) z1+z2 = z2+z1
g) z1. z2 = z2. z1
h) The complex number obeys the associative law of addition and multiplication.
i) (z1+z2) +z3 = z1 + (z2+z3)
j) (z1.z2).z3 = z1.(z2.z3)
k) The complex number obeys the distributive law
l) z1.(z2+z3) = z1.z2 + z1.z3
Check one example:
Question: Multiplicative inverse of 1 + i is ………..
Answer: https://www.doubtnut.com/question-answer/multiplicative-inverse-of-1-i-is--402475339
For more information on complex numbers visit Doubtnut It would be possible to clear all the queries in the best way that would help in proving to be of much use to you.
For me being an Electrical Engineer you find that complex numbers are applied to several problems from Power Calculations to Circuit Analysis and even in Control Systems. I think when used as a layer of abstraction above the real problem it can sometimes simplify your task.
There are a lot of answers on Quora to this question. Here is one I gave years ago: T Boyd Moore's answer to What are imaginary numbers used for in the real world? And without them, the quantum theory would be impossible to apply to the subatomic world.
There’s a great book, One, Two, Three... Infinity by George Gamow, with a section that shows what complex numbers can be used for. In it, treasure hunters find the island shown on their map, but the tree shown there is gone!
Distances from the tree show where “X marks the spot” of buried treasure. And the author shows how you can discover where the tree was, and hence where the treasure is, using complex numbers.
One Two Three... Infinity - Wikipedia
[PS—Among the other treasures in this book is an introduction to Georg Cantor's theory of three levels of infinity A great book “known for a quirky
There’s a great book, One, Two, Three... Infinity by George Gamow, with a section that shows what complex numbers can be used for. In it, treasure hunters find the island shown on their map, but the tree shown there is gone!
Distances from the tree show where “X marks the spot” of buried treasure. And the author shows how you can discover where the tree was, and hence where the treasure is, using complex numbers.
One Two Three... Infinity - Wikipedia
[PS—Among the other treasures in this book is an introduction to Georg Cantor's theory of three levels of infinity A great book “known for a quirky sense of humor and for memorable metaphors”. Yet it’s not too advanced for a high school student to understand.]
Complex numbers are a mathematical representation of two dimensional vectors that are used to analyze electrical circuit and the frequency response of systems.
so the basics of complex numbers comes from trying to solve equations like this : [math]x^2 + 1 = 0[/math]
It looks simple but that then means we get to [math]x^2= -1[/math]
So what number can you square to get [math]-1[/math]
clearly x can’t be 1: [math]1^2 = 1[/math]
x can’t be -1 : [math](-1)^2 = 1 [/math]
Mathematicians proved that there is no solution in any ‘normal’ number system - i.e. there is no negative or positive real number where [math]x^2 + 1 = 0[/math]
Mathematicians don’t like things with no solution - so they imagined a number which they called [math]i[/math] where
[math]i^2 = -1[/math]
Since they imagined [math]i[/math], they called [math]i[/math] an imaginary number.
With imaginary numbers Mathematicians also d
so the basics of complex numbers comes from trying to solve equations like this : [math]x^2 + 1 = 0[/math]
It looks simple but that then means we get to [math]x^2= -1[/math]
So what number can you square to get [math]-1[/math]
clearly x can’t be 1: [math]1^2 = 1[/math]
x can’t be -1 : [math](-1)^2 = 1 [/math]
Mathematicians proved that there is no solution in any ‘normal’ number system - i.e. there is no negative or positive real number where [math]x^2 + 1 = 0[/math]
Mathematicians don’t like things with no solution - so they imagined a number which they called [math]i[/math] where
[math]i^2 = -1[/math]
Since they imagined [math]i[/math], they called [math]i[/math] an imaginary number.
With imaginary numbers Mathematicians also decided to combine them with non-imaginary numbers - for instance
[math]3 + 2i[/math]
This is a complex number.
It turns out that complex numbers are pretty useful: for instance they allow mathematicians to solve equations such as : [math]x^2 -6x +13 = 0[/math]
Using complex numbers - the solution to this is that is either
[math]3 + 2i[/math] or [math]3 -2i[/math]
It turns out that complex numbers are very useful in a number of fields :
- They can be used to describe 2D co-ordinates and rotations in such a away that the relatively simple maths rules for complex numbers makes the result of those rotations pop-out of the maths
- They can be used to describe electrical currents through a circuit, and simultaneously represent both DC and AC currents and be used to simultaneously represent the static and inductive/capacitive loads in the circuit.
- They can be used with calculus to represent time dependent variations in active circuits (such as amplifiers).
Sorry for the electronics/electrics heavy answer but that was my major before i started writing software. There are many other uses for complex values as well.
Simplify computations involving real numbers.
Also used when designing and analyzing control systems.