The idea behind the Fourier Transform is rather genius; it proposes that any complicated wave can be decomposed into multiple sinusoidal waves with varying frequencies. What Fourier Transform does is that it predicts which frequency is likely to be equivalent to one of such sinusoidal waves. 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. Geometry taught me to justify my reasons for everything; trig taught me to convert the problem from one form to another; calculus taught me to take things to the extreme and see what shakes out; and complex numbers taught me there's always another dimension to things to take into consideration.
I never use complex numbers - ever - but my brain circuits are better at solving problems because I did them. Show 18 more comments. Active Oldest Votes. Improve this answer. I don't know how many times I have explained to people how to calculate the value of a percentage of a value, and I'm not a math teacher by a long shot. Add a comment. Complex numbers are the answer to what's the algebraic completion of polynomial roots? In the Preface to his Visual Complex Analysis , Tristan Needham writes: If one believes in the ultimate unity of mathematics and physics, as I do, then a very strong case for the necessity of complex numbers can be built on their apparently fundamental role in the quantum mechanical laws governing matter.
Daniel R. Collins Daniel R. Collins 20k 58 58 silver badges 99 99 bronze badges. I can take your entire wikipedia link to most people in my area , and I would think that a hardly any of them would recall what a complex number is, b wouldn't understand any of those example applications besides what the job is and knowing a bit about them or c use them because of their job, upon which I would not classify them as laymen.
That is a secondary issue. Mathematical truth is greater than workaday applications, and people have a mangled understanding of the discipline if that is not shared with them.
Mast 4 4 bronze badges. I could say sqrt -1 is any number I bloody well please, and be valid as you say: creative with numbers. However, the "cannon" complex numbers have a huge body of very consistent behaviors that make it valuable. This sort of "you can say anything you like, as long as it is consistent" attitude proved very helpful in later life as an Engineer understanding many really "special" formulations which exist only because they are useful, and consistent.
John Coleman John Coleman 1, 7 7 silver badges 12 12 bronze badges. The protagonist is a somewhat autistic female math graduate student living in Denmark but originally from Greenland, which is a Danish colony who gets involved with a murder mystery.
Math isn't the focus of the story, but there are occasional references to it, with the quoted passage being the longest. I further mention this application as the "easy way" to derive these identities any time I need to use one in a class. In linear algebra and differential equations, we discuss the interpretation of complex eigenvalues. J W 4, 2 2 gold badges 17 17 silver badges 41 41 bronze badges. I am not familiar with the term. For math departments this would be calc 1, 2, 3, ODE and typical probability and statistics.
The vast majority of students will be non math majors. The point is that having a lens of "needs of the math major" is obtuse since they are a tiny minority of the audience Lots of other cool things too. If I had it to do over again would do geology or geophysics. Sciencey but messy enough so that nobody really understands it.
And a connection to industry. And to outdoors. But whatever you like. But if you love pure math, that is fine. Someone has to Some examples of things I haven't had to do since high school: Name the 5 I think 5, maybe it changed? Aside: I don't consider having had taken those courses a negative by any means, just that they aren't "useful" in my line of work, and, I assume, in most people's daily lives As far as complex numbers explicitly, they show up again in Trigonometry, which immediately followed Algebra II in my HS curriculum, and appeared from time to time in succeeding courses in math and physics.
PGnome PGnome 1 1 silver badge 2 2 bronze badges. They even had a sewage system. Mesopotamia on the other hand Gene Gene 51 1 1 bronze badge. For every topic in math that you do teach, there's also a topic that gets crowded out of the syllabus.
Zach Haney Zach Haney 4 4 silver badges 7 7 bronze badges. Could you summarize the answer from Dr. You get the same error on overflow or division by zero. Show 1 more comment. Rehton Rehton 11 1 1 bronze badge.
Stephane Moreau Stephane Moreau 21 1 1 bronze badge. Valuable topic. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.
Email Required, but never shown. Featured on Meta. Now live: A fully responsive profile. Version labels for answers. Linked 2. Related Hot Network Questions. Question feed. When arithmetic operations were operated on natural numbers, the result wasn't one of the natural numbers.
When two natural numbers are divided, it always didn't include integer so, mathematics was extended to rational number. This is called real number system but its not more real than complex number system in any way. The square of the positive number is a positive number and square of the negative number is also a positive.
How can square of the number be a negative number?? So, there is no any real value of x satisfying this equation. What does mean? It means go 5 steps forwards and 4 steps in negative backward direction. So, negative number denoted the negative i.
Let us draw a Cartesian plane i. If, we get i we rotate in - 90 degree. Here, x is called real axis, y is called imaginary axis and plane is called complex plane or Argand plane. Consider a problem;. So, if. Isn't that fascinating? Division, however, becomes more complicated and requires using conjugates.
Multiplying complex conjugates causes the middle term to cancel out. When dividing complex numbers, determine the conjugate of the denominator and multiply the numerator and denominator by the conjugate. For example,. The absolute value of a number is considered its distance from zero on the number line. Because complex numbers include imaginary numbers, they cannot be plotted on the real number line. However, they can be measured from zero on the complex number plane, which includes an x axis for the real number and the y axis for the imaginary number.
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