Notices
Results 1 to 27 of 27

Thread: imaginary numbers

  1. #1 imaginary numbers 
    Suspended
    Join Date
    Nov 2011
    Posts
    276
    In what real world situation will I use imaginary or complex numbers (like )?

    They just seem so abstract and, er ... imaginary. Do engineers or computer technicians ever use them?


    Reply With Quote  
     

  2.  
     

  3. #2  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    They (actually, complex numbers) are an essential mathematical tool in most areas of engineering.
    A good BBC radio programme from some time back: BBC - BBC Radio 4 Programmes - In Our Time, Imaginary Numbers


    Reply With Quote  
     

  4. #3  
    Forum Sophomore
    Join Date
    Nov 2009
    Posts
    126
    Complex numbers along with vectors provide a bridge between geometry and algebra, so we get all sorts of very mathmatically rigorous extra modeling tricks out of it.

    One such is the anlysis of simple harmonic motion where complex numbers can be regarded as diplacement. Useful in electronics for studying oscilations.
    Reply With Quote  
     

  5. #4  
    Suspended
    Join Date
    Aug 2007
    Location
    grail search
    Posts
    811
    Basically, imaginary numbers provide for "imaginary dimensions" in geometrical analytical situations.

    Why the interest?
    Reply With Quote  
     

  6. #5  
    Forum Freshman
    Join Date
    Oct 2011
    Posts
    25
    One could argue that nought is an imaginary number as it has no value but is capable of making other numbers bigger just by presenting them together..nought is not a value , just a concept but it is still incredibly powerfull
    Reply With Quote  
     

  7. #6  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    Quote Originally Posted by pipster View Post
    One could argue that nought is an imaginary number as it has no value but is capable of making other numbers bigger just by presenting them together
    You are talking about multiplying by 10. That has nothing to do with imaginary numbers.

    nought is not a value
    Of course it is.
    Reply With Quote  
     

  8. #7  
    Suspended
    Join Date
    Nov 2011
    Posts
    276
    Okay, I understand how they might represent "imaginary" dimensions.

    And another question. Why is it called the imaginary unit?

    So as a unit does that signify the "imaginary" kind of number, as in . As if we would call the reals ?
    Reply With Quote  
     

  9. #8  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    Exactly right. One way to think of it is...

    Consider the "number line"; zero in the middle, +ve numbers to the right; -ve numbers to the left:

    -n .... -2 -1 0 +1 +2 .... +n

    Those are the "real" numbers.

    Now rotate the number line 90deg anticlockwise. Now you have:

    +n
    .
    .
    +2
    +1
    0
    -1
    -2
    .
    .
    -n

    These are the imaginary numbers.

    Now you can treat these as coordinates on a 2D plane; the "complex plane". Any complex number can be represented by a position on the plane (r,i).

    Now we can see that multiplying by a real number, stretches/moves along the real plane (multiplying by a negative number "flips" by 108deg).

    The clever thing is that multiplying by i rotates a point by 90deg. Multiplying by a complex number is a combination of these operations (possibly rotating by an arbitrary angle).

    Now, any position on the complex plane can also be represented by a distance from the origin (0,0) and an angle (phase). This is why complex math is so useful for signal processing.
    Reply With Quote  
     

  10. #9  
    Suspended
    Join Date
    Nov 2011
    Posts
    276
    Oh okay. So . I find that cool.

    So the unit is the most fundamental sense of quantity? I know our numerical digits (0, 1, 2...) aren't absolute, because of the base systems. And numbers only describe the unitary quantity, I guess.
    Reply With Quote  
     

  11. #10  
    Forum Freshman
    Join Date
    Oct 2011
    Posts
    25
    Quote Originally Posted by Strange View Post
    Quote Originally Posted by pipster View Post
    One could argue that nought is an imaginary number as it has no value but is capable of making other numbers bigger just by presenting them together
    You are talking about multiplying by 10. That has nothing to do with imaginary numbers.

    nought is not a value


    Of course it is.
    Oh ! Fair enough Point taken I apologise for my ignorance It was just a thought ... So nothing is something then?
    Reply With Quote  
     

  12. #11  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    [QUOTE=pipster;296722]
    Quote Originally Posted by Strange View Post
    So nothing is something then?
    Hmmmm... Depends on your definitions of "nothing" and "something". That is the trouble with language, it can lead to all sorts of ambiguities like that. That's why mathematics depends on formalising the definitions of things. If by "nothing" you mean zero (or the empty set), then this is a real mathematical vlaue that can be treated as any other. Whether that makes it "something" or not depends on what you mean by "something"
    Reply With Quote  
     

  13. #12  
    Forum Ph.D.
    Join Date
    Dec 2006
    Posts
    963
    Quote Originally Posted by Strange View Post
    They (actually, complex numbers) are an essential mathematical tool in most areas of engineering.
    A good BBC radio programme from some time back: BBC - BBC Radio 4 Programmes - In Our Time, Imaginary Numbers
    My education in basic maths just stopped short of studying complex numbers, but I was interested in them and also imaginary numbers such as (i) the square root of minus one.
    In particular I was fascinated by the fact that complex and, of course, imaginary numbers were not simply abstract mathematical entities, but had very useful practical applications. I fondly believed this showed the profound links between mathematics and the real world but was also an illustration of the famous quote, by Haldane, about the universe, or reality, being not only stranger than we imagine but stranger than we can imagine.
    I have to say that I was slightly, but not completely, surprised when I watched a recent BBC television programme introduced by the mathematician, and Oxford professor, Marcus du Sautoy.
    In this programme, which was one of a series, du Sautoy declared that complex, and hence imaginary numbers, were extremely useful, but not essential, mathematical tools that made a number of mathematical operations much less cumbersome and time consuming.
    Reply With Quote  
     

  14. #13  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    Quote Originally Posted by Halliday View Post
    In this programme, which was one of a series, du Sautoy declared that complex, and hence imaginary numbers, were extremely useful, but not essential, mathematical tools that made a number of mathematical operations much less cumbersome and time consuming.
    I think that is true of everything beyond addition!

    It is also true that you could write all software in binary. It is just so much easier when you have C++ and an extensive library of high-level functions.

    Doing the math of GR using just arithmetical operations would be possible but even more difficult than it is using shorthand like tensors, etc.
    Reply With Quote  
     

  15. #14  
    Suspended
    Join Date
    Aug 2007
    Location
    grail search
    Posts
    811
    Imaginary numbers are the closest thing to pure fabrication of what we already know. Using them with equations offers usual problems more solvable than what would otherwise not have otherwise been solved.

    Take the basic idea of catapulting a negative unit, like it has a family of complex numbers, "". Subunits of negative values where positive integers failed. The problem with using numbers is realising the limit of their absolute entity as a unit, divided, multiplied, and so on. Complex numbers accept the limitations of real numbers. Then they go beyond those confines.

    Tell me. Why does -1 x -1 = 1? Any way you can......
    Reply With Quote  
     

  16. #15  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    Quote Originally Posted by theQuestIsNotOver View Post
    Tell me. Why does -1 x -1 = 1? Any way you can......
    Because multiplying by -1 is "flipping" from going one direction on the number line to going the other way (a rotation of 180 on the complex plane). Doing that twice gets you back where you started.

    Similarly, multiplying by i is a rotation of 90 in the complex plane. Doing that twice is a rotation of 180, which is the same as multiplying by -1; in other words i x i = -1.
    Reply With Quote  
     

  17. #16  
    Suspended
    Join Date
    Nov 2011
    Posts
    276
    Also a note on multiplying by -1. Since multiplication is iteration of addition/subtraction, multiplying by -1 is a number minus itself then minus itself again.






    -1 minus -1 minus -1 is +1, which is just like flipping the sign.

    So it's nothing really special there. is interesting because it yields an entirely different kind of number. . And is just one :P
    Reply With Quote  
     

  18. #17  
    Suspended
    Join Date
    Aug 2007
    Location
    grail search
    Posts
    811
    Quote Originally Posted by Strange View Post
    Quote Originally Posted by theQuestIsNotOver View Post
    Tell me. Why does -1 x -1 = 1? Any way you can......
    Because multiplying by -1 is "flipping" from going one direction on the number line to going the other way (a rotation of 180 on the complex plane). Doing that twice gets you back where you started.

    Similarly, multiplying by i is a rotation of 90 in the complex plane. Doing that twice is a rotation of 180, which is the same as multiplying by -1; in other words i x i = -1.

    That's one way of trying to explain it.

    Another way: think of two negative axes though, right angles to one another; multiple them both out to get the surface area. It would be "1" as the concept of a surface area, not an axis, right?

    If one is to use mathematics geometrically, one needs to be consistent, right?
    Last edited by theQuestIsNotOver; December 23rd, 2011 at 02:45 AM.
    Reply With Quote  
     

  19. #18  
    Suspended
    Join Date
    Oct 2011
    Posts
    42
    Let me give you the definition of real numbers set of all rational and irrational numbers are called as real numbers. Imaginary numbers and infinity are not real numbers. We use real numbers to measure continuous quantities. The decimal numbers with an infinite sequence of digits to the right of the decimal point, are also real numbers.
    Reply With Quote  
     

  20. #19  
    Suspended
    Join Date
    Nov 2011
    Posts
    276
    Never mind about the two questions.

    I'm still surprised imaginary numbers are used in real-world situations. Most unexpectedly finance and marketing.
    Reply With Quote  
     

  21. #20  
    Suspended
    Join Date
    Jan 2012
    Posts
    6
    One example is i = sqrt(-1). Without this number, the polynomial x^2 + 1 has no zero.
    Last edited by marnixR; January 2nd, 2012 at 02:52 PM. Reason: deleted spam link
    Reply With Quote  
     

  22. #21  
    Moderator Moderator
    Join Date
    Jun 2005
    Posts
    1,620
    Quote Originally Posted by malikusmangee View Post
    One example is i = sqrt(-1).
    Well, it's no so much an example but more of a definition (though not a very good one: is usually preferred)

    Without this number, the polynomial x^2 + 1 has no zero.
    True, but why would that matter? Is it required that every real polynomial of degree has at at least 1 root and at most roots?

    PS I am being a little unfair, but I seem to be asking you to to prove the Fundamental Theorem of Algebra, a non-trivial task
    Last edited by marnixR; January 2nd, 2012 at 02:54 PM. Reason: deleted quoted spam link
    Reply With Quote  
     

  23. #22  
    Suspended
    Join Date
    Oct 2011
    Posts
    42
    your question is quite perfect .Even i have never thought this where is use of this imaginary number and also one question like that why peime numbers and what it's use .Even wre also use it and do examples of prime numbers .But no one know why do this .What it's use in feasture
    Reply With Quote  
     

  24. #23  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    Prime numbers are used in a number of places. One important one is encryption based on the prime factorization of large numbers. So, it is easy to encrypt and decrypt a message when you know the prime factors. It is hard to break the code because this would require factorizing a large number (the key) and finding the prime factors of a large number is hard.
    Reply With Quote  
     

  25. #24  
    Suspended
    Join Date
    Oct 2011
    Posts
    42
    Hi strange ,
    It's amazing to know about the prime numbers tha they are used for encrypting .But i want to ask whether it is possible to decrypt it with these numbers using prime factors .The main thing is these are used for this purpose and what is a prime number in math are used in such purposes .So please tell us how this is used in encrypting some how to me .Thanks .
    Reply With Quote  
     

  26. #25  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    There is a lot of detail (too much, maybe?) here: RSA (algorithm) - Wikipedia, the free encyclopedia

    This might be a simpler explanation: How Public Key Cryptography (PKC) Works (I haven't read that page though)
    Reply With Quote  
     

  27. #26  
    Suspended
    Join Date
    Oct 2011
    Posts
    42
    I am very eager to know the whole scinario of the Cryptography and the use of a list of prime numbers to be used in it to encrypt and decrypt .Thanks for the source you have given .I have gone through it and want more information regarding that .So please hel if you can provide me some more details .Thanks again for the reply .
    Reply With Quote  
     

  28. #27  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    17,036
    You will have to be more specific about what you want to know. The Wikipedia page on the RSA algorithm is pretty complete.
    Reply With Quote  
     

Similar Threads

  1. Computers and Imaginary Numbers
    By bgjyd834 in forum Computer Science
    Replies: 3
    Last Post: May 21st, 2011, 03:51 PM
  2. The imaginary unit
    By Sindrato in forum Mathematics
    Replies: 15
    Last Post: September 14th, 2010, 12:13 AM
  3. Imaginary Number raised to an Imaginary Number??
    By iamkennyk in forum Mathematics
    Replies: 2
    Last Post: May 5th, 2008, 10:55 AM
  4. More irrational numbers than rational numbers?
    By DivideByZero in forum Mathematics
    Replies: 12
    Last Post: March 6th, 2008, 09:01 PM
  5. Imaginary numbers
    By leohopkins in forum Mathematics
    Replies: 32
    Last Post: February 4th, 2007, 04:11 PM
Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •