Lost in Translation Posted August 31, 2018 26 minutes ago, OldDog said: Does this mean that any point on an coordinate system where the X and Y axis are real can be expressed with an imaginary (i) component? I don't know. Let's see what Wandelaar says. Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 Later today I will answer your questions. First I have some work to do... 1 Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 12 hours ago, Lost in Translation said: Does this mean that |1 + 1i| is the distance between (0,0) and (1,1), which is the square root of 2? Yes. 12 hours ago, Lost in Translation said: EDIT: Is the modulus |z| another way to say the length of the arrow? Yes. 1 Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 (edited) 11 hours ago, OldDog said: Does this mean that any point on an coordinate system where the X and Y axis are real can be expressed with an imaginary (i) component? In a trivial way: yes. Because even for complex numbers (= arrows) z that have Im(z) = 0 we can still write z = a + 0i . Edited August 31, 2018 by wandelaar Share this post Link to post Share on other sites
Marblehead Posted August 31, 2018 At the moment I don't even have sufficient knowledge to ask a question. I'm sure one will arise soon. Share this post Link to post Share on other sites
Lost in Translation Posted August 31, 2018 Nine pages of graph theory, elementary calculus, imaginary numbers, and calculating the length of the hypotenuse of a right triangle -- on a Taoism forum! Who would have imagined... ? 1 Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 1 minute ago, Lost in Translation said: Nine pages of graph theory, elementary calculus, imaginary numbers, and calculating the length of the hypotenuse of a right triangle -- on a Taoism forum! Who would have imagined... ? Well - after all we are dealing with imaginary numbers. So... 1 1 Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 I just now had an idea that might give us some pictures of functions (such as Re( ), Im( ), | | ) from the complex numbers to the real numbers. You know: a picture tells more ... What we need for that is a 3D grapher. We already introduced the Cartesian xy-plane for our complex numbers (= arrows), and adding one more perpendicular axis gives us the possibility to represent the values of the function above (or below) the arrowheads of our complex numbers. Share this post Link to post Share on other sites
OldDog Posted August 31, 2018 33 minutes ago, wandelaar said: In a trivial way: yes. Because even for complex numbers (= arrows) z that have Im(z) = 0 we can still write z = a + 0i . OK. I don't think we really answered your question about resolving the modulus of 1+ 0i ... which as I understand it would be the square root of 1 .. or 1. Share this post Link to post Share on other sites
OldDog Posted August 31, 2018 5 minutes ago, wandelaar said: What we need for that is a 3D grapher. Now that'll be interesting. A 3D plot on a two dimensional surface ... i.e. tablet. Sure that won't add to the confusion? Back later ... errands to run. Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 Added the modulus of a complex number to the picture: Share this post Link to post Share on other sites
Marblehead Posted August 31, 2018 35 minutes ago, wandelaar said: Well - after all we are dealing with imaginary numbers. So... Yeah! Great for our Hindu, Christian and Buddhist friends. Share this post Link to post Share on other sites
Marblehead Posted August 31, 2018 That graph sure helped me. (I sometimes lie.) My brain doesn't like this right now. Maybe this afternoon. Y'all go ahead. I'll catch up later. Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 OK Marblehead. Do the others now understand everything in the picture of my previous post? Share this post Link to post Share on other sites
OldDog Posted August 31, 2018 Got it. But seems like we have defined everything but the i component. I mean everything except the i component can be accounted for in standard (real) math ... arithmetic, geometry, algebra. Still don't understand much about i other than we call it imaginary or nonreal. At this point I don't think trying to perform operations on non-real/imaginary/complex numbers is going to inform much. I'll hide n watch. Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 (edited) One more function to go before we will introduce the sum and product for the complex numbers. DEFINITION The argument arg(z) of a complex number z = a + bi is the angle between the positive x-axis and the arrow from (0,0) to (a,b). The argument is usually measured in radians and chosen so that –π < arg(z) ≤ π . Edited August 31, 2018 by wandelaar Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 (edited) Here's the picture complete with all four functions: Re( ) Im( ) | | arg( ) Edited August 31, 2018 by wandelaar Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 Here is a nice 3D grapher to visualise our functions: https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/ Share this post Link to post Share on other sites
Marblehead Posted August 31, 2018 52 minutes ago, wandelaar said: Here is a nice 3D grapher to visualise our functions: https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/ Hi Wandelaar. I want to thank you very much for your effort with me and others in investigating this concept of Complex Numbers. I must admit that I have lost interest in going any further with it. But I did learn a little so my brain ain't dead yet. Again, thanks. You have been very kind. Share this post Link to post Share on other sites
OldDog Posted August 31, 2018 5 hours ago, wandelaar said: The argument arg(z) of a complex number z = a + bi is the angle between the positive x-axis and the arrow from (0,0) to (a,b). I am used to the term argument meaning something that is passed to the function when it is called ... a la software function. Are you saying the angle is passed to the function arg(z) or that it returns the value of the angle when called? Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 @ Marblehead Ah - it's a pity we didn't succeed. But I respect your choice to stop with this topic. 1 Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 @ OldDog It's just an unhappy coincidence that the word argument is also used for the independent variable of a function, there is nothing more to it then that. But we have to live with this fact because the name "argument" for the angle associated with a complex number (as far as I know) is used everywhere in texts on complex numbers. So indeed the function arg( ) returns the angle of the complex number that is fed in. Share this post Link to post Share on other sites
wandelaar Posted August 31, 2018 OldDog & LiT Do both of you know how to measured an angle in radians? Share this post Link to post Share on other sites
Marblehead Posted August 31, 2018 1 hour ago, wandelaar said: @ Marblehead Ah - it's a pity we didn't succeed. But I respect your choice to stop with this topic. Well, we had some success. Your time is appreciated. 1 Share this post Link to post Share on other sites
OldDog Posted August 31, 2018 Sorry to see Marblehead go. I'll stick with it for a while. Still confused over what exactly i is. And, no, have never used radians before ... just degrees. 1 Share this post Link to post Share on other sites
