so imaginary number give us mean to get oscillation or rotation of direction, if we keep on multiplying by imaginary number i. It would be easier to imagine the advantage of imaginary number if we can express fourier transform formula without ''i' and see how adding 'i' makes it more packaged and easy for maths manipulation.
Such a good video! Thank you so much! I've just started doing maths and my current book doesn't show how the imaginary numbers can be seen from a geometrical perspective in a coordinate system. This has helped me a lot, by making these numbers "more real" so to speak. Thank you.
i=sqrt(-1) by definition. i^2 = -1 is just a different way to express the relationship between i and -1. i^2 = -1 sqrt(i^2) = sqrt(-1) sqrt(a) = a^(1/2) (i^2)^(1/2) = sqrt(-1) (a^b)^c = a^(bc) i^(2 • 1/2) = sqrt(-1) i^1 = sqrt(-1) i = sqrt(-1) HTH
This is the first time I've seen i described geometrically, and you provided a very clear and thorough explanation. This has been very enlightening! Thank you
Thank you. The geometric explanation is nothing new, but I feel it is often overlooked in the race to explain these things in a more abstract way, the way maths is generally taught nowadays.
Sometimes, I just focus on the behaviors of numbers rather than questioning about what the numbers ARE. It's kind of like programming with an object-based computer language. God already create a library of mathematical objects. But without a documentation. We just need to discover them, identify their behaviors, use them to solve problems, and DO NOT ask about their implementations. Because that's beyond us.
The video is really good. I only would recommend if possible to notate -2^2 as (-2)^2 as - 2^2 may be understood to be - (2)^2 which is a negative number. This is a minor typo and is clarified by context. Great videos.
Thank you. I'm not sure it was Euler who gave it the symbol i. The symbol just stands for imaginary and it was Decartes who called them imaginary numbers. Whether it was Decartes who attached the symbol i or not, I don't know. As regards a proof, I think Euler's identity is the nearest one can get to a mathematical proof. Take a look at my video on the subject: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-sKtloBAuP74.html
@@MarkNewmanEducation Euler indeed was the first person in the historical record to use the letter _i_ to represent the square root of -1. He did this in 1777, in a paper which was later published in 1791. The title of Euler's paper is _De formulis differentialibus angularibus maxime irrationalibus, quas tamen per logarithmos et arcus circulares integrare licet._ That being said, it does not appear that Euler's use of _i_ caught on. There are plenty of big name mathematicians in the early 1800's who were still using √−1. It appears that Gauss was the next big name to use _i_ to represent √−1 in his 1831 paper _Theoria residuorum biquadraticorum. Commentatio secunda._ It appears that it was Gauss's use of _i_ that caused the symbol to catch on more broadly. But, I'm not an expert in math history! This is just the little bit of research I've done over the past several days into complex numbers. (For the record, you are correct that Descartes was the one to give the name "imaginary".)
Talking about geometry with "hidden dimensions" is still more mysterious and specific than it needs to be. Any time you have a system, with two dimensions, where the following operations are useful: (a,b) X (c,d) = (ac - bd, ad + bc) (a,b) + (c,d) = (a+c, b+d) you have a system that necessarily implies "i^2 = -1". Taking just the values of the form (a,0), they operate just like the real numbers: (a,0) + (b,0) = (a + b,0) (a,0) X (b,0) = (ab - 0*0, a*0 + b*0) = (ab, 0) And they also operate like a real-valued scalars: (a,0) X (c,d) = (ac - 0*d, ad + 0*c) = (ac, ad). So might as well just call (a,0) the real number "a". Then, every value can be represented by scaling the unit in each dimension by real numbers: (a,b) = a(1,0) + b(0,1) = a + b*(0,1) Now you can do arithmetic on values of this form, which is equivalent to the original operations on ordered pairs. (1,0) is just 1, but if you square the unit of the second dimension you get: (0,1) X (0,1) = (0*0 - 1*1, 0*1 + 1*0) = (-1,0) = -1, so call the unit "i" and you have the complex numbers: (a,b) = a + bi Nothing "hidden", mysterious or unbelievable about it. Just a convenient way to represent two dimensions when those two operations are useful, which is true of many real world physical systems.
Thank you very much. Never thought maths is this much beautiful. I'm ~40. Learned maths upto middle school. For some reason after 20 years I'm curious to learn maths. This is insightful well research video. ♥️🙏
Controversial: I am not sure. It took a long time for mathematicians to accept Imaginary numbers, but even though zero has been with us a long time, it too was not originally included in ancient numbering systems. Difficult: I would say imaginary numbers. It is the way they are taught that I have always found so difficult to understand. When I first heard of the geometric way of looking at them that I present in this video, it was a eureka moment for me.
That's very kind of you, but I have to say, to this day, I find maths a challenge; complex numbers, even more so. This video is the result of a Eureka moment when I came across a piece of research that emphasized how complex numbers can be thought of as a rotation. This sent me off in a whole direction of thinking of the imaginary number i from a geometric point of view and how multiplication, scaling, rotation, and reflection are all related to each other.
True... but I had a terrible time at school simply accepting that convention. Mathematics is very logical. There is a reason for everything. If one can understand the underlying logic then it makes it easier to use and adapt the methods for other applications.
@@MarkNewmanEducation Some of math is not logic, some of it is just conventions made up by men. In this case I don't think it's all that illogical - I believe "i" is simply an abbreviation for "imaginary." It is imaginary in contrast to the "real" numbers.
@@randyyates9837 Ahh... now I understand where your coming from. You thought I was trying to answer the question: why is the letter "i" is used rather than any other letter? That wasn't my intention in the video. I was trying to demonstrate an intuitive way of understanding what the imaginary number "i" is.
@@MarkNewmanEducation Lol I think it was the old school style filming plus you looming over a bunch of kids. No worries man, it was a great video and I learned something new.
The rotation is around the imaginary axis. On this particular graph, I have "borrowed" the z-axis to represent the imaginary axis. So when I rotate the shape it appears to rotate with the y-axis as it's pivot. I could have rotated around the x-axis, I just chose to rotate around the y instead. The main point is that one of the shape's dimensions moves through the imaginary axis.
there are no negative numbers; they only exist as differences between positive numbers. -c = a-b. b>a b-c = a a=a a-a=0 If there are no negative numbers, there are no square roots of negative numbers. i:= sqr(-1) i^2= sqr(-1)sqr(-1) = sqr[(-1)(-1)] = sqr(1^2) = 1 -1
