For this reason, we adopted a linear storytelling style, with alternating text explanations and data visualizations guiding the reader throughout the narration. The visualizations were designed to be tools for exploring and interacting with the narration, providing an intuitive way to understand the complex dynamics underlying the data. The initial section contains a general overview of the phenomenon and an explanation on the approach we took for our analysis.
The subsequent sections provide different perspectives on the same set of data: In order to visually identify the three main ideological roots, and make them intuitive and clear throughout the narration, we decided to use three different colours. Using variations of the primary RGB colors, we hoped that they could be seen as the decomposition of the colour black.
In this way, the colours represent three different faces of the same phenomenon. The first visualization that we used is called a bump chart. We used this chart to display the number of attacks from to by ideological root. In this visualization, we wanted to focus the attention on two periods, prior to and post , distinguished by a sudden change in the trends, in particular, an exponential growth in the number of attacks after This visualization gives a general overview of the trends before we examine the data in further depth.
The GTD data contained a row for each terrorist attack from to , and a extensive number of variables, including information about the time, place, terrorist group, weapons used, number of victims, type of victims, and so on, for each attack. After verifying the reliability and correctness of the dataset, we cleaned and filtered the data, reducing the number of variables, in order to focus only on the most important ones for our research. The goal of this analysis was to understand and visualize the countries that are most frequently attacked, in terms of the number of victims and number of attacks, as well as the evolution of these numbers through time.
We decided to visualize the distribution of the attacks using a world map. Doing so, let us show the number of victims per terrorist attack grouped by country, as well as allowing the reader to compare the two historical periods identified in the trend overview: We decided to use a bubble chart to display the number of victims, with clear relative proportions.
In order to visualize the bubble chart on the map, we used Carto , a software that allows to create geo-located visualization using json data. We cleaned and filtered the data in order to create two different datasets for the two different time periods and the software provided us the first drafts of the visualizations.
It’s not so ISIS: The complex shape of terrorism and its ideological roots, visualized through data
We then modified these drafts using Adobe Illustrator, creating the two overlapping layers on a fixed background map. A complicated issue, explains Nason, is one in which "the components can be separated and dealt with in a systematic and logical way that relies on a set of static rules or algorithms. Pumping crude oil from 6 miles below the surface of the Gulf of Mexico is complicated. So is making an electric car and a reusable rocket just ask Elon Musk. But once you figure out how to do these things, you can keep doing them at will.
On the other hand, a complex issue is one in which you can't get a firm handle on the parts and there are no rules, algorithms, or natural laws. A complex thing is much more challenging--and different--than the sum of its parts, because its parts interact in unpredictable ways.
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Managing people is a complex challenge. Maybe you'll get lucky and figure it out once, but whatever you did this time won't generate the same result next time. First, says Nason, understand and appreciate the differences between complexity thinking and complicated thinking.
The Critical Difference Between Complex and Complicated
How do we do this? Multiply by the opposite angle! Either way, the conjugate is the complex number with the imaginary part flipped:. What happens if you multiply by the conjugate? Without thinking, think about this:. The result has no imaginary parts, and is the magnitude squared. To reverse this effect we can divide by z , and to actually shrink by z we have to divide again.
We scaled both the top and bottom by the same amount, so the effects cancel. The result is to turn division into a multiplication in the numerator.
How to classify complex numbers (example)
The first should make sense. Another way to think about it: The second property is trickier. Well, complex conjugates are not a random choice, but a mirror image from the imaginary perspective, with the exact opposite angle. Here is what i tried to understand the intuitution for multiplication: Again, great explanation, thanks! Hi Ragavendar, thanks for the comment. You got it, by converting to polar coordinates you can see the sine and cosine addition formula emerge.
Khalid, You are always better explained!! Though most of us knew the concepts before, reading you is an experience by itself!! Hi Maheshexp, check out the article on complex numbers to see why they are needed. My aha moment, considering the use-fulness of imaginary numbers occurred when I pointed in a direction other than the 2 aforementioned ones. I think what throws people off is the addition sign. Yes, negative and complex numbers are a different way to talk about where things are. Even decimals are like this: Thank you for the comment, really glad it was helpful. However, we end up scaling the number by the size of 1-i.
Both have the same size just reflections of the same angle ]. So, we have to divide twice: Thanks for the explanation — very easily understood. Anyone have any ideas on how to visualize taking a number to the power of i? Thanks, glad you enjoyed it.
Constantly changing in a perpendicular direction will move you in a circle, leading to the full formula. Note that N or i is just a way to designate that the quantity is in a different, degree direction.
I hope this helps! This is just great! You explain really, really well. And i love your humour. Nevertheless…I m enjoying it now!!! The traditional difficulty in understanding the complex numbers is a man-made one. It reflects intellectual shortage in the course of definitions and thought construction. Much of that is due to the historic and gradual accumulation of mathematical knowledge which characterizes mathematicians more as a cult—a culture with rich inheritance of thought, history, and terminology. Imaginary things are understood as things that has no real reflection and exist only in the human mind.
Complex numbers have real applications in physics when rotational physical phenomena arises—things that has a cyclic nature such as an alternative current in electricity. All what an imaginary component in a complex number means is that the magnitude of the complex number is specially aligned at a certain angle relative to the conventional positive real line axis. Again the real line is a misname for real numbers.
So really, the complex numbers are nothing but real numbers in two-dimensional space with some closure field property pertaining to the roots of negative numbers. With this idea of rotation in mind, the study of complex numbers can be much easily understood. Thank you for your explanations.
Intuitive Arithmetic With Complex Numbers
Can you do an article on imaginary exponentiation? I get that multiplication by an imaginary is like rotation, but what about multiplying a real by itself an imaginary number of times!? Why do we take the conjugate of the denominator and not the numerator obviously, algebraically this is obvious, but intuitively? And why do we WANT to shrink the denominator by its modulus, why not shrink it by 56 or something arbitrary? Here is a crisp and dynamic word for complex numbers, hand both as a verb and a noun: It did not take him long to convince himself that zooms and rotations can be combined, in arbitrary order, without affecting the result.
Unfortunately, it is a useful term only for multiplication, not adding. But perhaps that is a feature rather than a bug, helping keep focus on the operations rather than the numbers. I love, love, love seeing how other people see these topics.
It's not that Complex — MARY YOUNG
Dear Kalid, Your blog is really mind-bending! I am sure one of your readers will be the next Great! Well, on sincere lines, I am really grateful to you for making this intuitive work available to all of us readers. When it comes to naming, one thought came to my mind — people call non-existing things like ghosts and spirits as supernatural things and they call numbers which are as real as Real ones imaginary! Just coz they are kinda intangible for the time being and in another dimension! Glad you enjoyed it — yes, I really hope that in years or 50, or 20! Crisis in maths - Page 2.
Thank you so much …. Reading your article has given me a good reason to start liking complex numbers … Math would be real fun if all concepts are taught this way in school …. I now see the light!!!! I totally hear you, complex numbers bothered me for years ever since first encountering them.
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