Organize your RSS Feeds with Feedly

If you read a lot of sites and blogs, and you subscribe to RSS feeds, you may want to organize your feeds with  Feedly. It is a website that creates a magazine-like start page for all RSS feeds.  It allows users to import feeds from Google Reader and categorize them.

Feedly is available as an Android app,  iPad app, and Google Chrome web app.  It is also available as a Safari and Firefox extensions.  In addition, Feedly can be integrated with Twitter, Facebook, and Tumblr.

What is i? What are imaginary numbers?

Squaring numbers always give a positive result or a 0.  For example, 3^2 = 90^2 = 0(-5)^2 = 25.  Using this argument, it follows that we can always get the square root of 0 or any positive number. Now, what about the square root of a negative number, say, -4?

Let’s see:  2 \times 2 = 4 and -2 \times -2 =4; therefore, no number exists that when multiplied by itself equals -4. In fact, it looks like we cannot find the square root of any negative number.

Now, what if — just what if — we invent square root of negative numbers? Nobody would stop us right? We start with -1.  Suppose - 1 has a square root, and we call it i. If \sqrt{-1} = i, then i^2 = -1.  If so, we can also answer the question that we have asked above: \sqrt{-4} = \sqrt{4(-1)} = 2iAs a matter of fact, we can generalize this operation for any negative number.  If a is positive, then -a is a negative number. So, \sqrt{-a} = \sqrt{a(-1)} = i\sqrt{a}.

But what kind of number is i? Surely, it is not a real number since we cannot locate it on the real number line. And, since we have all the real numbers and i is not a  real number, maybe, we should call i an imaginary number.

The Real Number Line

Imaginary numbers were ‘invented’ (or discovered if you prefer) because mathematicians wanted to know if they could think of square root of negative numbers, particularly, the root of the equation x^2 + 1 = 0 (that is, x^2 = - 1 which is the same as  finding the \sqrt{-1}).   Just like the many ‘radical’ ideas in mathematics,   it was not widely accepted at first, but eventually its invention proved to be very useful and has opened a lot of new ideas in mathematics.

In the next post in this series, we are going to discuss about the operations on imaginary numbers. .

Week in Review – Chinese New Year Edition

Good morning to everyone and a Happy Chinese New Year to all Chinese all over the world.  Here are the Math and Multimedia summary of posts for this week.

I. Mathematics and Multimedia

II. GeoGebra Applet Central

III. School of Freebies

IV. GeoGebra 4.0 Tutorials (Updated) 

That’s all for this week. If you would like to updated every time a post is made in Math and Multimedia, there are nine ways to do it.

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