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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. .

Math and Multimedia Blog Carnival 19

Welcome to the 19th edition of the Math and Multimedia Carnival. The entries in this edition were sent via email, others via the blog carnival site, and some were my own selection.

Scenes from the 2004 Carnivale in Venice (IMG_4683a)
Here are the entries for the Mathematics and Multimedia Carnival Chinese New Year Edition. » Read more

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