Imaginary Numbers Made Easy

I like math. Let me show you why…

Numbers are cool. They help us do stuff. For example, if you wanted to calculate how many days it would take you to read all my blog posts, and you read at a speed of, say, 1 blog post every 3 days, considering that I have exactly 10 published blogs to date, it would take you exactly 30 days (if my head is still on the right way around) to accomplish your task. That was mundane, but did you see my logic? 30 days is equal to 3 days, 10 times; or as mathematicians would call it 3 times 10.

I believe numbers are a creation of our minds, but thats more of a philosophical debate. Lets get down to the math, or history rather. People started using numbers when they saw similarities between two separate objects with similar features. An orange is round, orange coloured, and has a skin. Put a couple of them next to each other, and you suddenly have more than just single objects. You have a pair, or more lexicographically correctly, “two” of the same thing, “two oranges!” Now you can have three, four, five, etc. oranges and that would completely make sense, right? Stay with me.

It was soon of course that people realised that you can cut an orange in approximately half – lets just say it was exactly half for the time being. They didn’t know what half was; all they knew that it was not one of the numbers that they were used to using in counting. So they had to extend the number system to accommodate this. So if you put an orange next to another orange, it is no longer one orange, nor is it two, but it is somewhere in between. And hence a new type of number was born a fraction.

People quickly noticed that you don’t only have to “add” oranges, but if you are selling them, you are “subtracting” from the total amount.  If you had 5 oranges, and you sold 2, how many are left? Besides sounding like a child’s math problem, think of what we are doing. We can think of it as “subtracting” 2 from 5: we are left with 3. So now if a merchant wanted to sell 9 oranges, but only had 5, but urgently needed the money. He could strike a deal: sell 9, give 5, and deliver the remaining 4 later. But doesn’t that mean that he has no oranges, he has less than none! I think you can see where I’m going with this. Hence the birth of the negative number.

As you can see, the number line keeps on extending to accommodate for concepts that were initially not possible to grasp. Now imagine numbers not as a number of things, but a thing in its own right.

An now we bite into the meat of the situation. Adding 1 and 1 and 1 gives 3. This is the same as multiplying 1 by 3 (effectively adding 1’s this many times). Multiplying 2 and 2 and 2 is 8. This is the same as raising 2 to the power of 3 (multiplying 2’s this many times). Going backwards with multiplication is dividing 3 by 3 (asking what multiplied by 3 gives 3) yields the initial answer 1. Now rooting 8 to the component of 3 (asking what raised to the power of 3 gives 8) yields 2.

Think of raising 2 to the power of 2. You get 2 times 2 equals 4. Do the same with a negative 2. Knowing two negatives give a positive, hence you also get 4. Therefore the root with component 2 of 4 gives both 2 and negative 2 (written -2). Now lies the question: what raised to the power of 2 gives -4? A difficult question to answer, one that stumped centuries of mathematicians. But now, we introduce the concept of an imaginary number, which raised to the power of 2 will give a negative number. It is just as real as an any number, just as real as having -4 oranges, or 1.5 oranges. Imaginary is just a name, not a truth about the number system. Trust me, mathematicians are not very imaginative.

Thanks! Hope this helped broaden your imagination. C ya!


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