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Before we finish up our series and solve our problem from part 1,
let's talk about how complex numbers are the missing puzzle piece that make algebra complete.
Back in part 1, we saw how the definition of what a number is has evolved over time, beginning with the natural numbers.
The Egyptians figured out that these numbers are missing something, and it's pretty obvious today that the natural numbers are incomplete.
However, as we saw with complex numbers, it's not always obvious when your numbers are missing something.
Fortunately, there is a more sophisticated way to determine if we have all the types of numbers we need: the mathematical idea of closure.
Let's play a game.
I'll give you a set of numbers and an algebraic operation.
I want you to tell me if any two numbers in the set when combined with the operation
give a number that is not in the set.
Our first set is the natural numbers, and our operation is addition.
So the question is, "Are there any two natural numbers that when added together produce something that is not a natural number?"
After a little noodling, it should seem pretty reasonable that any two natural numbers added together results in another natural number.
Mathematically, we can say that the set of natural numbers is closed under addition.
Next, let's try the set of natural numbers and the operation of subtraction.
For some pairs of natural numbers like 6 and 4, things work out just fine.
6-4 is 2, which is a natural number.
But what about 4-6?
This results in an answer that is nowhere to be found in our set of natural numbers,
so our set is not closed under subtraction.
We need to expand our set to include 0 and negative numbers for this to be the case.
So the set of natural numbers is not closed under subtraction, but the set of integers is.
By expanding our number system, we can guarantee that any subtraction question we can ask will have an answer.
As we include more algebraic operations, we must continue to expand our number system.
Division requires us to expand our number system to include fractions, also known as a rational numbers.
'Rational' comes from the word 'ratio.'
Rational numbers are numbers that can be expressed as the ratio of two integers.
Venn diagrams are pretty useful here because they help us visually express the idea that one set includes another.
All integers are rational numbers because we can always express them as a ratio of two integers,
but not all rational numbers or integers.
Let's recap.
So far, we've made it to the set of rational numbers which includes numbers like 1, 0, -5, and -⅔.
What operations are the rational numbers closed under?
Well, any two rational numbers added together yield another rational number,
so we can say that the rational numbers are closed under addition.
We can say the same for subtraction, multiplication, and division.
Now, what about powers and roots?
Does a rational number raised to a rational power always yield a rational result?
It turns out that for problems like (2/9)², this is no problem; our result is rational.
Where we get into trouble our problems like 2^(½).
Raising something to the power of ½ is the same thing as taking the square root,
so this is equivalent to the square root of 2.
We'll save the full argument for another day,
but it turns out that √2 is not rational.
There are no two integers that when divided equally exactly the square root of 2.
Since numbers like this are not rational, we give them the name 'irrational.'
There's one more class of numbers that are even cooler than irrationals:
the transcendental numbers like π and e.
We'll also save these for another day.
So we've expanded our number system again to include irrationals and all these numbers taken together form what we call the real numbers.
Let's play our game one more time.
Our set is now the real numbers and our operation is taking roots.
Do we have closure?
Are there any real numbers that when we take some root yield a result that is not a real number?
The answer is that despite all the types of numbers we've included along the way,
we're still missing something.
We can write an expression only using real numbers and roots - for example, the square root of -9 that has no solution in the real numbers.
For this problem to have an answer,
we must expand our number system once more to include imaginary numbers,
and taking all our real numbers from before together with imaginary numbers,
we arrive at our broadest class of numbers: the complex numbers.
Initially, mathematicians were concerned that even complex numbers were not sufficient,
that problems like the square root of -i would result in an even more complex number,
perhaps even a 3-dimensional number instead of our 2-dimensional complex numbers.
Fortunately, this turned out not to be the case.
In fact we can evaluate the square root of -i, again using our good friend the complex plane.
Since -i has a magnitude of 1 at an angle of -90°,
We just need a number with a magnitude of 1 at an angle of -45°.
According to our unit circle, (√2/2)-(√2/2)i.
So the square root of -i is just another complex number!
There's no need for some wild new 3-dimensional number.
In fact, there's no operation in the world using addition, subtraction, multiplication, division, powers and roots that the complex numbers can't handle.
Imaginary numbers are the exact missing piece that makes algebra complete.