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We left off last time with Cardan and his broken formula for finding the roots of cubic functions.
Cardan knew that his problem had to have a solution,
but didn't know what to do with the square roots of negative numbers that kept popping up in his equations.
Cardan came close to finding a way to make his formula work, but got stuck in an algebraic loop,
where a bunch of work would just lead him right back to where he started.
It took one more generation of mathematicians to get to the bottom of this.
Cardan's student Rafael Bombelli made some incredible insights about what's really going on here.
Let's remember why Cardan was stuck. The square roots of negative numbers ask us to find a number, that when multiplied by itself will yield a negative.
Neither positive nor negative numbers will work.
Bombelli's first big insight was simply to accept that if positive numbers won't work
and negative numbers won't work, then maybe there's some other kind of number out there that will.
Now if there is some other kind of number out there, a good follow-up question is, "What are we going to call it?"
After all, we need to use it in our equations. Bombelli's approach was a very practical one.
Rather than dream up a new name and symbol, Bombelli simply let the square roots of negatives be their own thing.
In the past, mathematicians would have thrown in the towel here and declared the problem impossible,
but Bombelli was able to press on simply by allowing the square roots of negatives to exist.
Let's take a simple example of our new numbers: the square root of negative 1.
For being a new kind of number, it doesn't look very exciting and might kind of seem like our old numbers,
but remember, it does have the exact special property we need:
when we square it, the result is negative.
Further, since our number doesn't behave like a positive or a negative, it must be something new.
Now, if this all seems a bit fishy to you, like a slightly too convenient algebra trick, you're in good company.
In fact, it's hard to introduce imaginary numbers without them sounding like an arbitrary invention.
However, before we dismiss the square root of minus 1 as some abstraction invented to torture students, let's review what we've learned so far.
Cardan and Bombelli were genuinely stuck on a tough problem that they knew had a solution.
What Bombelli was able to see, is that if he extended the existing number system, as had been done so many times before, he could solve the problem.
Just as people needed fractions, zero, and negative numbers to solve new problems in the past,
to solve this problem, Bombelli now needed the square root of negative 1 to be its own, brand new number.
Let's make sure we're clear about what it means for the square root of negative 1 to be its own number.
If our new number is truly a discovery and not an invention, it should behave like the other numbers we already know about.
It should follow the established rules of algebra and arithmetic, and it turns out the square root of minus one does, for the most part.
Just as we can split apart the root of the product of two positive numbers, we can also split apart the square roots of negatives.
The square root of minus 25 splits into the square root of 25 times the square root of negative 1.
This process is important because it allows us to express the root of any negative using the square root of minus 1.
The square root of minus 25 becomes 5√(-1).
We can use this process to expand the root of any negative number, writing it as some number we already know about, times the square root of minus one.
Let's quickly make sure that our new numbers follow the same algebra rules as our old numbers.
In algebra problems with x, only like terms can be added and subtracted:
2x+3x is 5x,
but 2+3x is just 2+3x.
Likewise, 2√(-1)+3√(-1) is equal to 5√(-1),
but 2+3√(-1) is just 2+3√(-1).
Finally, unlike terms can be multiplied just as in algebra with x.
5 times x is just 5x,
and 5 times √(-1) is just 5√(-1).
Now, there are some cases where our new numbers behave a little strangely, but these can often be avoided by first separating out the square root of minus one.
Now that we have a grasp on how our new numbers work, we can see how they fix one of our problems from last time.
We now have a strategy for dealing with the roots of negatives.
We can evaluate the square root of negative 9 we were stuck on and obtain 3√(-1).
All this is important, but it isn't enough to solve Cardan's problem;
we still need to figure out how to deal with the cube roots of these numbers.
Bombelli was able to solve our problem through one more powerful insight, and that's what we'll discuss next time.