Imaginary Numbers Are Real [Part 08: Math Wizardry]
308K subscribers
SUBSCRIBED
More information and resources: http://www.welchlabs.comImaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering. Part 1: Introduction Part 2: A Little History Part 3: Cardan's Problem Part 4: Bombelli's Solution Part 5: Numbers are Two Dimensional Part 6: The Complex Plane Part 7: Complex Multiplication Part 8: Math Wizardry Part 9: Closure Part 10: Complex Functions Part 11: Wandering in Four Dimensions Part 12: Riemann's Solution Part 13: Riemann Surfaces Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: http://www.welchlabs.com/resources.
Caption author (Russian)
Caption author (English)
Caption author (Spanish (Latin America))
Caption author (Portuguese (Brazil))
Caption author (Turkish)
00:00
Let's solve a simple equation: x3=1. What value of x would make this equation work?
00:07
If you said x=1, way to go! 13 is 1, we have found an answer. Now, are there any more answers?
00:15
Back in part 1, we introduced the FUNDAMENTAL THEOREM OF ALGEBRA, which says that a polynomial must have as many roots as its highest power.
00:24
We can rearrange our equation as x3-1=0 to make it a more obvious polynomial, and since our highest power is 3, this equation must have 3 solutions.
00:35
There is an algebraic way to find all three answers, but it involves perfect cube factoring and the quadratic formula, and takes some work.
00:44
Instead, let's try to solve the problem visually using the complex plane. Our question in words is, "What numbers when multiplied by themselves 3 times equal 1?"
00:56
Let's think about this problem using the polar form of complex numbers we discussed last time.
01:01
We can think about 1 in the complex plane, as the number with a magnitude of one at an angle of 0, or 360.
01:09
When we multiply numbers in the complex domain, their magnitudes multiply and their angles add.
01:15
Our results should have a magnitude of 1, and since 1*1*1=1, if we give each of our axes a magnitude of 1, our resulting magnitude will work out to 1. Easy.
01:27
But what about our angles? We know that when multiplying complex numbers, our angles add, so we need an angle that when added together three times give 0, or 360.
01:38
360 seems a little more reasonable, so what's the correct angle here?
01:44
Since we're dividing 360 into 3 even parts, the right answer here is 360 divided by 3, or 120°.
01:52
When we put this together with a magnitude of 1, we have a second solution.
01:56
It's the complex number with a magnitude of 1 at an angle of 120.
02:01
This answer makes a lot of sense on the complex plane. Multiplying our answer by itself once results in a magnitude of 1 at an angle of 240°,
02:11
and multiplying by our answer again lands us exactly where we wanted to be - at a magnitude of 1 at an angle of 360°, also known as the real number 1.
02:22
This may seem like a strange answer for algebra class, so let's put it back in rectangular form.
02:27
Our good friend the unit circle can save us some time here. We would like to know the rectangular coordinates of a point on the unit circle at an angle of 120°.
02:38
According to the unit circle, our answer is -1/2 for the real part, and √3/2 for the imaginary part.
02:46
So now that we have an answer in rectangular form, let's try it.
02:50
If we multiply out our result, we get 1. Pretty cool, right?
02:54
We were able to solve a tough algebra problem visually using the complex plane, and get the same exact answer we obtained earlier through factoring.
03:04
Finally, we've only found two answers so far, and the FUNDAMENTAL THEOREM OF ALGEBRA demands that we have 3.
03:10
We find our missing answer going the other way around the circle. If we start at -120 and multiply our number together three times, we also land at the purely real number 1, so our missing answer is -1/2-(√3/2)i.
03:27
So we found all the answers to our problem x3=1, and shown how the complex plane allows us to find these visually.
03:35
In this case, using complex numbers saved us sometime over the algebraic approach, and for more complicated problems, the complex plane becomes even more useful.
03:44
For example, what if we change the power in our original problem to 8 (that is x8-1=0)?
03:54
We could try to factor this, or turn to our complex plane and see that just like our last problem, where we needed to divide the unit circle into 3 equal portions.
04:03
We need to in this case, divide our circle into 8 equal portions. So our solutions need to be at 45°, again, along the unit circle. All 8 answers, done.
04:15
Next time, we'll show how imaginary numbers are the missing piece of the puzzle that make algebra complete.
English