Complex Numbers - A History: Lesson 1 Part 1

How imaginary and complex numbers first came to be used. This is lesson one of a five part series.

My first lesson using Beamer.

[MikeAben]

Even the simple positive integers can be slippery. 1+1=2, right? Well, one drop of water added to another drop of water just makes one bigger drop; and 1 Litre of water added to 1 Litre of sand makes 1 Litre of mud, so 1+1=1. Looking at it another way, If 1 represents a measurement and we apply standard error, 1 can be as large as 1.5. If that is the case, now 1+1=3. Although physical models help, the idea of 1 is fundamentally abstract. It's best to concentrate on how to use it correctly.

[MikeAben]

Thanks for the thoughtful comments. Although this is more philosophy then math, I would argue that nothing in math actually exists in the real world. Math is constructed to model the real world, but I think any discussion of this being real and that not is often confusing and ultimately non-productive anyway.

[NewCalculus]

@21:40 You can't square sqrt(-1) because sqrt(-1) is not a number. Complex numbers are not numbers as they don't measure anything. The operation of squaring using indexes takes place without regard to that which is being squared, so care must be taken to ensure that that which is being squared is indeed a number!


[+] x [+] = [+]
[-] x [-] = [+]


So whatever the psychotic "number" sqrt(-1) might be, it immediately has a sign identity crisis. What we know is that there is no signed number which when multiplied by itself will produce a negative number.


Complex number theory is based on ill-formed concepts. It has no place in mathematics or any other field of rational thought. There is no mathematics that can't be done without complex numbers.

[carlfortytwo]

Then you weren't paying attention - they appeared in the process of solving the reduced cubic equations. Though they weren't understood for what they were at the time, they allowed the completion of the method to find a real solution to the cubic. Just watch it again, and look for the negative values under the square roots.

[MikeAben]

2 + sqrt(-1) is an example of a complex number that Cardan used to solve a specific type of cubic equation. As mentioned at the start of the video, this was more of a history lesson where I shied away from use of specific vocabulary and formal definitions. I get into that with the second video, which may be more what you are looking for.

[MikeAben]

(-1)^(1/2) works because its defined value is consistent with all mathematics. a/0 remains undefined because no one has come up with a way of defining it that doesn't create contradictions, and mathematicians loathe contradictions.

I'm glad you found the video useful. My advise is not to get too concerned with what is "real" and what isn't. Just note how the pieces all fit together so beautifully, and marvel at the fact that mathematics models the real world as well as it does.

Cheers

[mhmh]

An excellent discussion of this is also on the podcast - In our time with Melvyn Bragg

[potrkca]

thank you for an excellent presentation Mike.

[Melomathics]

At the end of the day thus, ''i'' may even actually exist in the real world, but just like extra spacial dimension, we just can't understand its ''physical'' meaning

The x³ = 15x + 4, equation you solved is a somewhat proof of this. If -1^1/2 in an equation doesn't mean that it's undefined or unsolvable, as it's the case of for example ''a/0'' (division by 0)

then we can't possibly reject the possibility that an actual physical value of '' i '' may exist somewhere in the Universe. Can we ?

[Melomathics]

This was awesome. I've been looking for some meaning as to why did they come up with ''i''.

As you said, ''i'' doesn't have any meaning at all in the ''real'' world (or at least in our ''know Universe''), but nonetheless we can still use it in calculation to actually SOLVE ''real'' world problems like the equation you gave.

Meaning seeing 1^1/2 in a calculation doesn't necessary mean its ANSWER doesn't have a real life representation.

Thanks a lot for this, : )

[MikeAben]

Thanks. I really got to get around to finishing this series.

[TheiLame]

Love the pace you are using..!! i can follow :)

[azscab]

Cool man. I've been interested in fractals for ages. My search for the origins has led me here...and I'm not disappointed. You got me hooked so on to part 2

[WDY]

and how complexnumbers was invented? and how do it work? i cant find in this lesson..

[dharma]

Complex numbers first arise from solving QUADRATICS, and not cubics. If the discriminant (b^2-4ac) of a quadratic is negative then you need complex numbers. You need to mention this in the video. You move on from the quadratic formula without insisting that the discriminant be non-negative.