Melvyn Bragg and his guests discuss imaginary numbers

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## Comments

Re David Nichoslon

"They all took to the idea of j and alternating current………..no-one told me where to go."

I'm sorry if what I wrote wasn't clear: What I meant was that, to some at least, the idea of the square root of a negative number can be "challenging" even repellent, separate from its utility.

I've tried to introduce the idea of sqrt -1 to a few more able kids, and they don't like it. By that I mean it upsets their pretty solid and dearly held ideas of what numbers are. Their reaction is similar to the Babylonians(?) who if I'm right gave up and ignored it because they didn't like it. *I think this is a good thing.* (Not the ignoring!) I'm not complaining that they tell me where to go. Far from it. Like I said, I think that introducing i upsets a few preconceived ideas, and the outraged, negative reaction is an initial defence against that.

As a pupil/student where I went from there was accepting it and using it, but never really understanding it. What is i? I've never really had that satisfactorily explained. I suppose I'm approaching it philosophically rather than practically.

Maybe everyone understands j, whereas i remains a total mystery :-)

Throughout this discussion I've been thinking about Quantum Mechanics: similarly simultaneously practically useful and philosophically incomprehensible.

I've been familiar with i for 30 years or so but I'm still not sure what it is, and this programme didn't help. That was a disappointment.

I'd say the aim of the programme should have been to inform, educate, and entertain the interested lay person. This subject should not be beyond the scope of that remit. The fact that there were nervous jokes about incomprehensibility from presenter and continuity announcer to me shows they failed, badly.

Like I said I think i could be a rich source of (sometimes hostile) debate. This programme was a wasted opportunity.

My two penn'orth.

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I can understand why some listeners found this programme difficult and in some cases boring, but the topic is far from either. The problem with decribing the historical developement is that the discovery of the the imaginary number 'i' came about through what, to non-mathematicians, are quite arcane topics, but the later discovery of its geometry is actually the route to understanding it in a much simpler way, and thereby to realising why it is so immensly useful.

From its geometrical properties discussed in the programme we can completely understand it by taking i, not as an actual number, but as a label which tells us that we are performing an operation of turning through a right angle (ninety degrees).

In this picture the ordinary multiplication operation, denoted by the sign 'x', tells us that we are we are expanding the length of whatever we are multipling, by the amount we are multiplying it by: thus if we start at 2 and then multiply by 3 we are expanding it to 6, 2 x 3 = 6. Thinking in distances along Melvin's east-west line if we start at 2 inches west from the centre of the line and the multiply be three we get to a position 6 inches west of centre. Thinking of the numbers as representing arrows lying on this line, our multiplication operation has expanded the arrow from a 2 inch one pointing west from the centre to a 6 inch one pointing west. If we multiply by a negative number we are moving in the opposite direction, thus 2 x (-3) takes us to - 6, which in this context is now an arrow 6 inches long pointing east from the centre. We can re-interpret the minus sign as really telling us, not something about a number different from three, but really about a different sort of multiplication, one that expands but only after rotating through 180 degrees. Thought about in this way we could really write it as 2 x- 3. We are not changing what the numbers mean, (so we don't really have to figure out daft things like what we mean by a minus orange); rather we are changing what the operation of multiplication means.

The two operations, 'x' and 'x-' will however never get us off this east-west line.

We now think of 'i' as also modifying what 'x' means into telling us that we rotate by 90 degrees before expanding.

Thus 2 x 3i = 6i {which I would like to write as 2 xi 3 = 6i} turns our arrow from pointing 2 inches west to 2 inches north and then expands by 3, i.e. to being 6 inches north of centre. We keep the label i with the 6 to tell us that we've rotated through 90 degrees and out now pointing north (just like we keep the minus sign on -6 to tell us that we have rotated through 180 degrees and are pointing east).

For this understanding to be clear we would be better off sticking the i alongside the multiplication sign, but, like with negative numbers we always just latch it onto the numbers; this has the advantage of ensuring that we know we are dealing with a rotated quantity, but to some extent obscures the fact that it was the operation that was different not the number. It is a label to indicate a different operation has been carried out - thus we are can consider that we now have three operations 'x' i.e. expand, 'x-' i.e. rotate through 180 degrees and then expand, and 'xi' i.e. rotate through 90 degrees and then expand. If we combine these last two, i.e. if we rotate through 90 degrees, then rotate through another 180 degrees, then expand, we have turned 270 degrees (or equivalently turned 90 degress the other way). Thus, 2 x -3i = -6i gives us an arrow that points six inches south of the centre of our line.

We now see why we can consider i as the square root of -1. What we are really doing with -1 is just rotating our arrow through 180 degrees i.e flipping into the opposite direction (the 1 just tells us we don't expand or contract, just stay the same size - only our direction has changed), and this reversed direction we can get to by rotating through 90 degrees twice i.e. we do xi and the xi again.

This gives us the original formulations of i as a number having the properties i squared = -1, i.e. i is the square root of -1. But we can see this as really just telling us the result applying this turning through 90 degrees operation twice.

If you then play with combining ordinary 'x' and 'x-' with 'xi', you will soon see that the combination of expanding (or contracting if you are multiplying by a number less than one) along the east-west line (ordinary 'x') and expanding/contracting by a different amount after rotating through 90 degrees can give us an arrow pointing in any direction in the east-west north-south plane. Whereas combining 'x' and 'x-' can only ever result in arrows that point one way or the other along the east-west line, 'x' and 'x-' and 'xi' can be combined to give us any general rotation in the whole plane.

It turns out that the famous exponential number, 'e', repeatedly multiplied by itself A times with our 90 degree turn each time - which we can write as exp(i x A} - gives us an arrow that has not expanded or contracted but has simply rotated by the angle A. In fact we have to express A not in degrees but in radians - where one radian is about 57 degrees - a 180 degree turn is 3.1.4159... = pi radians and so this gives us the equation they described as the most famous of all in mathematics

exp{i x pi} = -1.

We can easily see now why i is so useful. Anytime we want to describe something that changes its direction (and not just reverses direction) then the ideal operation to us is 'xi'. In the electricity example they kept quoting, the alternating current of our electricity supply originates in the turning of a dynamo, one complete turn gives us one complete cycle and if the rotor has turned through an angel of A radians then we model this using the factor exp{i x A}.

However, as well as well as being immensly useful for describing anything turning or spinning, we now know that such spinning lies at the very heart of our best explanation of the world:-

Quantum theory says that all particles carry with them a little abstract spinning arrow that turns at a rate related to their energy, E, such that in a given time, t, this arrow will spin through an angle A where A = E x t /hbar {where hbar is a feature of the universe, Planck's constant h divided by 2 x pi}. Thus when we describe a particle's state we must include a factor exp{i x E x t/hbar}.

The quantum world can be broadly summed up by saying that any particle can move from a to b by any path; along each different path its spinning arrow will have turned by a different amount when it gets to b (according to how the energy varied along the path); however we then have to add up all these different arrows from all these different paths nose to tail, in all these different directions and the resulting length of the overall arrow from the first tail to the final tip (doesn't matter what order we chose to add them, the end result will be the same), gives us the probablity that the particle is now at b. If all these arrows point in roughly the same direction then the final point will be a long way from the first tail and we get a large overall arrow and hence a large probablility that the particle will be at b; however if the angles turned along the different paths means that the final arrow point has come back around to be close to the starting tail, then the overall arrow is quite short and hence the probability is very low. If the paths differ by angles such that in the end they all come to lie on a big circle and we get back to the exact position the first arrow started from, then the particle has no probability of being at b.

It turns out that for large systems of particles such as any macroscopic object, where the energy is such that E x t is enormously bigger than hbar, then most the arrows spin so fast that most collections of paths turn many circles a huge number of times and the overall added up arrows never get far away from the first tail. Only at a very special position b do the arrows all line up away from the first tail and this become the only position at which we get any chance of seeing the object arriving - the conditions for this lining up turns out to be just those that pick out the paths of classical physics. This is why large objects obey Newtons laws. When we go the microscopic world our little arrows spin much slower and we start to get to approximate lining at lots of different b positions and we find this, to us used only to just the large object paths, much 'wierder' world of microscopic objects.

Without 'i', the mathematical descriptions of this real microscopic world would be extremely cumbersome.

Thus 'i' is relatively easily understood once viewed as changing the nature of the operation rather than changing the nature of the numbers themselves, and as such is of immense importance in the real world of engineering etc and also in the fundamental workings of the universe.

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Imaginary Numbers

Ahoy! At last a start on getting us ready for the multidimensional universe. Although the other shoe was not dropped in this discussion on Imaginary Numbers there was a strong hint that these values ARE real, but do not fit into our one dimensional arithmetic and 2 or 3 dimensional geometry because these are not as real as we assume they are. As soon as we go to 2 dimensions we are in trouble with ordinary numbers and fractions, as pointed out by the team, when we try to numerate something as simple as the length of the diagonal of a square of unitary side. The same problem arises with Pi, for the same reason. The Right Angle defines the relationship between the first 2 dimensions, just as it does with all the others, and infinities threaten unless curved circumscription is engaged.

2 dimensions can be drawn, 3 dimensions can be modelled in what we call reality or simulated in two dimensions thanks to the human brain, but after that most people cannot 'get their heads round' even a mental model. The drawings and models commonly used to explain more than 3D are 'pseudo-geometry' attempts to show in frozen form that which is, in reality, dynamic. Reality is multidimensional but we are part of the self-observing reality, which limits our perception and vocabulary. We can only use the products of the numbers from a reality beyond our normal perception. However, the Imaginary Numbers contain the clue to the nature of space-time. Those of you with the patience to listen to Richard Feynman's glorious failure to 'explain' magnetism [http://www.bbc.co.uk/archive/feynman/10701.shtml] can take heart that his brilliant, honest non-explanation is closely related to today's partial explanation of imaginary numbers, and to the quantum/relativity paradoxes, and all these problems can be clarified, probably quite soon. This will show us, as if we needed reminding, that having such knowledge will not alter the fundamental problems in managing human affairs, just add to extent of the adventures we can have and the mistakes we can make.

There are those who think that if and when we discover there is intelligent life elsewhere in the universe that this will change 'everything' in our lives here. As someone who has always opined that life elsewhere in the universe is an absolute certainty, I disagree. If evidence that intelligent life had existed elsewhere were to be announced with complete certainty tomorrow, it might not make the slightest difference to any social, economic, political, scientific or religious thinking here that could change anything as a result. Just something more to argue about.

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Wonderful programme - I bored my girlfriend with it on the way to work!

OK it could have been a bit better explained ... but full marks for tackling an important mathematical concept. Although I detect from some comments that some may have had their subjective sensibilities challenged a bit.... Just what IOT is for !

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It was a superb piece of radio. Although I didn't understand it completely I definitely glearnt a lot ,and it was so good to have a presenter who did us listeners the credit of assuming that if he could follow it so could we . Concept after concept were presented and explained lucidly by people who really owned their knowledge and thus were able to convey it to a lay audience. It put many of the previous contributors to shame. It was amazing how real expertise seemed to create so much time to work in. I am sure Mr. Brag and his guests new that they had done well as soon as they had finished, My wife and I burst into spontaneous applause at the sheer pleasure of hearing proper grown up radio

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I found this bewildering subject quite fascinating. The enthusiasm and enjoyment that Mathematics evokes in those who can handle it is such a pleasure to listen to. I found it very hard to really understand what I was hearing but an interesting spiritual parallel came to me while listening. If scientists find it possible to conceive of and use quite happily the imaginary numbers referred to as "i" in their calculations, because they manifestly solve real problems and allow us to understand complex invisible phenomena such as sound waves and radio signals and other dynamic systems in everyday life why do some scientists find God a difficult concept to accept? An invisible force, which, when applied to life's equations, manifestly solves the problems of human dynamics.

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Thanks to Displacement Act for the details of how complex numbers help us in everyday life; very interesting. I noticed that Melvyn did try, several times, to extract these ideas from the professors. As you say, not a good idea to have them all together, perhaps. I felt I needed an interpreter.

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DisplacementAct wrote "In fact, it's probably fair to say that almost none of the technology developed since the 40's would exist today if we didn't have an understanding of complex numbers."

I for one do not agree that it is fair to say that.

When John Bardeen, Walter Brattain and William Shockley developed the first transistor in 1947 they did not need complex numbers to design their device - and nearly all semi-conductor devices evolved from that first breakthrough. Probably the majority of people who are designing state of the art technology today dont use complex numbers(although they might understand them), whereas those who are designing the data transmission systems which link them together do.

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In Our Time - you're back. I've missed you. My morning cuppa was bliss. Euler's equation - aaahhhhh.

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Imaginary numbers are not the only imaginary concept we use to advantage. In the field of structural mechanics we can use imaginary forces and stresses to find out about real displacements and strains and imaginary displacements and strains to find out about real forces and stresses. These are all linked through a concept known a virtual (imaginary) work.

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Bragg: "Do imaginary numbers mean what you want them to mean?" Huh? That has to be the single worst question ever asked on this program. I'm surprised that Melvyn Bragg didn't do a little more research before hosting this program. Either that or his ability to comprehend mathematics is on a grade-school level.

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I partly agree with angryloner, i.e. I feel this wasn't well explained. The problem is that with all scientific editions of IOT they are trying to explain to too wide an audience. The attempts to explain to people who haven't a hope of understanding it, and probably aren't listening anyway gets in the way of targeting the material clearly to the more intelligent and educated audience.

Actually I think that M Du Sautoy is a good communicator and I enjoy his programmes. I feel that Ian Stewart is the worse offender in trying to explain in baby talk.

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I listen to this programme because it is challenging - unlike most of the other programmes - and because it tries to stretch our minds. The programme on imaginary numbers had my full attention, and I did follow what was being said, without having a degree in mathematics.

Please keep going. I've given up on most of BBC4 because it is facile and trite. In our time podcasts are among few things I still listen to.

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Knowing how little you know about something is the first step in the acquisition of knowledge.

I listened mostly in complete bewilderment to the first in a new series of Melvyn Bragg’s In Our Time on Radio 4 this morning. The discussion programme’s subject was imaginary numbers. It wasn’t so much highbrow as furrowed brow. I was lost within seconds but I stayed on for the full 45 minutes. Such is Bragg’s skill as a guide.

Underlying his work – and here I include his much missed television arts programme The South Bank Show – is the presumption his audience enjoy having their brains stretched.

This is old school. Addressing the contemporary life of their times George Bernard Shaw, Bertrand Russell, H. G. Wells, J.B. Priestley, and George Orwell didn’t temper their prose along class/education lines. Neither does Bragg.

There are many fine television documentaries to be found in our multi-channel universe but entertainment will always triumph over intellectual challenge. Where are today’s equivalent’s of Kenneth Clark’s Civilisation or Jacob Bronowski’s The Ascent of Man?

Men and women of letters do continue to campaign against injustices – Harold Pinter maintained a sense of outrage throughout his life.

Comparisons are odious but it is a fair bet that in future histories of English literature Pinter would be written the larger. I’ve seen Pinter’s plays but not read a single Bragg novel. But I believe the latter’s life has made the greater contribution to exercising the intellects of ordinary men and women in today’s Britain.

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All numbers are imaginary

and too many words are specious

Most people are funny some of the time

Radio is precious

If you were turned off maths

by arithmetic so vicious

don't put it down to the programmers

Blame the facetious teachers

And next time book an engineer

to tease out useful features

Like tensors, quaternions and magnetic fields

concepts are capricious

Sorry about the bad poetry, there's a Grouse in the house.

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I especially liked the idea that imaginary numbers make calculations easier, and that mathematicians move in and out of an imaginary world. Unfortunately, no one said when and why it was decided that a multiplier could be negative and that multiplying two minuses gives a plus. I would welcome any information on this subject.

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