Question #3 about Hardy's A Course of Pure Mathematics—complex number vs pair of real numbers

[MaxMath]

I'm intrigued by this paragraph of this book of Hardy (page 113 of pdf). Before this point of discussing complex numbers in a way familiar to most people, Hardy made a great effort to discuss "displacement", or a pair of real numbers. Particularly, he patiently showed the reader what the best way is to define the product of two displacements (or pairs of numbers). After lengthy discussions, Hardy now says—a complex number, in the form of x+yi, is merely a "technical" form of expressing a pair of two numbers. He also says—a complex number, or an imaginary number, is by no means less 'real' than an ordinary real number.

This may not be something revolutionary. But it is indeed very different from how complex number was introduced to me. While I hope I can find some clue when I read on, I really want to hear from people who study math as their major, or who are actually mathematicians, about why so much stress needs to be put on the fact that a complex number is only a pair of numbers. Why is this so important? He also seems to suggest that the word "imaginary" is probably not a very good term (and likewise "real").

 

[Steven G]

There is an obvious 1 to 1 correspondence between a pair of real numbers and a complex number.

(a,b)<=>a+bi

 

[MaxMath]

There is an obvious 1 to 1 correspondence between a pair of real numbers and a complex number.

(a,b)<=>a+bi

Indeed, that's very very obvious. But I'm really interested in why, according to Hardy, we should think of a complex number x+yi in terms of a pair of ordinary numbers, rather than a combination of a real part and an imaginary part, as the way it's normally introduced (at least the way I was taught). My question might be vague or somehow too big. But I want to see if someone deep in math can immediately point out something new to me.

In other words, I don't understand why Hardy chose to introduce complex numbers that way, and why he needed to take that much effort in choosing a 'proper' definition of the product of two displacements.

But thank you anyway!

 

[topsquark]

Indeed, that's very very obvious. But I'm really interested in why, according to Hardy, we should think of a complex number x+yi in terms of a pair of ordinary numbers, rather than a combination of a real part and an imaginary part, as the way it's normally introduced (at least the way I was taught). My question might be vague or somehow too big. But I want to see if someone deep in math can immediately point out something new to me.

In other words, I don't understand why Hardy chose to introduce complex numbers that way, and why he needed to take that much effort in choosing a 'proper' definition of the product of two displacements.

But thank you anyway!

He's not saying that you should or shouldn't, he's simply saying that you can. As long as you define
[imath](a,b) \oplus (c,d) = (a + c, b + d)[/imath]

and
[imath](a,b) \otimes (c,d) = (ac - bd, ad + bc)[/imath]
then you have two systems that behave indentically. There may be cases where you would want to use the "pair" definition over the a+ ib version, though I can't think of any Physical reasons.

These are called "representations" of the complex number system. There are more.

-Dan

 

[Dr.Peterson]

Indeed, that's very very obvious. But I'm really interested in why, according to Hardy, we should think of a complex number x+yi in terms of a pair of ordinary numbers, rather than a combination of a real part and an imaginary part, as the way it's normally introduced (at least the way I was taught). My question might be vague or somehow too big. But I want to see if someone deep in math can immediately point out something new to me.

In other words, I don't understand why Hardy chose to introduce complex numbers that way, and why he needed to take that much effort in choosing a 'proper' definition of the product of two displacements.

As he says, his goal is to eliminate the bias inherent in calling the numbers "imaginary". They were originally invented/discovered (and are commonly taught) as a sort of "what if": What if there were a number whose square is negative? As a result, these numbers seemed like a figment of mathematicians' imagination (though in a sense all of mathematics could be described that way). Defining them as pairs of real numbers with appropriate operations makes them concrete; we've actually constructed them starting with numbers we are familiar with, rather than just pretending they exist.

It happens that I went over this history in my article How Imaginary Numbers Became “Real”, where the definition you are asking about appears about halfway down.

 

[Steven G]

Indeed, that's very very obvious. But I'm really interested in why, according to Hardy, we should think of a complex number x+yi in terms of a pair of ordinary numbers, rather than a combination of a real part and an imaginary part, as the way it's normally introduced (at least the way I was taught).
In other words, I don't understand why Hardy chose to introduce complex numbers that way, and why he needed to take that much effort in choosing a 'proper' definition of the product of two displacements.

But thank you anyway!

The real answer for you is because he could (after all, there is that 1-1 correspondence). I guess that he doesn't like "imaginary" numbers. Complex Analysis has grown significantly since Hardy made this statement and possibly today he wouldn't make that statement

 

[blamocur]

Another $0.02 worth: one way to define new fields is by defining extensions of other fields. An extension is defined as a factor of all polynomial over some irreducible polynomial, which in the case of real numbers can be [imath]x^2+1[/imath]. The field of complex numbers is then a field of all (in practice only 1st degree) polynomials modulo [imath]x^2+1[/imath]. As an exercise one can try finding the remainder of the division of [imath](ax+b)(cx+d)[/imath] by [imath]x^2+1[/imath].

In a similar manner one can extend the field [imath]\mathbb Q[/imath] of all rational numbers by using irreducible (over [imath]\mathbb Q[/imath]) polynomial [imath]x^2-2[/imath]. This way one gets a field of algebraic numbers of the form [imath]u+v\sqrt{2}[/imath], where [imath]u,v[/imath] are rational numbers.

 

[khansaheb]

In electrical engineering, there are representations which make use of "phasors" .

The complex numbers can be graphically represented as pseudo-vectors. Division is defined in complex algebra - not so for "vectors".

 

[MaxMath]

Thanks for all the insights. These are the kind of things that I was after ...

He's not saying that you should or shouldn't, he's simply saying that you can. As long as you define
[imath](a,b) \oplus (c,d) = (a + c, b + d)[/imath]

and
[imath](a,b) \otimes (c,d) = (ac - bd, ad + bc)[/imath]
then you have two systems that behave indentically. There may be cases where you would want to use the "pair" definition over the a+ ib version, though I can't think of any Physical reasons.

These are called "representations" of the complex number system. There are more.

-Dan

Sure, he's not saying we should or should not. It's just my curiosity from sensing his tone and his taking the effort to stress something seemingly trivial or 'non-substantive'.

As he says, his goal is to eliminate the bias inherent in calling the numbers "imaginary". They were originally invented/discovered (and are commonly taught) as a sort of "what if": What if there were a number whose square is negative? As a result, these numbers seemed like a figment of mathematicians' imagination (though in a sense all of mathematics could be described that way). Defining them as pairs of real numbers with appropriate operations makes them concrete; we've actually constructed them starting with numbers we are familiar with, rather than just pretending they exist.

It happens that I went over this history in my article How Imaginary Numbers Became “Real”, where the definition you are asking about appears about halfway down.

Good to hear you went over this history and wrote an article that is related. I'm interested in math history and keen to understand what triggered the new ideas and new concepts, which I think are of utmost importance to truly understand them and, perhaps, to possibly think of different ways at those 'junctions' in the history which might have led down to different paths.

The real answer for you is because he could (after all, there is that 1-1 correspondence). I guess that he doesn't like "imaginary" numbers. Complex Analysis has grown significantly since Hardy made this statement and possibly today he wouldn't make that statement

Yes, clearly he didn't like the term "imaginary" and he probably had good reasons as a mathematician who was able to look into it deeper than most of us.

Another $0.02 worth: one way to define new fields is by defining extensions of other fields. An extension is defined as a factor of all polynomial over some irreducible polynomial, which in the case of real numbers can be [imath]x^2+1[/imath]. The field of complex numbers is then a field of all (in practice only 1st degree) polynomials modulo [imath]x^2+1[/imath]. As an exercise one can try finding the remainder of the division of [imath](ax+b)(cx+d)[/imath] by [imath]x^2+1[/imath].

In a similar manner one can extend the field [imath]\mathbb Q[/imath] of all rational numbers by using irreducible (over [imath]\mathbb Q[/imath]) polynomial [imath]x^2-2[/imath]. This way one gets a field of algebraic numbers of the form [imath]u+v\sqrt{2}[/imath], where [imath]u,v[/imath] are rational numbers.

This is a very interesting perspective and a nice analogy/example.

In electrical engineering, there are representations which make use of "phasors" .

The complex numbers can be graphically represented as pseudo-vectors. Division is defined in complex algebra - not so for "vectors".

Yes, there is such a thing as "phasor" in the electrical and electronics field. And to my limited knowledge about complex numbers, indeed that is one fascinating application use of complex numbers. Just to use loose language, the use of it unable us to describe AC circuits and their behaviour as if it was DC!