While reading in the magnificent book "Princeton Companion to Mathematics" I tend to ponder on the ideas lurking within. I was reading about solving polynomial equations (I.4 §1.2) and the author introduced square root of two in a rather intriguing way. What caught my attention is not only that we needed to prove that it exists (by the intermediate value theorem), but also that it is defined by the sole property that is the number that squares to two (if it exists) and cannot be defined by other means. He intended to highlight that numbers (or solutions) are opaque and all we care about are their properties. More on that in the next paragraph.
Extrapolating on this line of thought, I ended up thinking about the fraction 2/3. It is, in fact, defined to be the number that, when multiplied by 3, gives you 2. That number doesn't exist in the set of integers, much as the same way square root of two doesn't exist in the set of rational numbers. Now think about the number 4/6. It is the number that when multiplied by 6 gives you 4. Although it might look intuitive to someone who went through elementary education, it is still an interesting thing to highlight: it turns out that both numbers, 4/6 and 2/3, are in fact the same number. What we have just stated in the last sentence is a property of both numbers, both of which being solutions to 3x = 2 and 6x = 4, respectively. But behind that, it is actually a property of the solution of the two equations. It shows that their solutions are the same number, even though these solutions (2/3 and 4/6) might look syntactically different. Put in more accurate words: they define the same number.
What is intriguing about this is that the number i (the square root of negative one) is no more artificial than √2. It is just a number, that similarly to √2, which does not exist in the set of rational numbers, is a number that doesn't exist in the set of real numbers. Moreover, as being the square root of two (and nothing more) is the defining property of √2, and as being the number that multiplied by 3n gives 2n (for positive integer n) is the defining property of the number 2/3, so is being the square root of negative one is the defining property of i. All further properties could be derived from this property alone. This process is abstract in nature; we may not know what the square root of two is. For instance, did you even need to know that √2 = 1.414... in order to perform calculations in high school ? No. All you did need to know was that it squared to two. Quick, what is the solution to 3 + √341 x = 344 ? Simply enough, it is x = √341. Did you need to know the value of √341 to do that ? (I hope not!) Actually, if you've dealt with the value instead, things would have been more confusing and uglier.
The study of this kind of properties of numbers which are defined this way is heart to Abstract Algebra. In fact this is exactly why it is called "Abstract" and "Algebra". Abstract Algebra may look like this: if we take the field Q (the field of all rationals) and adjoin the number √2 to it (in a process called "field extension"), we end up with the new field Q(√2) which have different structure, fully determined by the properties of √2.
Extrapolating on this line of thought, I ended up thinking about the fraction 2/3. It is, in fact, defined to be the number that, when multiplied by 3, gives you 2. That number doesn't exist in the set of integers, much as the same way square root of two doesn't exist in the set of rational numbers. Now think about the number 4/6. It is the number that when multiplied by 6 gives you 4. Although it might look intuitive to someone who went through elementary education, it is still an interesting thing to highlight: it turns out that both numbers, 4/6 and 2/3, are in fact the same number. What we have just stated in the last sentence is a property of both numbers, both of which being solutions to 3x = 2 and 6x = 4, respectively. But behind that, it is actually a property of the solution of the two equations. It shows that their solutions are the same number, even though these solutions (2/3 and 4/6) might look syntactically different. Put in more accurate words: they define the same number.
What is intriguing about this is that the number i (the square root of negative one) is no more artificial than √2. It is just a number, that similarly to √2, which does not exist in the set of rational numbers, is a number that doesn't exist in the set of real numbers. Moreover, as being the square root of two (and nothing more) is the defining property of √2, and as being the number that multiplied by 3n gives 2n (for positive integer n) is the defining property of the number 2/3, so is being the square root of negative one is the defining property of i. All further properties could be derived from this property alone. This process is abstract in nature; we may not know what the square root of two is. For instance, did you even need to know that √2 = 1.414... in order to perform calculations in high school ? No. All you did need to know was that it squared to two. Quick, what is the solution to 3 + √341 x = 344 ? Simply enough, it is x = √341. Did you need to know the value of √341 to do that ? (I hope not!) Actually, if you've dealt with the value instead, things would have been more confusing and uglier.
The study of this kind of properties of numbers which are defined this way is heart to Abstract Algebra. In fact this is exactly why it is called "Abstract" and "Algebra". Abstract Algebra may look like this: if we take the field Q (the field of all rationals) and adjoin the number √2 to it (in a process called "field extension"), we end up with the new field Q(√2) which have different structure, fully determined by the properties of √2.
2 comments:
it's interesting that the first example in the book is also the first question -topic- presented to me as a student in faculty of science in an a mathematical analysis tutorial. I'm confused now because i know that we dealt with abstract definitions of numbers during our study so we never considered their value. However, i feel that i haven't been thinking of the definition of irrational or imaginary - i know you hate these 2 definitions :D - in the same way you proposed.
I think my article is not about something "new" that we haven't been doing before. But I think the real value in what I said is that I merely made it explicit and brought it into our conscious minds. It is simply a different way to look at it.
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