Okay, so why do engineers use j instead of i for the square root of -1?
Clearly, they’re operating in a different subfield of H from the rest of us.
Thanks for all the responses, folks! If I seem slow, keep in mind I’m taking time to digest.
We invented the symbology, the grammar, syntax, perhaps most of the logic–and I realize there are valid number systems which are complete inventions–but all of them? This doesn’t seem right at first glance. The numbers (and at least some of the rules that govern them) are there whether we observe them or not. Two apples is always one more than one apple. 7 was prime before we even showed up, right? It just is. So at the very least…we didn’t really invent all of the natural number system, did we?
Sure, but that’s not the same as saying it isn’t there. Again, 7 is prime. But it’s also 7 + 0i, right? They’re the same number.
Sorry, yes: familiar numbers == real numbers. Positive, negative, rational, irrational, fractional, and zero.
Yes, I’m talking about quantities. Not just counting numbers–to me, zero is a quantity. -1 is a quantity. You might not be able to see them like you can see 1, but they’re definitely implied by things you can see, touch, count, and measure. Quantities, yes–but am I missing something? Isn’t a real number fundamentally a quantity? Certainly I understand that we have numbers that are symbols and not acutal quantities, and that those have valid uses and can be used to describe or measure the world. But all numbers? Aren’t there some numbers that were already there when we invented the symbols for them?
I guess that’s what I was going for…the first sentence. There’s no way to visualize i like we can visulize 2. You couldn’t take a picture of it, or even describe it generally as a vanishingly small or incomprehensibly large amount. And you can’t represent a real quantity uniquely and completely as a non-zero complex number?
Yes, this is what I was thinking about. Restricting the discussion to algebraic operations. But you single out “rooting” as the operative basis for imaginary numbers. It’s only an unclosed operation if we need every number to have a root, right? At one point, we decided every number has to have a root. It’s easy to see that the square root of 4 is two. It’s more difficult to picture the square root of two, but you can do it. You can understand it as a quantity, even if it’s not rational.
Consider the primes again. In the beginning, when there were only positive whole numbers, we knew about the primes. We knew every number has two trivial factors–itself and one. And we knew that some numbers had other factors, and some didn’t, and we called them composite and prime. We also knew that some numbers had what we would later call perfect roots and some didn’t. 36 is a square number; it has a whole number as its root. 37 doesn’t.
We then expanded our number system to include negative numbers. But we restricted the qualifier “prime” to apply to only positive whole numbers. -7 isn’t prime, but it does only have two positive divisors. But we didn’t make a similar distinction about negative numbers not being able to have roots. We left that door open, and so we had to close it, with i. Why did we do that?
Think about it another way. 7 has no perfect square root, right? But it does have a square root. But it’s only a root because we call it a root, because we make that distinction between perfect roots and other roots. In fact, it’s a number that, when multiplied by itself equals seven…which by that definition, should make it a factor of seven as well. But we don’t make a distinction between perfect factors and other factors–a factor by definition is a whole number.
Does this make any sense? At the heart of my question is where, exactly, did we make the conceptual leap from real-world-quantity to conceptual symbol? When did we start inventing numbers because the systems we’d constructed weren’t closed?
In the sense that 2 + 0i apples represents two apples. My question is, can you show me 2 + i apples?
Equal seems like a much more significant condition than identical. I don’t understand the technical reasons, though. Would |-2| be identical to 2?
You lost me there. If x is a real number >= 1 (not saying that you meant that), wouldn’t it have two roots in R?
By “roots of x[sup]2[/sup] + 1” he meant “values of x for which x[sup]2[/sup] + 1 = 0” (which would mean x[sup]2[/sup] = -1).
It is. I don’t know why this looks better to me:
e[sup]iπ[/sup] = -1
…Maybe I’m a glass-is-double-half-empty kinda guy. I don’t know what this is saying about i, though. If e is about 2.7 and pi is about 3.14, then i has to be a pretty wacky number to make that come out to -1!
No, but that doesn’t mean that imaginary numbers are useless for modeling physical phenomena. cf. the responses from electrical engineering-types you’ve gotten here.
I don’t even want to get into expressions here, so let me just say that the value of |-2| is identical to 2 iff the 2s in each expression are of the same type.
7 is a prime in the integers. 7 is not prime in the complex numbers.
No.
Got it. But:
x[sup]2[/sup] + 1 = 0
x[sup]2[/sup] = -1
x = sqrt(-1)
…so x = i, right? How is that bringing anything new to the table, because i isn’t in **R[/R] to begin with. I get that we can solve for x over C but not R…
Of course not, we already know how useful they are–the question is, does it mean it’s useless to try to visualize it as a quantity? The answer seems to be yes.
:eek:
Why? Because it has other complex factors?
Can you quantify the negatives? In other words show me a negative apple. In your example of 2 is three less than 5 exemplifying -3, please pick up one of the -3 apples. As pointed out by others, i is used in electrical engineering.
This can be generalized to any extension field of the reals. For example the quaterions, R^n, etc.
No. See number 2
As others have explained, none.
Nope. The complex numbers are the algebraic closure of the reals, thus anything involving polynomials need complex numbers
If you really want to visualize imaginary numbers, you might get something out of this book: Imagining Numbers (particularly the square root of minus fifteen). (Personally, I’d rate it so-so as popular math books go: not bad, but not great. You can see it got mixed reviews on Amazon.com.)
Yes. It turns out that a real integral prime is a complex prime if and only if it is of the form 4n+1. So, 3 and 5 are real and complex primes but 2 and 7 are not prime when considered over the complex integers. 2 = (1+i)(1-i) and 7 = (4+3i)(4-3i).
Of course that’s Z[[symbol]i[/symbol]] rather than C. 7’s not prime in C is that it has a reciprocal, and numbers with reciprocals aren’t prime.
I think you have your examples wrong. (4+3i)(4-3i)=25, not 7; and the complex primes with zero imaginary part are those whose magnitudes have the form 4n+3 (these being the numbers which cannot be written as the sum of two integer squares). 5=(2+i)(2-i), for example.
Auggh!
You are right. Also I said 3 is not a complex prime. Actually it is since 3 = (2+i)(2-i) = (1+2i)(1-2i)
Or, to take a historical approach to it: What number system you need depends on what you’re using it for. If, like the earliest mathematicians, your interest is sheep, you need only the counting numbers. You can’t have a fractional sheep (well, you can, but it won’t do much good), nor a negative sheep, nor an imaginary sheep. So for a shepherd, the counting numbers are plenty.
Now, advance civilization a millenium or so, and folks are starting to settle down and grow crops. The counting numbers are still useful, as you can have three bushels of wheat, or four bushels, or seventeen bushels. But you can also have three and a half bushels, or seventeen and a third bushels. So now you need fractions, too. But it still doesn’t make sense to talk about negative or imaginary bushels of wheat.
Make a little more progress, and you get to money. You can have an integer number of dollars, and you can have a fractional amount of dollars. But now we have something new: You can have negative dollars, that is to say, be in debt. So now we have to extend our number system yet again. When you have money, you need fractions and negatives. But you still can’t have imaginary dollars (or if you do, some folks from the Secret Service would like to have a chat with you).
But now suppose that you start developing inductive circuits, or quantum physics. You’ll find that now, in addition to a positive or negative and possible fractional wavefunction, you can also have an imaginary wavefunction. So for quantum physics, you need the full set of complex numbers.
But in all this, we can suppose that you get tired of this whole business, and retire to your sheep ranch. You still only need the positive integers to count your sheep.
Sequent writes:
> Does this make any sense? At the heart of my question is where, exactly, did
> we make the conceptual leap from real-world-quantity to conceptual symbol?
> When did we start inventing numbers because the systems we’d constructed
> weren’t closed?
Well, if you take the view that the natural numbers are given to us by nature (i.e., are “real”) and everything else is just some system was have created, then we began inventing numbers at the point we created fractions. However, even the natural numbers are a mental construction in some sense. Consider the following scenario: You are new to the world and have no mental constructs to deal with anything, although you have all your senses. You suddenly appear in a room filled with many piles of objects. There are a number of ways of classifying all the piles so that you can say that some piles are like other piles. For instance, you could say that some piles are composed of just blue objects, some just of red objects, some of objects of mixed colors, etc. In some piles, the objects are very large, but in other piles the objects are very small. In some piles, the objects are roughly spherical, in other piles the objects are long and thin, in other piles the objects are a mixture of different shapes, etc. Some piles are tall and other piles are short. If you pick up the piles, you can feel that some are heavy and others are light. Some piles have very hard objects in them (like ball bearings), while other piles have soft, squishy objects in them (like grapes), and other piles are a mixture of objects.
The number of objects in a pile is just one more way to classify the kinds of piles. Some of the piles have one object in them, some have two objects, some have three, etc. I would say that the idea of “number” is not an obvious concept but a mental construct that we impose on the world. I would say that the natural numbers require a certain amount of mental manipulation of the world. Fractions require a certain further amount of mental manipulation. Real numbers require further manipulation. Complex numbers require still further manipulation and so on. The natural numbers are not “natural” and are not “given.”
i is used for electrical current.
To add another point of view:
Complex numbers are necessary to ‘close’ a system under exponentiation (that also is closed under subtraction and addition). But there’s no physical reason to have a closed system – it’s just that esthetically, mathemeticians like a theoretical symbol-manipulating system that’s closed. So mathemeticians defined complex numbers in a way that made everything pretty.
Complex numbers are not necessary for any physics engineering calculations or theories. Nobody can measure 6.2i units, therefore no physics model requires complex numbers to make predictions.
However, it turns out that complex numbers (as mathemeticians have defined them, again for mostly esthetical reasons) can be useful in calculating and writing many physical laws. In many cases, two different quantities change together in such a way that if you pick the right physical properties to associate with the ‘real’ and ‘imaginary’ parts of a complex variable, then you can simplify the equations and often the calculations.
But just like vectors, you can rewrite any set of physical equations that uses complex variables using only real variables; you’ll just need twice as many variables and at least twice as many equations.
Sequent,
I’m curious about your terminology in this thread. It seems to me that you’re using the expression complex to mean complicated, or at least more involved than the set of real numbers. The way I understand it, a complex number is one that has a real number part and an imaginary number part (remember that imaginary is just the name for it, it’s still really a number). That is to say, the expression complex number already has a particular meaning in math.