Actually, never mind. I was being dumb.
That was the equation that hooked me on math.
Another cool quirk: What is i^i (i raised to the i power)?
Hint: it’s a real number.
Further, the complex numbers are complete, in a sense that most of the other categories of numbers you know of aren’t. Think, for instance, of the natural numbers (the nonnegative integers). If you add two natural numbers together, you always get another natural number, and if you multiply two natural numbers together, you also always get another natural number. Well, so far so good, and that’s a rather nifty property to have.
But what if you subtract them? Well, sometimes you still get a natural number, like 10 - 7 = 3. But sometimes you can’t get a natural number, like 2 - 5. So if we want to be able to always subtract numbers, we have to extend our numbers to include a new kind of numbers, and so we have the integers. Whenever you add, subtract, or multiply two integers, you always get another integer, and that’s even niftier than the natural numbers.
But we still can’t always divide them: Some divisions of integers, like 12 ÷ 3 = 4, give us other integers, but others, like 5 ÷ 7, don’t. So now we have to extend our numbers again, to the rational numbers. Take two rational numbers, and add, subtract, multiply, or divide them, and you’ll always get another rational number.
…but we can’t always take square roots. For that, we have to first of all include at least some irrational numbers, for the likes of sqrt(2). Technically, we only need a minuscule proportion of the irrational numbers for square roots. But we could also ask for certain properties of limits, and then we need all the real numbers.
But even with all the real numbers, we still can’t deal with all square roots. So now we need the complex numbers.
But what can we do to the complex numbers? Amazingly, even though complex numbers were invented to deal with one particular sort of thing we couldn’t do, square roots of negative numbers, they also turn out to deal with everything else we can think of: You can take square (or cube or any other) roots of complex numbers, you can take them to any complex power, you can take trig functions or inverse trig functions of them, or logarithms, or whatever else you can think of, and you still always get another complex number.
Well, OK, sometimes you get a set of complex numbers. But that’s nothing new; it already happens with square roots of integers (both 3 and -3 are square roots of 9, for instance). And, OK, you still can’t divide by zero. That, you deal with by changing your expectations of what a number system ought to do, not by changing your number system.
But you can do anything else. Inverse sine of 2? Log of -1? i^i? Not a problem, any of it.
Another pretty simple argument for e^iπ = -1:
One definition of e is that it’s the number such that e = (1 + 1/n)^n for large n. And from there it’s pretty easy to work out that e^x = (1 + x/n)^n for large n.
So e^iπ = (1 + iπ/n)(1 + iπ/n)(1 + iπ/n) … n times. We know from the basic properties of complex numbers that multiplying them together adds their angles and multiplies their magnitudes. The magnitude of each component is very nearly 1, and so multiplying them together is also 1. The angle is π/n, and n copies of that leads to a final angle of π radians.
What’s the complex number with an angle of π radians and magnitude of 1? Why, it’s -1 of course.
I think that’s just a historical quirk. One could have a more complete system by including the multiplication inverse of zero.
Please expand on this interesting idea.
Something like:
x / 0 = ∞
x / ∞ = 0
x · 0 = 0
x · ∞ = ∞
x + 0 = x
x + ∞ = ∞
x − 0 = x
x − ∞ = -∞
0 / ∞ = 0
∞ / 0 = ∞
0 · ∞ is undefined
0 + ∞ = ∞
0 − ∞ = -∞
0 / 0 is undefined
∞ / ∞ is undefined
0 · 0 = 0
∞ · ∞ = ∞
0 + 0 = 0
∞ + ∞ = ∞
0 − 0 = 0
∞ − ∞ is undefined
if zero times infinity is undefined, x/0 must also be undefined.
If “undefined” is a possible answer, then your number system isn’t complete in the sense that you’re using it. So just adding a single infinity doesn’t seem to help much.
Various hyperreal number systems do allow operations like ∞ · 1/∞ to give sensible results, but they still don’t allow dividing by zero (although they do allow dividing by a number that “rounds” to zero when converted to a real number).
The trouble with this is that you cannot perform any algebraic manipulation on any of these expressions without trivially allowing contradictions or nonsensical results to emerge. Infinity does not equal infinity in any useful sense here.
Undefined is what it says. There is no defined value. Which means in many cases you can choose any value, including infinity. It becomes possible to prove black is white with fatal consequences at pedestrian crossings.
Can you explain more?
I did say “more complete”. It helps some. Something similar is commonly implemented on computers because it’s useful.
Please show an example.
That’s not quite true. Pleonast is basically describing the projectively extended real line. If you follow the rules, it gives either sensible or undefined results, just like the real numbers. And it does have some uses. It just doesn’t achieve completeness in that every division between two numbers in the system gives another number in the system. And in fact it adds some undefined cases, such as that the system is no longer complete under subtraction (unlike the reals, rationals, integers, etc.).
As far as I know, there isn’t actually any number system that fully solves the division by zero problem, even the surreal numbers.
IEEE floating point representation includes a positive and negative infinity. It also contains an undefined number, called NaN for “not a number”. And yes, it is useful, in particular because if any bad operation sneaks into your calculation, every operation downstream will come out as NaN. You know that the calculation went wrong but didn’t have to error-check each step of the way.
Numbers of the form eix are not supposed to form a field like the set of all real numbers or all complex numbers. What they form (under multiplication) is a group. So there is no “division by zero problem”, not that there is anyway in a field since one simply cannot divide by zero, not sure who ever claimed it was supposed to be possible.
It is true that on a standard-conforming (IEEE 754) computer, the default result of an operation like +1.0/+0.0 will be +infinity, but you can interpret that as a type of overflow rather than meaning that it makes sense to divide real numbers by zero.
That’s just another way of saying that the complex numbers under addition is a group, since the exponential converts addition to multiplication. It’s also another way of saying that the complex numbers without zero is a group. All elements of a group have an inverse, so it can’t include zero, but all other axioms are satisfied. e^x is always non-zero of course, so one can just see it as a strange way of excluding zero from your set.
I don’t think anyone said that division by zero must be possible, but it is awkward and seemingly asymmetrical that we have to carve out an exception for zero. There are ways to include a multiplicative inverse of zero, but then it causes other problems.
Chiming in to agree that thinking of i as a ‘sideways’ number is much better than calling it ‘imaginary’. (Maybe someone can come up with a latin word that more or less means sideways but begins with i, so we can pretend that’s what the i is for?).
Just like -1 is a ‘backwards’ number, and just as imaginary or legitimate as i. After all, you can’t have -1 apples in your hand any more than you can have i apples. Negative numbers only make sense when you think of them as having a direction (backwards), and complex numbers only make sense when you think of them as having a direction that includes a sideways part as well as a forward-back part.
You can extend the real numbers e.g. to a “wheel” where division by any element is no problem. Note that 1/0 and 0/0 are distinct elements in the resulting algebraic structure. We could think of them as ∞ and NaN, perhaps. This is not at all nonsensical, but for sure there are problems like that x – x is not equal to 0 in general.
Groups like the non-zero complex numbers are usefully thought of as Lie groups, from which point of view the complex numbers under addition can be identified as its Lie algebra.
But it is still fair to say that complex numbers are, in and of themselves, for real legitimate numbers that need not be thought of merely as pairs of real numbers.