Slashdot Mirror
Professor Comes Up With a Way to Divide by Zero
54mc writes "The BBC reports that Dr. James Anderson, of the University of Reading, has finally conquered the problem of dividing by zero. His new number, which he calls "nullity" solves the 1200 year old problem that niether Newton nor Pythagoras could solve, the problem of zero to the zero power. Story features video (Real Player only) of Dr. Anderson explaining the "simple" concept."
The article and Slashdot's synopsis don't make note of it, but Dr. Anderson isn't claiming to have discovered something new in dividing any number other than zero by itself. The video linked in the article shows him saying that 1/0 = infinity, and -1/0 = -infinity, but 0/0 = capital phi (nullity -- we'll ignore the fact that this usually means the golden ratio in mathematics). Math isn't my area of study so I don't know why 0/0 specifically is so important... the article certainly is very much a fluff piece. Anyone feel like explaining the importance of 0/0?
No, it made sense when he wrote it. If there are zero comments, then there are zero posters. So that's 0/0.
Uh... are you joking?
Imaginary numbers (specifically, complex numbers, which consist of a sum of a real and an imaginary number, and which comprise the "complex plane") are INCREDIBLY important in the "real world."
I'm just a chemist, not a mathematician, but I am well aware that imaginary numbers are critical in the Fourier transforms used every time I take an IR or NMR spectrum.
Ever do electrical engineering? Circuit analysis is made a great deal easier when you can treat circuit elements in terms of complex numbers. All that "impedance" stuff you hear about capacitors and the like that makes it possible to apply Ohm's Law to LRC circuits.
These also are not merely made up properties, they are fundamental to mathematics and thus (if one believes that math is the language of the universe) physics. For example, certain integrals necessarily yield imaginary results. These integrals are not of some ethereal interest, but appear throughout quantum mechanics. This is why the amplitude of a wavefunction (used, for example, in molecular modeling that allows for practical achievements like better medicines) is not the square of the wave function (or, for that matter, its absolute value) but the product of the wavefunction and ITS COMPLEX CONJUGATE.
If you'd like more examples of the utility of complex numbers and other "random rules," check out Boas' "Mathematical Methods In The Physical Sciences."
I hate to put it this way, but "It'll make sense when you're older". And by older, I mean when you take a higher math course. What is the square root of -1 equal to then? Nothing? Something? Saying it's "imaginary" is merely a construct that allows us to muck with things. We could say they're "happy fun times" numbers, with the symbol "hft", and it'd mean the same thing.
At first, numbers were integers - what you could count on your fingers. (N) Later on, numbers were fractional - in order to express the sharing of things. (Q) Later on, numbers were negative - in order to express debt. (Z) Even later on, some numbers were found not to be fractionar (the first proved was square root of 2). Enter R However, not every polinomial equation has its solutions as real numbers (see x^2+1=0). The solution to this equation was named i, with the property that i squared is -1. It was called imaginary because no real number had such property, and it is as real as a figment of your imagination ;)
While other real numbers can be aproximated by integers, negative integers and fractional numbers (with better and better accuracy), i has no aproximation in any of the previous pools of numbers.
In engineering, a useful aproximation for pi is 3. There is no aproximation of i as an integer.
Infinity isn't a real number. Ergo, it cannot be the limit of a sequence, as the definition of a limit include the priviso that it is a real number.
You can only perform the substitution lim x->a f(x) = f(a) when f is continuous at a. f(x) = 1/x is (very trivially) not continous at a = 0.
Damnit, why is this sort of thing spilling over from sci.math now?
After all, I am strangely colored.
This fantastic new math is also helpful in solving this intractable problem: http://mcraefamily.com/MathHelp/JokeProofFactoring .htm
How cool is that?
Seriously, it's hard to take someone like this seriously when he uses ignorant scare tactics such as his autopilot example. Either he's performing self aggrandizing hand waving, or he really is completely ignorant about the real world. Trust me - we do account for division by zero in autopilot systems. And - believe it or not - not only does the computer not "stop working" but we actually get a result back. It's called NaN. Furthermore, not only are our systems built with robust libraries that allow us to carry on (no pun intended) we also write downstream code to mitigate propagation of these types of errors. [see Celarier, Sando for a good example of this].
Trust me. This is an inactive account. Regardless of what the
The first paper he describes as:
The second paper he says:
Submitter couldn't be bothered to do the research, but there is a paper written by this guy about the concept.
"Elmo knows where you live!" - The Simpsons
Actually, Mr. L'Hopital pretty much bought his theorem. Rather, Mr. Bernoulli would be the one saying something.
You simply don't define infinity and -infinity as numbers.
_ line
Well, not Reals, at any rate:
http://en.wikipedia.org/wiki/Extended_real_number
http://en.wikipedia.org/wiki/Real_projective_line
Tweet, tweet.
Actually, in his own paper here:
a chineVIII.pdf
http://www.bookofparagon.com/Mathematics/PerspexM
he agrees with closely what you just said. It says nullity*14 = nullity, he's created a new mathematic Field. It still might be a load of hot air but this paper is at least more rigorous than the video's 0/0 = nullity craziness.
If you don't know what a Field is its kind of like a different universe for numbers with different rules than the one's we're taught in school. Another example of a Field would be discrete mathmatics which is used for encryption.
Mathematicians are in the business of (among other things) taking mathematical equations that are currently unsolvable and finding ways of solving them.
At one point, taking the square root of the number "-1" was a totally unsolvable problem. It didn't make any sense, because a negative number times a negative number yielded a positive number. So to handle this, they made up a number i, defined it as the square root of -1, and found that hey, this number was consistent and worked with all already extant mathematical rules. Suddenly, you can make sense of equations involving that number.
As far as imaginary numbers being useful is concerned, well, as people mentioned, they're critical in electrical engineering. One of the most impotant numbers is exp^(ix) which is equal to "cos(x)+i*sin(x)". Fourier proved a long time ago that any periodic signal can be expressed as a sum of sines and cosines. Which means that any signal can be expressed as a sum of exp^(ix)... which is exactly what the Fourier transform does. It takes an input signal and transforms it into a sum of sines and cosines.
The Fourier transform is absolutely critical in electrical engineering. It is a transform that takes a time domain representation of a signal and converts it to a frequency domain representation. It makes hard problems easy, and totally intractable problems tractable.
If you take the output of one system ( f(x) ) and tie it into the input of another (g (x) ), the resultant output in the time domain is given by an annoying process known as convolution (integrate f(x)*g(x+t) dt from negative infinity to infinity). In the frequency domain, you just multiply the two functions, a process which is much easier.
Also, modelling the signal in the frequency domain allows you do look at what the components are of a signal by their frequency (obviously). You can see how much power is in any given frequency, for example. This is pretty useful when looking at RF signals, for example -- you can see what signals exist on what RF frequencies, what needs to be filtered out, etc.
Also, in the frequency domain, there are characteristics you can look for that will imply stability or instability in a system that you can identify at a glance; in the time domain, the only way to find out these characteristics is to do some rather annoying proofs. For example, in the time domain, if I want to prove that a system is BIBO stable (ie, bounded input always yields bounded output), I have to actually *prove* that the system will never go higher than a certain point. In the frequency domain, I can simply look at the the poles -- what values of frequency will cause the denominator of the frequency domain representation to go to zero -- and apply a couple of rules that tell me whether the system is stable based upon where the poles are.
Imaginary numbers are absolutely critical to electrical engineering. You can't do anything beyond the most trivial of things without them.
Sheesh, I can make an easier proof:
1 = 1
1+1 = 1+1
2 = 1
What I skipped a step? So did you only mine was more obvious.
In your last step You divided the entire function by (a^2 - ab).
When dividing by a function you must acount for the case where the function equals 0.
Hence, your proof is only valid for all numbers where (a^2 - ab) != 0
However, your proof states that a = b which breaks the above statement.
Hence, the proof has an empty set, which means its not valid.
I hoping you knew this and playing the fool but there's some people who weren't taught that step.
I blame the education system for spreading dumb things like this.
I thought he lived circa 500 BC, which would make the problem at least 2500 years old, not 1200, if he were working on it.
This has to be a hoax of some kind. I can't believe they let people this dumb teach math.
The same sort of manipulation this guy does can easily be applied to show that 0 = nullity.
0=0^1=0^-1 * 0^2 = 1/0 * 0*0 = 1/0 * 0 = 1/0 * 0/1 = 0/0 = nullity.
How can someone who is supposedly trained and licensed do this to kids.
If you liked this thought maybe you would find my blog nice too:
While a math person would strangle another math person for saying something like that, I was a math/physics major, so I'll tell you that at least in the sciences, you're dead on. It so happens that a lot of really messy operations (particularly trig ones like sines and cosines) over the real numbers look really clean once you realize they are just the real/imaginary parts of simple imaginary functions.
...), which is an infinite sum that only converges to a finite value if the real part of s is greater than 1 (for example, if it's zero, we have zeta(0) = 1+1+1+1+...). We can define its analytic continuation for other values, though, and prove interesting and unintuitive formulae like 1+2+3+4+5+... = -1/12 (which is, amazingly enough, actually somewhat relevant in physics when you look at the Casimir effect or string theory - it's the reason that in bosonic string theory you need 26 dimensions for quantum consistency, as in 2(left/right moving waves)*12(magic number from the zeta formula which counts energies of each mode) = 24, the number of degrees of freedom of a 2 dimensional string world-sheet).
Another way to think of it is that complex numbers are just a really special way of dealing with 2-dimensional geometry, where scaling and rotation are represented by complex multiplication. i corresponds to a 90 degree rotation, which is why i^2 = -1 (i.e. a 180 degree rotation). It's also why you can arbitrarily choose whether i is a clockwise or anticlockwise rotation as long as it's a consistent choice: two -90 degree rotations are equivalent to two positive ones (um...I hate to even bring it up, but that's actually not true in physics, where we have spinors - imagine a book attached to a ribbon which is attached to a table, and imagine turning the book 360 degrees; the ribbon is now twisted, and without further rotation it can't be untwisted, but if you rotate it another 360 degrees, you can undo the twisting without moving the book, by sort of pulling the loop of ribbon over the book - try it out if you're confused. That's the essence of a spinor, that a single full rotation leaves it in the "opposite" state, and that it leaves you confused).
Now I'll take off the science hat and put on the math one...the reason mathematicians love complex numbers is that if you have a function f(z) that is a function of the complex number z = x + iy (where x and y are both real), but not a function of x or y alone (i.e. f(z) = z+z^2+e^iz qualifies, f(z,x,y) = x - y + z does not), there are many subtle and powerful qualities that that function must possess. The one that comes up a lot is that you can do a Taylor expansion of the function and it "works" within a well defined range of values; another nice thing is that integration of the function along closed paths is all but trivial (it's always zero unless it encloses a "pole," i.e. a place where the function blows up in a certain way). As it turns out you can also take a function that you've defined along a single line (or piece of a line) and use its Taylor expansion to extend it to the whole complex plane. This is especially nice for functions like the Riemann zeta function (zeta(s) = 1/1^s + 1/2^s + 1/3^s +
So in summary, complex numbers are very important because they give us so many results that we could not even approach any other way (I haven't even mentioned the more subtle ones, esp. having to do with prime numbers!). To the contrary, the stuff that this professor is pushing seems entirely useless, more of an attempt to push a new term rather than a new concept. Mathematicians have understood infinity and what you can and can't say or do with it for a long time; anything you could even try to explain to a bunch of schoolchildren is either wrong, old news, or irrelevant.
The answer to a / 0 is defined as the limit for a / x when x approaches 0.
So you've proved that f(x) = 0/x is continuous?
lim x->0 (23 / x)
lim x->0 (-5 / x)
Neither of these exist.
Play Command HQ online
To be fair it's not entirely uncommon for mathematicians to invent new concepts. Take as the primary example the square root of -1, this is the imaginary number i. So having a symbol to designate dividing by zero quite sensible, does it help the maths, well no because once you divide by 0 algebra stops making sense eg.
1 x 0 = 0
(1 x 0)/0 = 0/0
and
2 x 0 = 0
(2 x 0)/0 = 0/0
It then follows that
(1 x 0)/0 = 0/0 = (2 x 0)/0
so you have
1 x 0/0 = 2 x 0/0
cancelling the x 0/0 you have
1 = 2
(there are more elegant proofs than this i just can't remember them this morning)
"Anything else divided by zero can be defined as giving infinity or -infinity, which can be used in further calculations just fine, even coming to the correct result."
False. While in some applications it may be useful to allow a divide by zero to go to +- infinity, this wreaks havoc with a ton of other applications. /0 is undefined for a very good reason.
For your arctan example - arctan *is* in fact undefined at 90 + 180n degrees, where n is a whole number. tan = opposite / adjacent, when the x component of your vector is 0, tan does not exist.
If we were to divide by a number *approaching* zero, however, we could very well end up with +- infinity, which in itself is a concept and not an actual number. In these cases it is often necessary to know which direction you're approaching from. Take the function 1/x for example. If you were to divide by 0- (that is, a negative value that is infinitely close to zero), it would be -INF. If you were to divide by 0+, it would be INF.
It is important to know that 0- and 0+ are not zero. These concepts need to stay very clearly separate. A divide by zero should stay undefined, not arbitrarily pinned to +-INF.
The limit of a constant over x as x approaches zero would depend on which direction you're approaching x from. For 23/x, if you approach 0 from the left, you get -inf, and if you approach it from the right you get a positive inf. Really, though, the behavior is better defined as an unbounded number approaching positive or negative infinity.
lim x->0+ (1/x) = inf
lim x->0- (1/x) = -inf
Select "Plain Old Text" instead of "HTML formatted"
(note that plain text according to slashdot is not plain text at all, but rather html with carriage returns automatically replaced with <br>, so html tags are still interpreted, and you have to use < and > to show angle brackets. Yes this is braindead.)
Be wary of any facts that confirm your opinion.
I say you should've read up on the subject first. (But then again this is Slashdot, after all.) There are some papers available. So, at least the children weren't his first audience but merely the strange byproduct you get when you contact the media.
His stuff is still a bit weird though -- if his stuff were really such a groundbreaking mathematical discovery, it wouldn't have been published in a journal of the International Society for Optical Engineering...
Maybe, though, you can derive more consistent rules for using the IEEE 754 NaN and Inf numbers using his findings, but I'd think he still has a long way to go to prove his findings useful enough for that.
Again:
</tag> closes a tag.
<tag/> is a tag that has no content inside it (<tag/> == <tag></tag>).
So, <p/> is an empty paragraph.
factor 966971: 966971
In contrast, this guy's "nullity" compares true to itself. He explicitly states (some other poster quoted this elsewhere) that he wanted to avoid the situation of a variable not comparing true to itself. I have to agree with this, though imagine the IEEE standard insists the opposite be true for a good reason.
If you speak about limeses, then it depends how you go toward some value (toward 0 in this case).
For instance, both functions f1(x)=sin(x) and f2(x)=x are 0 for x = 0, but
lim x->0 (sin(x)/x) = 1, as we know.
If you take function like f1(x) = x*sin(x) and other one f2(x) = x then
lim x-> f1(x)/f2(x) = 0.
In these two cases, "0/0" have different values.
When you use division in limeses, the path you take is important, i.e. functions that describe in which way you go toward 0. That's why other posters mentioned continuity and other stuff related to functions, and not related to numbers.
Big breakthrough would be to solve lim x->0 f1(x)/f2(x) for f1(x) = 0, f2(x) = 0.
No sig today.
Absolutely. It's also possible to extend the real number system to support something else physicists use all the time, infinitesimals and infinites:s
http://en.wikipedia.org/wiki/Non-standard_analysi
http://en.wikipedia.org/wiki/Hyperreal_numbers
Once you can get your head around ultrafilters, it's really a cool system and, like complex numbers, can allow you to arrive at conclusions that you would have a hard time arrive at without them. But like complex numbers, they don't "really exist". They're just a useful model that helps us solve and understand real-life problems.
The problem isn't that people haven't figured out ways of dividing by zero, the problem is that there are many different ways in which you could reasonably define division by zero, and they are not mutually consistent. Wikipedia lists some of them.
I think the GP was refering to the hardware level, not an abstract software layer. Where traditonal computers, even those with modern math extensions dont know what an imaginary or complex number is. Normally, two floating point values are used to represent complex arithmetic, however its not a native operation, and still requires some software logic to be accomplished.
correction you cannot say 2 = 1 here because 2*0 = 1*0 and dividing both sides by 0 gives you 0/0 on both sides which is inderterminate.
So sadly your above assumption holds false.
-- "Genius is 1% inspiration and 99% perspiration" - TAE --
No falling out of the sky, at least for that.
Another routine divide-by-zero occurs when you attempt to calculate the amount of flavor in the crap sandwich they serve as a snack--but I digress.
No, he didn't prove that f(x)=0/x is continuous. He simply stated that it has a hole discontinuity (which occur when a value of a function is not defined, but a limit exists at that point), not an asymptotic one (which occur when a value of a function is undefined, and a limit is either undefined, positive or negative infinity at the point). There is one other type of discontinuity, a displaced discontinuity. For example, consider the piecewise defined function f(x)={0/x for x != 0, 1, x = 0}. The function is defined at x=0, but its value does not equal the limit at that point.
It has been a nervous year, with people beginning to feel like Christian Scientists with appendicitis.
a/0 is undefined. The end. The limit of a/x as x approaches 0 from above is positive infinity, but the two statements are not the same. Division by 0 breaks a number of rules and can be used to "prove" all sorts of things that aren't true. If 1/0 == inf, then inf * 0 == 1, but any number multiplied by 0 is 0, so inf * 0 == 0, therefor 0 == 1. It just doesn't work. a/0 is undefined. As best as I can tell all this guy is doing is assuming that 1/0 == inf, -1/0 == -inf, and calling 0/0 something else.
-matt
Uhm... The reason IEEE does not have consistent rules for this is because different definitions make sense in different domains. We already know several useful definition, it's nothing new.
- These characters were randomly selected.
These days your grahpics card does 3d transformations, even 4d with the physics models.
Many, many of these in parallel.
Good old 2d cartesian coordinates/imaginary numbers are no problem.
The real problem is irrational numbers.
PI, sqrt(2) sqrt(3), e
Complex numbers are useful becuase they are useful in equations and can be used to generate real answers.
I've read his "technical" paper and all it says, in a lot of mathematical jargon, is that once you divide by zero anywhere in an equation the result is 'undefined' only he has now given 'undefined' a new mathematical symbol and a funky name.
Unlike an imaginary number which can give a real single value when used in an equation (e.g. 2i^2+4 = 2) once you divide by zero anywhere in an equation you result can be anywhere in an undefined space between infinity and negative infinity. He calls this space Nullity
So his invention is actually not a mathematical one, it is a gramatic one. Nullity = Undefined, Undefined = Nullity.
Quantum Physics a.k.a. sub-molecular statistics
What is needed is a precedence for those two rules.
...), which makes some sense, because 0 is after all, special.
...) for the series x/0 and the point of TFA is x/0 altogether.
/. standards).
1. 0/x gives 0.
2. if x != 0, x/x gives 1
That way, 0/0 = 0, 1/1 = 1, 2/2 = 1... Which makes practical sense. The series looks like (... 1, 1, 1, 0, 1, 1, 1
That still doesn't provide for a general x/0 solution. Given the above, we'd still be at (... ?, ?, ?, 0, ?, ?, ?
Since x/0 == 1/0 * x we must define 1/0 in order to define x/0. How do you define 1/0 ?:
a. Is 1/0 part of x/0 -- then we must define x/0 to define x/0 -- a circular problem.
b. Is 1/0 an arbitrary exception to x/0 -- if so, is that by arbitrary definition? What would it be defined to be?
c. Is 1/0 a fuzzy set of [0...1] or [0...x] or maybe [0...+inf] -- if so, does it collapse into an integer/real/imaginary/complex or something when observed, like a quantum state?
That's the problem as I see it. I'm not a mathematician, but this issue fascinates me and I've read and thought about it quite a bit (yes, I know that's geeky even by
Time for a word problem... Okay, imagine you walk down the street and find a $10 bill on the sidewalk. Currently, that $10 is divided by zero -- no one owns any part of it. How do we solve the problem in the real world? Well, if you're walking alone, you do what we call claiming it, which is adding 1 to the denominator. So it's 10/1, yielding you $10. If you and a friend are walking together and decide to share the bounty, that's 10/2, yielding you $5. If nobody ever claims that $10, where's the value? The whole $10 is useless until there's a claim on it. So before it is seen in the street, it's worth $0 to everyone (no matter how many people there are that haven't seen it -- up to an infinite number of people that haven't claimed it have zero value out of it). Once someone sees it, it's worth $10/x, with x being the number of people splitting the claim.
If you skimmed that, you might ask, "10/0 == 0 * +inf ??? What is 0 * +inf ??? is that 0 ???" _But_, we don't count he people who _don't_ claim it, just like we didn't for 10/1. The number of people who don't claim it still get 0 no matter whether someone has claimed it or not. Maybe it's 10/0 == [0...10] ?
Reasoning from the money example, I'm tempted towards the following ideas, but I wouldn't claim this is really my philosophical stance on the idea. It's just musings. The $10 on the street is really not worth anything to anyone until it is claimed. But is that the same as zero? Is the lack of a divisor really the lack of value? The $10 bill is still worth a total of $10, but it's not worth that _to_ anyone until it is claimed. It's like there really is a quantum set/fuzzy set here of [0...10], and the condition of its collapse is that there must be a non-zero denominator. If someone needs $10, and people step forward to chip in towards that need, that's a negative $10. So -10/2 (two people share the cost) is -5 (they each chip in $5). Until the debt is covered, it's -10/0 which is -10/0 == [0...-10]. So that leads me to say it's really x/0 = [0...x] and not anything to do with absolute values. Is saying that x/0 = [0...x] the same as saying it's undefined? Or is calling it a superposition of all number from 0 to x that is waiting for a denominator to collapse a definition in itself? I'm not comfortable making that call, even just playing around with the concept. I don't have the math background to try to prove these intuitions and musings, but it'd be fun to see someone work on it.
After years of being a mathematician, I can report that I have yet to see "0" used to mean "nothingness".
if a=b, then (a-b) = 0. going from the fifth line to the sixth line, when you divided out (a-b) from both sides, you were, in fact, introducing a nullity.
do not read this line twice.
ok, lets assume 0/0 = 1
since 0 * 2 = 0 => (0 * 2) / 0 = 1
just apply commutative property => (0/0) * 2 = 1
since 0/0 = 1, substitute => 1 * 2 = 1
an now you get => 2 = 1
I'm pretty sure they weren't called imaginary numbers because they were useless - they were invented to solve the general cubic equation!
To know recursion, you must first know recursion.
It wasn't until I started using/programming computers (at around age 23) that I started to really use some of the useful concepts like algebra through assigning variables etc. I always quite liked algebra at school.
I think that if an enlightened teacher had mentioned that in some cases 1+1=11 (binary) or 9+7=10 (hex) then maybe conventional decimal might have made more sense, and been interesting as a *subset* of mathematics rather than the be all and end all, "coz I say so".
"one could technically define a set of numbers which includes +=infinity"
Technically you could not do this. Remember, infinity is not a number, it is a concept meaning an unbounded limit. There are rules for including it in algebraic equations, but it is still not a "number."
100/0 != 10/0 != 1/0 != 0/0
but he uses the same identifier for all of them
Actually, he doesn't. He uses "infinity" for the first 3 and "nullity" for the last one.
so that would mean:
(100/0) / (1/0) = 1
No, according to his axioms, infinity/infinity = nullity, not 1
That goes against the principle of:
infinity / (infinity - 1) != 1
There is no such principle!
AccountKiller
Anyway the proof as I know it is this: Define 0 as a number. Define a successor function which takes a number as input and produces a number as output. Then start defining some labels like 1 (doesn't really have to be 1, could be the Symbol formerly known as Prince... just a label... still the same crazy music genius... this, it would be nice if were explained more...) is the Successor of 0, 2 is the Successor of the Successor of 0, 3 and then 4 in the same way. Then finally define + as the following construction: 0 + any number = that any number and S(x) + S(y) = x + S(S(y).
2 + 2 = 4
S(S(0)) + S(S(0)) = S(S(S(S(0)))) by definitions above.
S(0) + S(S(S(0))) = S(S(S(S(0)))) by the second rule of +
0 + S(S(S(S(0)))) = S(S(S(S(0)))) again by the second rule of +
S(S(S(S(0)))) = S(S(S(S(0)))) by the first rule of +
QED
Anyway, ask some 6 year old who knows how to count on their fingers... they'll show you that (holding two sets of fingers on either hand and then counting the "successors" by dropping fingers as they go.)
People who quote themselves bug the crap out of me -- Me.
This is incorrect. Just because a physics equation uses an i doesn't prove anything. Complex numbers are a tool for expressing physical models in a simpler way. They are not fundamental to physics - far from it. Geometric Algebra can be used to express all the laws of physics without resorting to imaginary numbers. You can indeed do QM without complex numbers and in fact it's a bit easier if you do. See, for instance, Chris Doran's book on Geometric Algebra.