What does $3+2i$ apples mean? Can the simple counting analogy with apples be extended to complex numbers?

Question

Please read, or at least skim the question. Past attempts at answering the question have ignored vital constraints provided below.

Natural Numbers

Imagine I have $n$, a positive natural number, of apples. We know $n$ can be partioned into a series of additions by $1$. Then having $n$ apples means one can map each apple to a a partition of $n$.

For example, I have two apples $a$ and $b$. $2=1+1$, since I can literally draw a line between each letter and partition. Thus the mathematical concept, $2$ leapt to the physical realm, i.e what do I associate with having two apples in hand? Thus counting how many apples I have has been described after I determine the unique number $n$.

Zero

Imagine I have $0$ apples. This corresponds to having a symbol $0$, but having no objects to draw a line to. The meaning is whatever I associate having no apples to mean.

Negative Numbers

Imagine I have $-3$ apples. Very similar to a positive number, I associate each partition to an apple that some other person/entity/process desires from me. But I associate these to apples that I owe, rather than to apples I own.

I have $-2$ apples means, whatever I associate having to owe $2$ apples to mean.

Rational Numbers

Imagine I have $\cfrac{a}{b}$ apples. $a$ and $b$ are integers at this point, $b$ doesn't equal $0$, however. $a$ counts how many pieces of Apple I have. $b$ counts how many apples pieces are needed to make a full Apple. The division operation already has meaning, so it won't be discussed here. We'll settle with correspondence

$\cfrac{a}{b} \sim$ $a$ out of $b$

The mathematical correspondence is that you can have $a \cdot b^{n}$ apples and always partition them into a positive natural number of apples of size $b$.

Thus having $\cfrac{a}{b}$ apples, means whatever I interpret having $a$ out of $b$ apple pieces to mean.

Real Numbers

Through a limiting procedure, that could theoretically be performed, all real numbers can be given an interpretation in the limit.

Of course, we can also observe that the circumference of an apple is transcendental, and the diagonal length is irrational, theoretically at least. We have no evidence suggesting for, or against.

Caveats

Ambiguities can arise, but they are inherent to the observer's opinion, not the number. For instance, what one person calls an apple, someone else may not. In addition, it may be impossible to literally have an irrational number of apples, but procedures can implemented that converge to the true interpretation.

Complex Numbers and Question

So, if I have a complex number of apples, what does that mean?

I want the answer to be in this form.

If you have $z$, a complex amount of apples, then it means whatever you interpret ($X$) apples to mean.

Where $X$ can be a string of English words that corresponds to the mathematical concept. However the English string must be correspondence to an actual action, or theoretical sequence of realizble actions, that can be taken to verify the mathematical truth.

To avoid cheap answers, such as interpreting $z$ as apples and oranges, apples with spin...the algebraic operators must also have an interpretation that is physically realizble.

In the form,

$z \ $ * $w$ means whatever you interpret (Y) to mean.

Once again, the English string must be verifiable in the physical world.

Problems with Current Answers

Some don't answer the question in the form required. Others don't present a proof that this task is impossible. Some suggest interpretations that can't be verified.

Answer Format

The answer should either be sentences in the form described above followed by a description of how to verify the statements, or an argument explaining why this can't be done.

Answer 1

the English string must be correspondence to an actual action, or theoretical sequence of realizble actions, that can be taken to verify the mathematical truth.

There is no reason to believe that a statement having a mathematical form should be expressible in English ( or any informal language ). The languages of mathematics are required and were invented precisely for the purpose of expressing mathematical concepts without ambiguity or potential misconception.

There is no reason to assume that a mathematical statement must have a "real world" physical incarnation or be described by actions in the real world. Indeed many mathematical concepts, when being taught, invariably require the use of the word "imagine" to begin an explanation.

Many mathematical concepts start in the real world but their rules have been applied to a logical extreme that often "discovers" domains which are outside the normal human conception of existence. Complex numbers are a prime example. While we can make use of these to explain physical phenomena, this generally takes us into realms outside of the everyday, common sense, understanding of our world.

Answer 2

(You might mean a few different things by "number" when you talk about "number types" here. I'll assume you mean "elements of a field" -- things you can add, subtract, multiply, and divide.)

I'd disagree with your statement that "all other number types can be fit into this form" i.e. that they can usefully describe something about apples. For an example that has nothing to do with the complex numbers, take the field $\mathbb{F}_2$, which consists of: $\{0, 1\}$, with the rules $0 \cdot 1 = 0$, $0 + 0 = 0$, etcetera, but also $1 + 1 = 0$.

Now $1 + 1 = 0$ doesn't describe any property of apples (i.e. discrete objects). But it does happen to describe logical statements pretty well, if you interpret $a + b = 1$ as "either $a$ is true, or $b$ is true, but not both" and $a \cdot b = 1$ as "both $a$ and $b$ are true".

So I reject your challenge on the following basis: what's so special about apples? All the time we need to model phenomena that don't come in discrete chunks.

Just like $\mathbb{F}_2$, complex numbers don't describe apples very well. But they do describe two-dimensional arrows (vectors), in the following way.

You can associate every complex number with a point in the plane by the diagram shown above: the vertical axis is the imaginary part, the horizontal axis is the real part. Then draw an arrow from the origin to the point. (If you've seen vectors in $\mathbb{R}^2$, these are just like those.)

You can associate an angle with each arrow by measuring (counterclockwise) from the horizontal axis. That angle, along with a length, uniquely defines a complex number.

Now, we could have done all that with $\mathbb{R}^2$, pairs of real numbers. But the complexes come with a multiplication defined

$$(a + bi)(c + di) = ac + adi + bci + i^2bd = (ac - bd) + (ad + bc)i$$

and this interpretation lets us use it: when we multiply any two complex numbers $z$ and $q$, the result $zq$ represents the arrow that has:

• angle equal to the sum of their angles
• length equal to the product of their lengths

Now that we have this, we can define all the other operations too. $z \cdot 10$ means scaling the arrow by a factor of $10$. The square root of an arrow means: what arrow is halfway between me and the horizontal axis?

Here's an example from Kalid Azad.

Suppose I’m on a boat, with a heading of 3 units East for every 4 units North. I want to change my heading 45 degrees counter-clockwise. What’s the new heading?

Our current heading is represented by the number $3 + 4i$. Changing it by $45$ degrees is the same as multiplying by $1 + i$, since that arrow makes a $45$-degree angle. So the answer is:

$$(3 + 4i)(1 + i) = 3 + 3i + 4i - 4 = -1 + 7i$$

or "one unit West for every 7 units North".

Answer 3

As detailed laboriously in the comments, the request of the OP is most likely not possible as written. I've included my thoughts below, as I believe this to be the closest approximation to an answer. (The OP is free to disagree, of course, but I still think that some people will find this helpful.)

The existence of a nontrivial automorphism of $\mathbb{C}$ limits the physicality of interpretations of the complex numbers: there is no way, not even algebraically, to use real numbers to distinguish between $i$ and $-i$. This is a stark contrast to the case of real numbers, which can be distinguished algebraically using only the natural numbers.

(There are of course trivial interpretations, like calling a positive imaginary apple an orange, so that $3+2i$ apples means $3$ apples and $2$ oranges. But this is poorly-behaved under scalar multiplication and not really a good-faith answer, despite having the right properties as a vector space over $\mathbb{R}$.)

I think the best we can do is allow ourselves to think in terms of waves, which naturally have a magnitude and a phase. This is roughly inspired by Feynman's description of photons as having small spinning clocks whose position affects their interactions. Of course physical apples do not behave like waves, but if there were such a thing as complex apples, I would expect them to behave in the following way:

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I would interpret $3+2i$ apples as $\sqrt{3^2 + 2^2} = \sqrt{13}$ apples, but with a phase shift of $\arctan{\frac{2}{3}} \approx 33.7^\circ$ to the apples' wave function. (Water or sound waves are enough to make this metaphor work; quantum mechanics is not necessary.)

In general, apples of roughly similar phase will tend to combine constructively to produce more apples, while apples of roughly opposite phases will tend to combine destructively and product fewer apples. Orthogonal apples will combine according to Pythagoras.

If you acquired $\sqrt{13}$ apples with exactly the opposite phase of $180^\circ + \arctan{\frac{2}{3}} \approx 213.7^\circ$, i.e. $-3-2i$ apples, the apple waves would cancel and you would be left with $0$ apples.