Arbitrariness or Ambiguity in the Definition of the Imaginary Unit in Mathematics

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People's first introduction, in middle school, to complex numbers typically involves the definition of "i" as "the square root of minus 1". This is often the student's first introduction to an abstract mathematical concept—"Let's define 'i' this way, and see what we can do with it." They of course find out that they can do a lot with it, a journey that continues into graduate school. (Depending on the teacher's inclination for teaching careful mathematical thinking, negative numbers might also be introduced in an abstract mathematical way—"Let's define '-3' as that which, when added to 3, gets zero, and see what we can do with it.")

Once students become skilled at doing arithmetic with complex numbers, and they become familiar with the complex plane, and with the fact that every nonzero number, including negative numbers and complex numbers, has two square roots,[1] which are negatives of each other, they notice that the idea that started the whole development has a seeming ambiguity. The number -1 has two square roots, and, in the language of complex numbers and the complex plane, one of them is "i" and the other is "-i". In the language of the complex plane as it is derived from the choice of "i", the first is "upward" and the second is "downward".

So which one did we mean when we defined "i" in the first place? Usually the students don't worry about this, because they don't have the required mathematical sophistication. And they weren't ready for definitions in terms of abstractly constructed vector spaces. If two stubborn people adamantly insist that each other's definition is upside-down, there's not much we can say about it.

Later, when students are able to handle the concept of an abstractly defined vector space, complex numbers are redefined as elements of a two-dimensional real vector space, with "1" defined as the vector (1, 0) and "i" defined as the vector (0, 1). Once the students reach this point in their education, they are already familiar with the concepts of real and imaginary parts, and of arithmetic operations in terms of those parts, so they have no trouble transitioning to the vector space formulation and to the arithmetic operations being defined in terms of vectors.

With this new definition of "i", the conflict between our stubborn people can be seen for the triviality that it is. Both agree that "i" is the (0, 1) vector, because that is its definition. Their only dispute is which direction that vector points on the graph. Which way people hold a piece of paper has no mathematical significance.

But, just to play devil's advocate, what would happen if we were told that we were using the wrong definition of "i", and that we had to change "i" to "-i" in all our formulas and theorems? Everything would still work. We would just be flipping one component of the vector space, and vector spaces are invariant under this.[2]

Here is an example. Euler's formula is

If we change the sign of , we get

Does that follow from the original equation? Well, using the formula

we get

So the question is whether

is equal to

but, because everyone agrees that ,

Also, Euler's equation is

If we have to change the sign of , it becomes

See also


  1. This is a case of De Moivre's Theorem.
  2. Actually, it's not quite that simple. Once one gets past middle school, there are issues of "handedness" or "chirality", that show up most famously in the "right hand rule" for magnetism. Beyond high school, concepts like "pseudo-vectors", "pseudo-scalars", and "pseudo-tensors" come into play. But these issues don't arise when the dimension is less than 3.