Why Complex Numbers Are Not an Ordered Field

The concept of an ordered field is fundamental in abstract algebra and analysis. A field is said to be ordered if it can be equipped with a total order that is compatible with the field operations of addition and multiplication. For any two elements a and b in the field, one of the following must hold: a b, a b, or a > b. Additionally, the order must satisfy the following properties:

While real numbers (#x1D444;), which form an ordered field, can be easily ordered by magnitude, complex numbers #x1D45C; cannot.

The Incompatibility of Complex Numbers with Ordered Field Properties

Consider the imaginary unit i. If i were to be assigned a specific order, say i > 0 or i

If i > 0, then multiplying both sides by i gives i2 > 0. But i2 -1, which implies -1 > 0, a contradiction.

Alternatively, if i 0. Multiplying both sides by -i gives (-i)(-i) > 0, which simplifies to -1 > 0, again a contradiction.

These contradictions demonstrate that complex numbers cannot be ordered in a consistent manner with the field operations of addition and multiplication.

Components of Complex Numbers

It's important to note that while the components (real and imaginary parts) of complex numbers can be ordered, an overall order for the entire complex number system is not possible.

For instance, even real components can be ordered, as in the comparison between -10000000000.0000000001i and 0. However, this ordering of components does not extend to the complex number system as a whole. We cannot derive a coherent order for complex numbers such that if -10000000000.0000000001i > 0, -1 should also be greater than 0 in the same ordering.

Ordering Complex Numbers vs. Real Numbers

In contrast, the real numbers form an ordered field. This means that the real numbers can be placed on a number line, where each number has a unique position, and the order is consistent with the field operations. The real numbers satisfy all the properties of an ordered field, which is why they can be used in various mathematical and practical applications.

However, as demonstrated, the complex numbers do not possess these properties, making them an example of a field that is not an ordered field. The imaginary unit i and any complex number cannot be ordered in a way that is compatible with the field's operations.

Understanding why complex numbers cannot be ordered provides insight not only into the structure of complex numbers but also into the limitations of ordered algebraic systems.