Equivalence of Definitions of Complex Number

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Theorem

The following definitions of the concept of Complex Number are equivalent:

Definition 1

A complex number is a number in the form $a + b i$ or $a + i b$ where:

$a$ and $b$ are real numbers
$i$ is a square root of $-1$, that is, $i = \sqrt {-1}$.

Definition 2

A complex number is an ordered pair $\tuple {x, y}$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows:


Complex Addition

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.

Then $\tuple {x_1, y_1} + \tuple {x_2, y_2}$ is defined as:

$\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$


Complex Multiplication

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.


Then $\tuple {x_1, y_1} \tuple {x_2, y_2}$ is defined as:

$\tuple {x_1, y_1} \tuple {x_2, y_2} := \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$


Scalar Product

Let $\tuple {x, y}$ be a complex numbers.

Let $m \in \R$ be a real number.


Then $m \tuple {x, y}$ is defined as:

$m \tuple {x, y} := \tuple {m x, m y}$


Proof

Since:

$\tuple {x_1, 0} + \tuple {x_2, 0} = \tuple {x_1 + x_2, 0}$
$\tuple {x_1, 0} \tuple {x_2, 0} = \tuple {x_1 x_2, 0}$

we can identify a complex number (definition 2) $\tuple {x_1, 0}$ with the real number $x_1$.

Specifically, we can define an isomorphism between the set of complex numbers (definition 2) of the form $\tuple {x, 0}$ and the field of real numbers.


Now, we define $i = \tuple {0, 1}$.

Then:

\(\ds x + i y\) \(=\) \(\ds \tuple {x, 0} + \tuple {0, 1} \tuple {y, 0}\)
\(\ds \) \(=\) \(\ds \tuple {x, y}\) Definition of Complex Addition and Definition of Complex Multiplication


Finally, we see that:

\(\ds i^2\) \(=\) \(\ds \tuple {0, 1} \tuple {0, 1}\)
\(\ds \) \(=\) \(\ds \tuple {0 \cdot 0 - 1 \cdot 1, 0 \cdot 1 + 1 \cdot 0}\)
\(\ds \) \(=\) \(\ds \tuple {-1, 0}\)
\(\ds \) \(=\) \(\ds -1\)

Thus we can say that $i = \sqrt {-1}$.

$\blacksquare$


Sources

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