Real Multiplication Identity is One

Theorem

The identity element of real number multiplication is the real number $1$:

$\exists 1 \in \R: \forall a \in \R: a \times 1 = a = 1 \times a$

Proof

From the definition, the real numbers are the set of all equivalence classes $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.

Let $x = \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right], y = \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$, where $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ and $\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$ are such equivalence classes.

From the definition of real multiplication, $x \times y$ is defined as:

$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n \times y_n} \right \rangle}\right]\!\right]$

Let $\left \langle {1_n} \right \rangle$ be such that $\forall i: 1_n = 1$.

Then we have:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left[\!\left[{\left \langle {1_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]\)	\(=\)	\(\displaystyle \left[\!\left[{\left \langle {1_n \times x_n} \right \rangle}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left \langle {1 \times x_n} \right \rangle}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Similarly for $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {1_n} \right \rangle}\right]\!\right]$.

So the identity element of $\left({\R^*, \times}\right)$ is the real number $1$.

$\blacksquare$

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): Chapter $1 \ \S 1$