The Ring of Real and Complex Numbers

# The Ring of Real and Complex Numbers

Recall from the Rings page that if $+$ and $*$ are binary operations on the set $R$, then $R$ is called a ring under $+$ and $*$ denoted $(R, +, *)$ when the following are satisfied:

• 1. For all $a, b \in R$ we have that $(a + b \in R)$ (Closure under $+$).
• 2. For all $a, b, c \in R$, $a + (b + c) = (a + b) + c$ (Associativity of elements in $R$ under $+$).
• 3. There exists an $0 \in R$ such that for all $a \in R$ we have that $a + 0 = a$ and $0 + a = a$ (The existence of an identity element $0$ of $R$ under $+$).
• 4. For all $a \in R$ there exists a $-a \in R$ such that $a + (-a) = 0$ and $(-a) + a = 0$ (The existence of inverses for each element in $R$ under $+$).
• 5. For all $a, b \in R$ we have that $a + b = b + a$ (Commutativity of elements in $R$ under $+$).
• 6. For all $a, b \in R$ we have that $a * b = b * a$ (Closure under $*$).
• 7. For all $a, b, c \in R$, $a * (b * c) = (a * b) * c$ (Associativity of elements in $R$ under $*$).
• 8. For all $a, b, c \in R$ we have that $a * (b + c) = (a * b) + (b * c)$ and $(a + b) * c = (a * c) + (b * c)$ (Distributivity of $*$ over $+$).

We will now look at two very basic rings - the ring of real numbers and the ring of complex numbers.

## The Ring of Real Numbers

The set of real numbers under the operations of standard addition $+$ and standard multiplication $*$ form a ring, $(\mathbb{R}, +, *)$. To verify this, let $a, b, c \in \mathbb{R}$.

Clearly the sum of any two real numbers is a real number, i.e., $(a + b) \in \mathbb{R}$, so $\mathbb{R}$ is closed under $+$. We are also familiar that addition of real numbers is associative as for $a + (b + c) = (a + b) + c$. The additive identity is $0$ since $a + 0 = a$ and $0 + a = a$. For every real number $a$, the additive inverse is $-a$ since $a + (-a) = 0$ and $(-a) + a = 0$. Also, addition of real numbers is commutative since $a + b = b + a$.

We also have that the product of any two real numbers is a real number, i.e., $(a * b) \in \mathbb{R}$ so $\mathbb{R}$ is closed under $*$. We also have that standard multiplication is associative on $\mathbb{R}$ as $a * (b * c) = (a * b) * c$.

We lastly note that multiplication distributes over addition since:

(1)
\begin{align} \quad a * (b + c) = (a * b) + (a * c) \end{align}
(2)
\begin{align} \quad (a + b) * c = (a * c) + (b * c) \end{align}

Therefore $(\mathbb{R}, +, *)$ is a ring.

## The Ring of Complex Numbers

It can rather laboriously be shown that $(\mathbb{C}, +, *)$ is also a ring too where for $x, y \in \mathbb{C}$ where $x = a + bi$, $y = c + di$, we define addition of complex numbers by:

(3)
\begin{align} \quad y + z = (a + bi) + (c + di) = (a + c) + (b + d)i \end{align}

And we define multiplication of complex numbers as:

(4)
\begin{align} \quad y * z = (a + bi) * (c + di) = ab + adi + bci + bdi^2 = (ab - bd) + (ad + bc)i \end{align}

The reader is encouraged to verify this.