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)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)And we define multiplication of complex numbers as:
(4)The reader is encouraged to verify this.