Algebraic structure for addition and multiplication of rational numbers

The operations of addition and multiplication have the following properties.

For the addition;

  1. Associative Property: given any three rational numbers $a,b$ and $c$, it is satisfied that: $$a+(b+c)=(a+b)+c$$

  2. Commutative Property: for any pair of rational numbers $a$ and $b$ it is saitisfied that: $$a+b=b+a$$

  3. Neutral Element: a rational number, $0$, which, added to any other real number $a$, has $a$ as result: $$a+0=a$$

  4. Opposite Element: for any rational number $a$ there is another rational number, which we called $-a$,. When adding them up, the result we obtain is the neutral element $0$. We call $-a$ the opposite element of $a$.

All these properties, can be summarized by saying that the set $\mathbb{Q}$ is a commutative group or Abelian group with the addition operation.

For the multiplication:

  1. Associative Property: given three any rational numbers $a,b$ and $c$, it is fulfilled: $$a\cdot(b\cdot c)=(a \cdot b) \cdot c$$

  2. Commutative Property: For any pair of rational numbers $a$ and $b$ is fulfilled: $$a \cdot b=b \cdot a$$

  3. Unit Element: There exists a rational number $(1)$ which if multiplied by any other real number $a$, gives the same number as a result $a$: $$1 \cdot a=a$$

  4. Inverse Element: for any rational number $a$ there exists another real number, which we so-called $a^{-1}$, or $\dfrac{1}{a}$, which if multiplied will give us the unit as a result $(1)$.

Let's observe that all these properties also define the set of rational numbers as an Abelian group with the multiplication operation.

We also have to mention a last property that relates to the addition and product of rational numbers:

Practice exercises