The operations of addition and multiplication have the following properties.
For the addition;
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Associative Property: given any three rational numbers $$a,b$$ and $$c$$, it is satisfied that: $$$a+(b+c)=(a+b)+c$$$
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Commutative Property: for any pair of rational numbers $$a$$ and $$b$$ it is saitisfied that: $$$a+b=b+a$$$
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Neutral Element: a rational number, $$0$$, which, added to any other real number $$a$$, has $$a$$ as result: $$$a+0=a$$$
- 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:
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Associative Property: given three any rational numbers $$a,b$$ and $$c$$, it is fulfilled: $$$a\cdot(b\cdot c)=(a \cdot b) \cdot c$$$
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Commutative Property: For any pair of rational numbers $$a$$ and $$b$$ is fulfilled: $$$a \cdot b=b \cdot a$$$
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Unit Element: There exists a rational number $$(1)$$ which if multiplied by any other real number $$a$$, gives the same number as a result $$a$$: $$$1 \cdot a=a$$$
- 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:
- Distributive property of the product regarding addition: given any three rational numbers $$a,b$$ and $$c$$, it is satisfied that: $$$a\cdot (b+c)=a\cdot b + a \cdot c$$$ This property, together with all the others of the addition and the product, defines the rational numbers as a structure that we name a commutative field with a unit.