Arrangement of the real numbers

Arrangement of the real numbers

In the set $$\mathbb{R}$$ we have defined a relation of order that we denote $$ < $$ intuitively, if $$a$$ and $$b$$ are two real numbers, we will write $$a < b$$ if, when drawing them on the real line, point $$a$$ is on the left of point $$b$$. We will then say that $$a$$ is less than $$b$$.

$$a\leq b$$ is usually used to indicate that number $$a$$ is less than or equal to $$b$$. It is also said that $$\leq$$ is an inequality symbol and that $$ < $$ is a strict inequality symbol.

It is said that this relation is of total order $$\mathbb{R}$$: that is, considering two different real numbers $$a$$ and $$b$$, we always have $$a < b$$ or $$b < a$$. Or,in other words, $$a$$ and $$b$$ are always comparable.

Properties of the arrangement

The operations with real numbers and the arrangement of these are related by the following properties:

• Monotonicity of the sum: an inequality won't change if adding the same value to both members, in other words, if $$$a < b$$$ then for any real number $$c$$, it is satisfied that: $$$a+c < b+c$$$ Also it is satisfied if the inequality is not strict: $$a\leq b \Rightarrow a+c \leq b+c.$$

• Monotonicity of the product by a positive number: an inequality does not change if we multiply both members by the same positive number, that is, if $$a < b$$ and $$c$$ is a positive real number $$(c > 0) $$, it is satisfied: $$$a\cdot c < b\cdot c$$$ Also it is satisfied if the inequality is not strict: $$a\leq b$$ and $$c\geq 0 \Rightarrow a\cdot c \leq b\cdot c.$$

• Antimonotonicity of the product by negative numbers: all inequalities change if we multiply both members by the same negative number, that is, if $$a < b$$ and $$c$$ is a negative real number $$(c < 0)$$, it is satisfied that: $$$a\cdot c > b\cdot c$$$ Also it is satsifed if the inequality is not strict: $$a\leq b$$ and $$c\leq 0 \Rightarrow a\cdot c \geq b\cdot c.$$