Order and Absolute Value


Order of Real Numbers

Let $ a$ and $ b$ be two real numbers. Then
$ a$ is less than $ b$, written $ a<b$ if $ a$ is to the left of $ b$.
$ b$ is less than $ a$, written $ b<a$ if $ a$ is to the right of $ b$.

Variations of $ <$ and $ >$:
$ \leqslant$ -- less than or equal to
$ \geqslant$ -- greater than or equal to
$ \not<$ -- not less than
$ \not>$ -- not greater than

An inequality is a statement involving symbols $ <$, $ >$, $ \leq$, $ \geq$, $ \not<$, $ \not>$, $ \not\leq$, $ \not\geq$. Ex. $ 2<3$, $ 5 \leq 5$, $ 3\not\leq 2$.

The inequality $ a<c<b$ says that $ c$ is between $ a$ and $ b$.

Absolute Values of Real Numbers

The absolute value of a real number is the distance from the number to 0 on the number line. The absolute value of $ a$ is written as $ \vert a\vert$. For all real numbers

$\displaystyle \vert a\vert=\begin{cases}a  , \quad a\geqslant 0  -a  ,\quad a\leqslant 0\end{cases}$

Ex. $ \vert 3\vert=3$, $ \vert-2\vert=-(-2)=2$

Properties of absolute values For all real numbers $ a$ and $ b$:

1. $ \vert a\vert\geq 0$
2. $ \vert-a\vert=\vert a\vert$
3. $ \vert a\vert\cdot \vert b\vert=\vert ab\vert$
4. $ \frac{\vert a\vert}{\vert b\vert}=\Big\vert\frac{a}{b}\Big \vert$, $ b \ne 0$
5. $ \vert a+b\vert\leq \vert a\vert+\vert b\vert$ (triangle inequality)

Distance between two points on a number line Let P and Q be two points on the number line with coordinates $ a$ and $ b$ respectively. Then the distance $ d(P,Q)$ between $ P$ and $ Q$ is $ d(P,Q)=\vert a-b\vert=\vert b-a\vert$.



Subsections


Department of Mathematics
Last modified: 2005-08-30