The real number system can be visualized as a horizontal line
that extends from a special point called the **Origin** in both directions
towards infinity. Also associated with the line is a unit of length.
The origin corresponds to the number 0. A positive number * x*
corresponds to a point * x* units away from the origin to the
right, and a negative number * -x* corresponds to a point on the
line * x* units away from the origin to the left. All of this is
illustrated in the above Figure.

We said that the number * corresponds* to a point on the real
number line, but actually there is no useful distinction between a
real number and its corresponding point on the real number line. Hence
we may also say that a real number *is* on the real line, and a
point on the real number line *is* a real number.

We say that a number * x* is **greater
than** a number * y*,
in symbols

Similarly, we say that * x* is **less than
** *y*, in symbols

Now get ready for a bit of convoluted logic that often confuses students in Math 1010.

A true statement such as

- is true and is false
- is true and is false
- and are both true.

because equals . That's one of the two parts of the meaning of . It may be tempting but would be wrong (false) to make the following statement: is false since is not greater than .

Here is another example. It is a true statement that Napoleon was a man or a woman because he was a man. The fact that he was not a woman does not make the statement false.

The utility of the symbol is more apparent when we don't know the particular relationship. For example we may know that , even if we don't know whether or . Indeed, one major technique in mathematics to establish equality is to show separately that one quantity is greater than or equal to the other, and also that it is less than or equal to the other.

If *x* is less than *y* then we also say that
*x* is **smaller** than
*y*. For example, *-25* is smaller than *2*.
Similarly we define the phrases
**larger**,
**no smaller**,
and **no larger**.

.

Slightly more
The **distance** between two real numbers * x* and * y* is
the absolute value of their difference. For example, the distance
between
* 3* and * 5* is (i.e., 2) which makes perfect sense
if you visualize this on the real number line. A more complicated
example is provided by the distance * d* between * 3* and
* -2*. We have