Comparison of Real Numbers - Two Ways
various kinds of numbers
Along a line, given the origin and a positive unit distance from it,
real numbers may be employed as coordinates for points on the line
Example 1. Look at the two positive numbers ${}^+10$ and ${}^+15$.
The second is 5 units.
more than the first in length. The second equals the first plus 5 units. On the line,
the second is more to the right.
Example 2. Look at the two negative numbers ${}^-10$ and ${}^-15$. The second
is 5 units
more than the first in length, but it equals the first - 5 units. On the line,
the second number is more to the left.
In comparing the four numbers ${}^+10$ and ${}^+15$, ${}^-10$ and ${}^-15$, we
may order by length or may order them by position. The plus and minus fifteens
have the same length, and their common length is more than the common length
of the plus and minus tens. But
${}^+15$ is 5 units more [to the right] than ${}^+10$
${}^+10$ is 10 units more [to the right] than ${}^-10$
${}^-10$ is 5 units more [to the right] than ${}^-15$
In algebraic terms, if A - B is positive for a first real numbers A and second
real number B, then on a coordinate line,
A is to the right of B by A-B units. But if B - A is positive then B is to right
of A by B-A units. Here A - B and B-A when calculated have opposite signs, but are
equal in length.
the length of the first is five less than than the length of the second
\[ \mbox{length}({}^+10) = 10 \mbox{ is 5 less than } 15 = \mbox{length}({}^+15) \]
The first number +10 is less than than the second number 15 by 5 units
\[{}^+10 = {}^+15 - {}^+5 \]
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