Question 752581
To graph and compare real numbers you just plot them on a number line, like this:
 
{{{number_line( 600, -10, 10, -1, sqrt(3), 6.5, -60/7,1-sqrt(19) )}}}
 
The number line is like a ruler, with positive and negative integers marked at regular intervals. Negative numbers are to the left of zero, positive numbers are to the right of zero.
TIP: That is easier to do with paper that already has regularly spaced markings. Grid paper is a good choice. Paper with lines works well if you turn the paper so that the lines run up and down rather than side to side.
 
COMPARING:
The numbers to the left are smaller than (less than) the numbers to the right, so
{{{1-sqrt(19)<-1}}} and {{{sqrt(3)<6.5}}}.
You can turn that around and say that a number than is to the right is larger (greater) than all the numbers to its left.
Negative numbers are to the left of zero, so they are less than zero:
{{{-9<0}}}, {{{-60/7<0}}}, {{{1-sqrt(19)<0}}}, {{{-1<0}}}.
Positive numbers are to the right of zero, so they are less than zero:
{{{2>0}}}, {{{sqrt(3)>0}}}, {{{6.5>0}}}, {{{8>0}}}.
 
ABSOLUTE VALUE:
Negative numbers are to the left of zero, positive numbers are to the right of zero.
The absolute value of a number is the distance between the number and zero in the graph,  and It is always a positive number.
For example,
{{{-1}}} with {{{abs(-1)=1}}} is {{{1}}} unit to the left of zero.
{{{-60/7}}} with {{{abs(-60/7)=60/7}}} is {{{60/7=8&1/7}}} units to the left of zero.
 
PRACTICAL TIPS:
To figure out where to place a number like {{{-60/7}}} or {{{1-sqrt(19)}}} you can calculate an approximate value as a decimal.
{{{-60/7}}}= -8.571428571428571428571428571428....
so you can use {{{-60/7=-8.6}}} (rounded) to figure out where to plot that value.
The point for {{{-60/7}}} must be {{{8.6}}} units to the left of zero. That's a distance a litle longer than {{{8.5=8&1/2}}} units, so you place it a little to the left of the point that is halfway between -8 and -9.
{{{1-sqrt(19)=-3.36}}} (rounded)
Your calculator may give you the result of that calculation as {{{3.358898944}}}, and you know that it is an irrational number with infinite non-repeating digits, but for your purposes, {{{-3.36}}} is a good approximation, accurate enough.
So, when graphing by hand, you plot the point for {{{1-sqrt(19)}}} at a distance of about {{{3.36}}} units to the left of zero. You can pretend that {{{-3.36}}} is
{{{-3&1/3}}}= -3.333333333... and plot {{{1-sqrt(19)}}}= approximately{{{-3.36}}} {{{1/3}}} of a unit to the left of {{{-3}}}.