story12, thanks for your interesting post. Here is your starting comment:
If we divide 1 by b, we know that as long as b is greater than zero, the smaller b becomes, the larger the value of 1/b becomes. For example, 1 divided by 2 is 1/2, 1 divided by 1 is 1, and 1 divided by 1/2 is 2. As the value of b approaches zero, the value of 1/b approaches infinity. So, the smaller we can make b, the larger the value of 1/b should become. Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity. Yet, if we replace b with negative 2, then we have negative 1/2 for the value of 1/b, which is less than zero. Therefore, a negative number can simultaneously be larger than positive infinity, while being smaller than zero.
But, how can this be?
This points to what is fun (I think) about limit theory in calculus -- the odd behavior of numbers when we get close to zero or infinity. If I consider b approaching zero from the right (picture the line of numbers with zero in the middle, positive numbers to the right, and negative numbers to the left), 1/b approaches infinity. At zero I don't have a number, since division by zero is undefined. But close to zero on the left, I have a "large" negative number (infinite in absolute value). That is, if b approaches zero from the left, 1/b approaches "negative infinity", with b=zero again not yielding any number at all.
Taking this further, 1/b approaches zero from above as b gets larger and larger (approaches infinity), and it also approaches zero from below as b gets negatively larger and larger (approaches negative infinity). These limits are called asymptotes -- they're never reached, but can be approached as closely as I want by making b as large (positively or negatively) as I want.
Focusing on what you say here, story12:
Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity. Yet, if we replace b with negative 2, then we have negative 1/2 for the value of 1/b, which is less than zero. Therefore, a negative number can simultaneously be larger than positive infinity, while being smaller than zero.
But, how can this be?
This does seem odd! However, this "function", 1/b, where b varies from negative infinity to positive infinity, is discontinuous
at b=0. There are many such functions in math -- sometimes called "ill-behaved" functions. In this case, as b moves through zero, the function "violently flips" from one extreme to the other, and has no definition at zero. In your illustration, b moved from positive to negative through zero, causing 1/b to "flip" suddenly from extremely large positively, to extremely large negatively.
If I draw the function 1/b as a graph with a vertical line at b=0 for values of 1/b and a horizontal line for values of b, which divides my paper into four quarters, or "quadrants", the graph looks like two dishes, exact replicas of each other, but trapped in different quadrants. See this web site for the graph: http://www.google.com/search?client=saf ... 8&oe=UTF-8
Math isn't always according to my everyday intuition. For example, why does the infinite sum 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... grow larger and larger, going to infinity, while the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2^n + ... approaches 2 but never quite "gets there"? Here, 2^n means 2 multiplied by itself n times.
Cool stuff. For some of us