LFN Staff Emeritus
Join Date: Apr 2002
Something to ponder
Does .99 repeating (.99 to an infinite number of 9s) equal 1? It does in fact equal 1. I'm just going to say that here. Now, as to why...I will give a very simple why and then a detailed why.
First of all, I will give you a basic situation in which to determine how .99 repeating could equal one. Imagine that a baseball is being thrown from first base to second base. For the case of this situation let's pretend that it is being thrown in slow motion so we can carefully observe the ball. Now, in order for it to reach it's destination, the ball at some point will cross the halfway point in between the two bases.
As the ball continues along its path, it crosses the halfway point in between the halfway point of the two bases and second base. However, it's speed remains constant.
As the ball continues further, it crosses yet another halfway point
and so on. As the ball continues to travel, the speed remains the same, as the ball is crossing halfway points faster and faster and faster.
Now, using simple logic, one could attempt to declare that the ball would NEVER reach its destination because there will ALWAYS be another half for the ball to have to cross.....even if the number is so minuscule and small that it could never be possibly measured by mankind. However, in the real word, the ball in fact reaches its destination, despite the outcry of simple logic. The halfway mark starts at .5, then moves onto .25, and so on. Eventually the halfway mark is such a large decimal number that at SOME point the number actually breaches and goes to 1. Otherwise, no object, or person, would ever reach any kind of destination.
The fact of the matter is, in some cases, an infinite series of numbers that go into infinity can actually have a sum result generated. That's right. In SOME cases, you can add ALL of infinity and get a true number result. For this example's sake, we are going to use a normalized rate, which is an unchanging number that is multiplied by the previous answer.
For example, 1, 2, 4, 8, 16... this infinite series goes on forever, and as you can tell, the constant being multiplied by the previous is 2. 1 * 2 = 2. 2 * 2 = 4. 4 * 2 = 8. 8 * 2 = 16, and so on. Now, rates by no means have to be whole numbers. What happens if we start at 10 with a rate of .5? 10, 5, 2.5, 1.25, .625, .3125, and so on. Now, despite that these numbers will go into INFINITY...you'll constantly be dividing by half, you can actually get a REAL SUMMATION of ALL the infinity numbers. The formula for this is
while -1 < rate < 1 (in our example rate = .5 which is between -1 and 1)
First number (in this example, 10)
-------------- (divided by)
1 - rate [in this example, (1 - .5)]
so our problem would be
10 divided by (1 - .5)
Do we get an error? Do we get an infinite number that goes on forever? Actually, no. If we were to take our example of 10 (10), then adding 5 (15), then adding 2.5 (17.5), and so on to infinity, we would eventually have a total sum of 20. Crazy, eh?
What does this prove? At some point, a decimal number going on into infinity and the difference between it and another number much like it becomes so immeasurable that there is no difference, at all and it equals each other. Despite there always being a halfway point the ball breaches, for the halfway point eventually is SO small that the halfway point ceases to exist and the ball reaches its destination.
So yes, .9 may not = 1. .9999999 may not = 1. But eventually .99999999999999999... to infinity becomes so small it breaches the immeasurable point and it equals 1. If it didn't, then no object in time or space would be able to ever reach its destination.
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