For some reason some people have an extremely difficult time accepting that 0.9 repeating is equal to 1. Not that it merely "approaches" 1, but that it's exactly equal to 1. They are just two different ways to write down the same value.
Some people are so incredibly obsessed with trying to prove that they are not equal that they will go to incredible lengths to try to do so. They will start arguing semantics, they will try to muddle the definition of an infinitely repeating decimal, and some may even attempt to invent completely new mathematics in order to somehow make the two things not equal. They are so obsessed with this that there's absolutely nothing you can tell them that would convince them otherwise. Nothing. You can try, but you will fail.
Regardless, even if it's rather moot (and will never, ever convince these people), here are two slightly different approaches at showing the equality of the two things. Instead of trying to prove it yourself, try to make them do the work.
Approach 1: Repeating decimal patterns as a fraction
It's a well-known result that every real number which decimal representation has an infinitely repeating pattern starting at some point after the decimal point is a rational number, and this is actually relatively easy to prove. And, in fact, this is a (well-known) one-to-one relationship: In other words, if the decimal representation of a number has an infinitely repeating pattern after the decimal point (not necessarily starting immediately after the decimal point, but from some point forward after that), it is a rational number, and if it doesn't have such a pattern, it's an irrational number.
This can be more succinctly (and mathematically) expressed as: A real number is rational if and only if its decimal expansion is eventually periodic.
And since such a value is a rational number, it can be written as a fraction, ie. the ratio between two integers. And, indeed, all values whose decimal representation has an infinitely repeating pattern starting at some point after the decimal point can be written as a ratio of two integers, ie. a fraction.
As an example 0.4 repeating is a rational number and can be written as 4/9.
Since this is a proven mathematical fact (and it's actually relatively easy to prove yourself), that means that 0.9 repeating is also a rational number which can be written as a fraction, ie. the ratio between two integers.
So the question is: Given that proven mathematical fact, find out what those two integers are. In other words, what is the fraction that gives 0.9 repeating.
If you want to present the argument to someone succinctly, it could be something like this:
"It's a known result that a real number is rational if and only if its decimal expansion is eventually periodic. This is easy to prove. That means that 0.9 repeating is a rational number. This also means that, as a rational number, it can be written as the ratio of two integers. Calculate what those two integers are."
Approach 2: Calculate the difference
If two values are equal, then their difference, ie. one subtracted from the other, is 0, pretty much by definition.
If two values are not equal, then their subtraction will differ from 0, again pretty much by definition.
Thus, if the real number 1 is different from the real number 0.9 repeating, calculate their difference, ie. the result of their subtraction. If they are indeed not equal, then the result has to be a real number that's different from 0. What is that real number?
(Note: There is no such a thing as "the smallest real number larger than zero". Such a thing does not exist, and it's logically and mathematically impossible to exist, especially in the set of real numbers. This is a quite famous and extremely trivially provable fact of arithmetic.)
This could be succinctly presented as:
"By axiomatic definition, if two real numbers are equal, their subtraction results in 0. Conversely, if two real numbers are not equal, their subtraction results in a non-zero real number. Calculate the subtraction of 1 and 0.9 repeating."
No comments:
Post a Comment