In Math Fractions Are Strange But Decimals Are Even Stranger - Part I
Most people agree that fractions are a bugbear in mathematics.
Wherever they rear their ugly heads, they wreak havoc on both novice and expert alike.
During my years teaching mathematics, I have seen many a mistake directly attributable to fractions.
Yet with all the wacky considerations of fractions, decimals prove to be far stranger when you plumb the depths of this curious arena.
In this two-part series of articles, we are going to examine some of these "outer-worldly" qualities.
As you may already know, fractions, percents, and decimals are three different ways of expressing the same idea.
A fraction expresses the ratio of two integers a/b; a percent expresses the portion of the whole with respect to itself in parts per hundred, as in 55%; and a decimal expresses a number with respect to a specific tolerance, or to a number of significant digits.
In turn, decimals can come in three forms: terminating, non-terminatingrepeating, and nonterminatingnon-repeating.
The terminating decimals really do not present that much curiosity other than being able to represent a measurement to any arbitrary precision.
For example, 1.
4123, could be thought of as a measurement, in which the 3 in the ten-thousandths place is the digit in question.
If we had a more precise measuring tool, perhaps we could take this same measurement to one in which the digit in question would be the hundred-thousandths place.
So much for terminating decimals.
Where things get really strange-indeed bizarre at times-is in the realm of the non-terminators, so to speak.
This is one area where mathematics starts to open its arms of mystery and to provide us with clues as to the wonders of the universe.
In this series of articles, we are going to experience this wonder with the non-terminating repeaters.
To start, let us take the repeater 0.
333...
Most of you know this as the ordinary fraction 1/3.
Or take 0.
6666...
Most of you know this as 2/3.
How we convert from the decimal to the fraction is not usually learned until one studies at least Algebra II in high school, or perhaps later comes across this idea as an infinite series in elementary calculus; but you are going to learn it here so that you can experience one of the wonders of decimals.
In demonstrating this method, let us take the non-terminating repeater 0.
999...
What this decimal consists of is an infinite string of 9's.
Yes infinite.
Not to seem repetitive, but in order to appreciate the strangeness of this whole concept, I must underscore the idea of "never-ending-ness.
" Now what comes as a shock is that this decimal 0.
999...
is the very same thing as the number 1.
Yes that is correct.
The never-ending, repeating decimal 0.
999...
is the exact same thing as the number 1 (which by the way can be considered the fraction 1/1).
Wow! How could something which seems to always be short from 1 by just a little bit, be actually exactly equal to that very thing? In Part 2 of this article, we examine how this is so, and demonstrate the method which proves this very fact.
Yes fractions can be strange, especially when we consider that they are infinitely dense along the number line, yet still leave gaps which can be fallen through; but decimals are still stranger.
Stay tuned for Part 2...
Wherever they rear their ugly heads, they wreak havoc on both novice and expert alike.
During my years teaching mathematics, I have seen many a mistake directly attributable to fractions.
Yet with all the wacky considerations of fractions, decimals prove to be far stranger when you plumb the depths of this curious arena.
In this two-part series of articles, we are going to examine some of these "outer-worldly" qualities.
As you may already know, fractions, percents, and decimals are three different ways of expressing the same idea.
A fraction expresses the ratio of two integers a/b; a percent expresses the portion of the whole with respect to itself in parts per hundred, as in 55%; and a decimal expresses a number with respect to a specific tolerance, or to a number of significant digits.
In turn, decimals can come in three forms: terminating, non-terminatingrepeating, and nonterminatingnon-repeating.
The terminating decimals really do not present that much curiosity other than being able to represent a measurement to any arbitrary precision.
For example, 1.
4123, could be thought of as a measurement, in which the 3 in the ten-thousandths place is the digit in question.
If we had a more precise measuring tool, perhaps we could take this same measurement to one in which the digit in question would be the hundred-thousandths place.
So much for terminating decimals.
Where things get really strange-indeed bizarre at times-is in the realm of the non-terminators, so to speak.
This is one area where mathematics starts to open its arms of mystery and to provide us with clues as to the wonders of the universe.
In this series of articles, we are going to experience this wonder with the non-terminating repeaters.
To start, let us take the repeater 0.
333...
Most of you know this as the ordinary fraction 1/3.
Or take 0.
6666...
Most of you know this as 2/3.
How we convert from the decimal to the fraction is not usually learned until one studies at least Algebra II in high school, or perhaps later comes across this idea as an infinite series in elementary calculus; but you are going to learn it here so that you can experience one of the wonders of decimals.
In demonstrating this method, let us take the non-terminating repeater 0.
999...
What this decimal consists of is an infinite string of 9's.
Yes infinite.
Not to seem repetitive, but in order to appreciate the strangeness of this whole concept, I must underscore the idea of "never-ending-ness.
" Now what comes as a shock is that this decimal 0.
999...
is the very same thing as the number 1.
Yes that is correct.
The never-ending, repeating decimal 0.
999...
is the exact same thing as the number 1 (which by the way can be considered the fraction 1/1).
Wow! How could something which seems to always be short from 1 by just a little bit, be actually exactly equal to that very thing? In Part 2 of this article, we examine how this is so, and demonstrate the method which proves this very fact.
Yes fractions can be strange, especially when we consider that they are infinitely dense along the number line, yet still leave gaps which can be fallen through; but decimals are still stranger.
Stay tuned for Part 2...
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