The Hindu – Arabic Series

At first glance, the topic of series seems to be something encountered late in the year of a BC Calculus course. But everyone uses series any time they use numbers, which is to say very often. Let’s look at this particularly important series.  

The way numerals were written way back when was clumsy. If you don’t believe me, try multiplying or dividing with Roman Numerals. Around AD 1200, Leonardo of Pisa introduce the Hindu-Arabic system for writing numbers to Europe. He learned this system during his travels in the Middle East. Leonardo is also known as Fibonacci although he was not given that name until the 1500s. While he is better known for his famous sequence, I think improving the way numbers are written is a much more important contribution to mathematics (even though it would have caught on eventually). This is the system used world-wide today.

A page of Fibonacci’s Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled with Latin numbers and Roman numerals (black) and the value in Hindu-Arabic numerals (red). (Some numerals are slightly different than those in use today.)

Hindu – Arabic notation is a shorthand for a series, sometimes finite, often infinite.  It is a sophisticated idea when thought of in modern terms. This “new” system is a place-value system: the value of each digit depends on its position relative to the decimal point, defined by a sequence. For example,

\displaystyle \begin{array}{l}456.789=4(100)+5(10)+6+7(\tfrac{1}{{10}})+8(\tfrac{1}{{100}})+9(\tfrac{1}{{1000}})\\\end{array}

This notation has advantages over other methods of denoting numbers. The decimal representation of a number is unique (well almost, as we’ll see below), whereas every Rational number may be written as many different fractions. Certainly, makes computation easier. It also makes finding approximations and arranging numbers in order easier than writing them as fractions.

But other things also happen.

The Hindu-Arabic decimal system revealed that all Rational numbers written in this notation are repeating decimals. A repeating decimal is an expression containing a string of one or more digits that repeated forever. For example, 1/3 = 0.333333… with the “3” repeating forever and \displaystyle \frac{{241}}{{55}}=4.38181818... with “18” string repeating forever. (Some numbers repeat zeros forever; they are a special case called terminating decimals.)

The decimal form of a fraction may be found by using the division algorithm. Since only those numbers less than the divisor may appear as “remainders,” eventually one of them will appear again after which the succeeding digits will repeat.

Conversely, any repeating or terminating decimal can be written as a quotient of integers. This example shows the procedure.                                                              

Let \displaystyle n=4.3818181818...

Then \displaystyle 100n=438.18181818...

Subtracting the first from the second

\displaystyle 99n=433.8

\displaystyle n=\frac{{433.8}}{{99}}=\frac{{4338}}{{990}}=\frac{{241}}{{55}}

So, all the Rational numbers can be written as a repeating or terminating decimal, and conversely all repeating or terminating decimals are Rational numbers. Numbers that cannot be written as repeating or terminating decimals are exactly the Irrational numbers.  

An Irrational Number – the Diagonals of a Square

The length of the diagonals of a square is a non-repeating decimal. That is, the length must be expressed as an infinitely long decimal that contains no string of digits that repeats. \displaystyle \sqrt{2} is an Irrational Number and there are a lot of others like it!

By the Pythagorean Theorem the diagonals of a square with sides of one unit have a length denoted by \displaystyle \sqrt{2} – the number whose square is 2. In a previous post, I showed a way, one of several, to find closer and closer decimal approximations to \displaystyle \sqrt{2}. The table below shows the results.

n = decimal places
Ln < \displaystyle \sqrt{2}Gn > \displaystyle \sqrt{2}
Each number in list Ln was produced by taking the preceding number and affixing the digits 0, 1, 2, … , 9 to it, squaring that number, and finding the largest whose square was less than 2. The Gn is the next number, the smallest number with a square greater than 2. The numbers in Ln have squares less than 2; the numbers in Gn have squares greater than 2.

The Ln list is a sequence of numbers that has two important properties easily seen from how it was developed: (1) it is non-decreasing – each number is greater than or occasionally equal to the preceding number because each time we append an extra digit we get a greater number, and (2) the list is bounded above – the numbers never exceed 100, or 15, or \displaystyle \pi , or 2, or in fact any number from the Gn list. The smallest number they never exceed is \displaystyle \sqrt{2}. We know this because this is how the list was developed.

The number that’s between the two lists is the number we’re looking for is \displaystyle \sqrt{2}, but we can never find an “exact” decimal representation. The two lists give better and better approximations to \displaystyle \sqrt{2}. They close in on it. But neither gets there.

So, how do we know that number exists?

The Axiom of Completeness

All the decimal numbers, the Rational Numbers, and the Irrational Numbers (and no others), make up a set called the Real Numbers.

If this list above can be continued forever, it will never get to \displaystyle \sqrt{2}. To handle this kind of situation a new rule (called an axiom) was imposed.

The Axiom of Completeness: Every non-decreasing sequence of Real numbers that is bounded above converges to – gets closer and closer to – its least upper bound.

Since this is an axiom, it is not proved; it is just accepted as fact.

The axiom says that even though there is no decimal to represent it, the number nevertheless exists.

Someone made the axiom up. This is very different than observing and naming a property of numbers like the commutative property or the associative property. It doesn’t have to be true, but it seems very reasonable, and no one has ever found a counterexample. [1]

In the example, the least upper bound is \displaystyle \sqrt{2}. How do we know that? Because that’s what we made the sequence to do. Any other method of “finding” \displaystyle \sqrt{2}, and there are many, gives us, not just the same kind of thing, but the exact same list! Creepy, isn’t it?

Thus, \displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\left\{ {{{L}_{n}}} \right\}=\sqrt{2}

Not only are all Irrational Numbers handled the same way, but all repeating decimals are also. They never repeat digits and, however you find their decimal approximations, the same thing happens: you have a non-increasing sequence of numbers that is bounded above and, by the Axiom of Completeness, converges to the fraction.

I mentioned above that the Hindu-Arabic system gives a unique expression for every number. Not quite. Consider 1/3 = 0.3333333 ….  Since three times one-third is one, it must be that 3 times this decimal which is 0.9999999…. = 1.

At first, I disliked decimals because they were not “exact.” I got over that. For the cost of the Axiom of Completeness, we have a system for writing numbers that makes computation easy (if, often, only to a very good approximation). It’s worth the cost.

But does \displaystyle \sqrt{2} exist? Is it really there?

[1] As a corollary to the axiom, there is a theorem that says a non-increasing sequence that is bounded below, such as Gn, converges to its greatest lower bound, again \displaystyle \sqrt{2}.

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