Real Numbers, Decimal Representation

Appendices, Volume 3, Why Slopes and More Math.

Some fractions can be written in the form

m
10^k

where m is a natural number and k is an integer. Such fractions have a finite decimal expansion.

Theorem: If a fraction r can be written in the form p/q where p is an integer and q is given by a products of 2s and 5s (i.e. has no other prime factors), then p/q can be written in the form

m
10^k

Now fractions with denominators with prime factors other than 2 and 5 do not have finite decimal expansions. They have periodic decimal expansions. For example

Here the infinite decimal expansion may be found by long division. Long division is done until the expansion starts to repeat.

Arithmetic with fractions can be done directly and exact without decimal expansions, or approximately with decimal expansion. In approximate calculations, only finitely many decimals are used - the more, the better, for the sake of accuracy. It can be shown that arithmetic with periodic decimal expansions produces results with periodic decimal expansions. Error control with approximate arithmetic depends on the continuity or error control analysis of addition, subtraction, division and multiplication.

The whole number 1 = 1.000 and the repeating decimal expansion 0.99999 give two decimal representations of the same number. The first expansion 1 = 1.000 (finitely many zeroes or none) is finite and exact. The second decimal expansion 0.9999 (9 recurring) represents a sequence of fractions 0.9, 0.99, 0.999, 0.9999, whose limit equals 1. When a number has a finite and an infinite decimal expansion, the finite one is simpler to use, but both are valid.

The square root of 2 is not a fraction. But there is a sequence of decimal numbers

• 1.41421
• 1.414213
• 1.4142135
• 1.41421356
• 1.414213562
• 1.4142135623

whose squares have the limiting value 2. The error (difference between) the limit 2 and the square decreases as more and more decimal places are used.

On a coordinate line, any line segment whose length can be approximated by an infinite decimal expansion is considered to be a real number.

Here continuity or error control arguments allow us to do arithmetic with infinite decimal expansion and compute the results with unlimited error control to an unlimited number of places. We assume that each finite and each infinite decimal expansion gives us a real number.

Cauchy Sequences

Imagine we have an infinite sequence of numbers g(1), g(2), g(3), ... This sequence is said to be a Cauchy sequence if the one of the following properties holds:

• (Decimal Perspective): For every whole number k, there exist a whole number m such that g(p) will agree with g(q) to k decimal places when p > m and q > m
• (Decimal Free Perspective): For every positive number E > 0, there exists a whole number m such that |g(p) - g(q)| <e if p > m and q > m.

Both conditions are equivalent. Each implies the other. Again, why depends on how you think of the real numbers.

Each infinite decimal expansion can be thought of as a Cauchy Sequence in which the k-th term gives the limit, a real number, to say k-decimal places.

Now every Cauchy Sequence has a limit L. To show this, we assume that specifying in principle how to compute the decimal expansion of L determines the value of L. (The number pi = 3.14... is an example of real number that can be computed to million of decimal places. The number pi is given by the limit of this decimal expansion.)

Now if we have a Cauchy sequence g(1), g(2), g(3), ... , how do we determine the first k decimal places of a limit L. The answer is simple. According to the decimal perspective we may compute L to k-decimal places because

For every whole number k, there exist a whole number m such that g(p) will agree with g(q) to k decimal places when both p and q are greater than m.

So given k, we may choose or find in principle, a whole number m with the property that g(p) and g(q) will agree to k decimal places whenever both p and q are more positive than m. Take the decimal expansion of g(m+1) to k decimal places. This decimal expansion to k places tell us how to compute L to k decimal places. Since k can be as large as we like, that is, arbitrary, we can in principle determine every digit in the decimal expansion of a number L. Simply go far enough along the sequence. By this construction, a limit L of the Cauchy sequence g(1), g(2), g(3) can in principle be computed. That is enough to say the limit L exist at least in principle.

The argument using decimal free perspectives of real numbers is more complicated.

The Role of Decimals

The decimal-free set theoretic view of mathematics reached it almost final form in the 1920s. It took another 30 years, that is, until the 1950s, for the set theoretic view of mathematics to be adopted in mathematics departments. The modern mathematics movement in the 1960s was intended to spread or provide a setting for the teaching of the set theoretic perspective.

The set theoretic perspective began about the mid 1800s, and it was used in the period 1900 -1930 to provide a strict thought-based foundation for computations --- the arithmetic based part of mathematics --- a foundation (hopefully) free of contradictions and inconsistencies. This set theoretic perspective was not developed for ease of exposition. The initial aim in studying sets was not to provide a foundation for arithmetic based mathematics. In the set theoretic approach to mathematics after arithmetic (counting included), the decimal perspective of real numbers was not necessary. So it was put aside.

In contrast, the common knowledge of mathematics is based on counting, a decimal knowledge of arithmetic and real numbers, and the use of simple formulas. This common knowledge is introduced and hopefully explained in elementary school in a thought-based manner. The common knowledge presently encompasses counting, arithmetic and the use of simple formulas.

The decimal expansion of real numbers provides a concrete sense of convergence. Unfortunately, in the zeal to derive the set theoretic perspective from first set-theoretic principles or assumptions about real numbers in our high schools and colleges, the decimal perspective was put aside at least partially. That is, while the decimal representation of whole numbers and real numbers was employed in computational examples in algebra, trig, chemistry, physic, business and calculus, the chains of reasoning emphasized in algebra and calculus typically made no mention of decimals (nor units). Decimals (and sometimes units) were used in many computational subjects yet not recognized nor sanctioned in math courses axioms.

Courses on analysis (advanced calculus) could be made more accessible to students by detailing in them a set-theoretic justification of decimal expansions and their convergence of the latter. Before and after this, courses that discuss the decimal and decimal-free perspective would be agreeable both to students of analysis and students who just assume the convergence of decimal expansion. Ease of exposition is the motivation for this suggestion.

Remark: A mathematics or science student could follow the more accessible decimal perspective in a calculus and then in a later analysis (or advance calculus) course, meet the set theoretic perspective justification and/or reformulation of the decimal arguments. Does rigor in haste lead to rigor mortis?

Remark (for advance students): Appendices in Volume 3, Why Slopes and More Math, provide the decimal and decimal free perspective of the basic theorems in calculus. For instance, the Bolzano Weierstrass Theorem that every infinite set in a closed interval has a limit point can be viewed as consequence of the Pigeon hole principle. The leftmost limit point has a decimal expansion computed to k decimals by covering the interval with nicely aligned subintervals of length 10**(-k) and locating the leftmost one with infinitely points (or concluding that such an interval most exist. The latter must contain the leftmost interval of length 10^-k-1. Thus a sequence of nested intervals with left end points converging to the "lim inf" (technical expression) of the set is obtained.

Remark (also for advanced students): Lexicographic ordering of points and intervals with sides of length 10^-k should extend this argument to bounded infinite sets or sequences in Rⁿ

If a bounded region B is covered and partitioned by intervals whose sides have length 10^-k, and an infinite S in B is given, then there must be a lexicographically least interval with infinitely many points of S inside it. As k increases, these intervals will be nested, and the lexicographical least corners of the k-th interval will yield the decimal expansion of a limit point -- approach the limit in a lexicographically increasing fashion. Exercise: Verify the details and show that writer has not made any mistakes. (In this process we could define the lim inf of a set in R^{^}n with respect to the lexicographic ordering of points.

Here (x₁, x₂, ..., x_n) is lexicographically > (y₁, y₂, ..., y_n) iff there is a whole number k (1 < k < n) such that x_m = y_m if m < k and x_k < y_k.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science