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Posted 25 November 2008

Decimal representation of real numbers

Contents

Notation and terminology. 1

Variations in usage. 2

How to think about the decimal representation. 2

Decimal representation and geometric series. 2

Notation and terminology

A real number has a decimal representation.  It gives the approximate location of the number on the real line.

Examples

¨  The rational number 1/2 is real and has the decimal representation 0.5.  The rational number  has the representation . 

¨  The number 1/3 is also real and has the infinite decimal representation 1.333… This means there is an infinite number of 3’s, or to put it another way, for every positive integer n, the nth decimal place of the decimal representation of 1/3 is 3.

¨  The number  has a decimal representation beginning 3.14159…  So you can locate  approximately by going 3.14 units to the right from 0.  You can locate it more exactly by going 3.14159 units to the right, if you can measure that accurately.  The decimal representation of  is infinitely long so you can theoretically represent it with as much accuracy as you wish.  In practice, of course, it would take longer than the age of the universe to find the first  digits. 

Bar notation

 It is customary to put a bar over a sequence of digits at the end of a decimal representation to indicate that the sequence is repeated forever.  For example,

and 52.71656565… (65 repeating infinitely often) may be written . 

A decimal representation that is only finitely long, for example 5.477, could also be written .

Terminology

The decimal representation of a real number is also called its decimal expansion.  A representation can be given to other bases besides 10; more about that here.

Variations in usage

Approximations

If you give the first few decimal places of a real number, you are giving an approximation to it.  Mathematicians on the one hand and scientists and engineers on the other tend to treat expressions such as " 3.14159" in two different ways.

¨  The mathematician may think of it as a precisely given number, namely 314159 / 100000, so in particular it represents a rational number. This number is not , although it is close to it.

¨  The scientist or engineer will probably treat it as the known part of the decimal representation of a real number. From their point of view, one knows 3.14159 to six significant figures.

Abstractmath.org  always takes the mathematician's point of view.  If I refer to 3.14159, I mean the rational number 314159 / 100000.  I may also refer to  as “approximately 3.15159…”.

Integers and reals in computer languages

Computer languages typically treat integers as if they were distinct from real numbers. In particular, many languages have the convention that the expression ‘2’ denotes the integer and the expression ‘2.0’ denotes the real number.   Mathematicians do not use this convention.  They regard the integer 2 and the real number 2 as the same mathematical object.  (Well, most of them do, anyway.)

How to think about the decimal representation

¨  The decimal representation is not the number, any more than an Exxon sign is the Exxon corporation.  It is a representation of the number.  (Duh).   It is good to know the representation, or the first part of it, since it allows you to place the number in approximately the right place on the number line (or to approximate a distance of that length). 

¨  The notation  denotes a decimal representation of .  This decimal representation contains an infinite number of 3’s after the decimal point.   It is wrong to think of it as “going to infinity” or “going on for ever and ever”.  It is not going anywhere.  It already has all of the 3’s.  It is a static mathematical object, not a changing process.  More here.

Decimal representation and infinite series

The decimal representation of a real number is shorthand for a particular infinite series (MW, Wik).  Let the part before the decimal place be the integer n and the part after the decimal place be

where  is the digit in the ith place.  (For example, for ,   and so forth.)  Then  the

the decimal notation  represents the limit of the series

Example

The number  is EXACTLY equal to the sum of the infinite series.  If you stop the series after a finite number of terms, then the number is approximately equal to the resulting sum.   For example, 42 1/3 is approximately equal to

The inequality below gives an estimate of the accuracy of the approximation above:

Agitated objection

When I think about  I can’t visualize an infinite number of 3’s all at once.  I can think of them only as coming into the list one at a time.

Sharp rejoinder:

You are not being asked to visualize all the 3’s at once, but just to accept the fact that the notation  denotes all the 3’s at once, and that that is the decimal representation of .  Live with it. 

In ordinary English the “…” often indicates continuing through time, as in for example

“They climbed to the top of the ridge, and saw another, higher ridge in the distance, so they walked to that ridge and climbed it, only to see another one still further away…”

But you should think of the decimal representation of  as a complete, infinitely long sequence of decimal digits, every one of which (after the decimal point) is a “3” right now.  You should similarly think of the decimal expansion of  as having all its decimal digits in place at once, although of course in this case you have to calculate them in order.  Calculating them is only finding out what they are.   They are already there.

Important:  This description is about how a mathematican thinks about infinite decimal expansions.  The thinking has some sort of physical representation in your head that allows you to think about to the hundred millionth decimal place of  or  even if you don’t know what it is.  This does not mean that you have an infinite number of slots in your brain, one for each decimal place!  Nor does it mean that the infinite number of decimal places actually exist “somewhere”.  After all, you can think about unicorns and they don’t actually exist somewhere.

Exact definitions

Both the following are true:

(1)  The numbers 1/3,  and  have infinitely long decimal representations, in contrast for example to , whose decimal representation is exactly 0.5. 

(2)  The expressions “1/3”, “  ”and “  ” exactly determine  the numbers 1/3,  and : 

a)     1/3 is exactly the number that gives 1 when multiplied by 3. 

b)      is exactly the unique positive real number whose square is 2.

c)      is exactly the ratio of the circumference of a circle to its diameter.

These two statements don’t contradict each other.  All three numbers have exact definitions.  The decimal representation of each one to a finite number of places provides an approximate location of that number on the real line.  On the other hand, the complete decimal representation of each one represents it exactly, although you can’t write it down.

Example

A teacher may ask for an exact answer to the problem “What is the length of the diagonal of a square whose sides have length 2?”  The exact answer is .  An approximate answer is 2.8284.

Different decimal representations for the same number

The decimal representations of two different real numbers must be different. However, two different decimal representations can, in certain circumstances, represent the same real number.   This happens when the decimal representation ends in an infinite sequence of 9’s or an infinite sequence of 0’s.

Example

These equations are exact.   is exactly the same number as 3.5.  (Indeed, , 3.5, 35/10 and 7/2 are all different representations of the same number.)   

Two proofs that  

The fact that  is notorious because many students simply don’t believe it is true.  I will give two proofs here.  There is much more detailed information about this in Wikipedia.

Proof by formula

This proof uses geometric series and requires understanding limits and infinite series.  The main theorem about infinite geometric series is that, for , this exact equation holds:

The series represented by  is .  So here a = 9  and .   Then by the main theorem,

This is an exact equation.  It says  is 1, not that it “goes to 1” or is “nearly 1”. 

Proof using the Archimedean Property

The proof in this section (suggested by Maria Terrell) requires less theoretical machinery than the previous proof.   However, you still have to believe that , which means  by definition, converges to a real number.

The Archimedean Property says that if r is a real number then there is an integer n bigger that r. 

Lemma   If r is a positive real number, there is an integer n such that . 

Proof of Lemma   

a)     If r  is positive then so is . 

b)     The Archimedean Property says that there is an integer n such that . 

c)     That means there is an integer n so that  by a standard rule about inequalities. 

   End of proof.

The contrapositive of the Lemma says:

Lemma   If r is a real number for which for every integer n, , then r is not positive.

Proof that    

a)     To prove  is the same as to prove that .   

b)     Let k be any positive integer and let t = .   So .  For example, for k = 3, .

c)     Then .   For example,  .

d)     Then  since all the terms in  are positive (remember  ).

e)     t  is the sum of the first k terms in  (which are all positive).  By c) and d), for all integers k,  

f)      If n is any integer, then there is an integer k such that  (let k be the number of digits in n, for example).  This means that . 

g)      since it is the absolute value of something.

h)     Now e) and f)  prove that for all integers n,  .

i)      The contrapositive of the Lemma means that  can’t be positive, but , so only possibility left is that , and that is what we had to prove.

Explicit description

Here is an explicit description of all pairs of decimal representations that represent the same number:

Theorem

Let  and , where all the d_i and e_i are decimal digits, and suppose that for some integer  the following four statements are all correct:

a) d_i = e_i for ;

b) d_k = e_k + 1;

c)  d_i = 0 for all i > k; and

d)  e_i = 9 for all i > k.

Then m = n.  Moreover, if the decimal representations of m and n are not identical but do not follow the pattern described by a) through d) for some k, then .