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Back to the real numbers chapter

How to think
about the decimal representation

Decimal
representation and geometric series

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 *n*th 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.

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 .

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.

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

¨
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

**Abstractmath.org always
takes the mathematician's point of view. ****If I refer to 3.14159, I mean the
rational number **

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.)**

¨
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.**

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 *i*th place. (For example, for ,
and so forth.)
Then the

the decimal notation represents the limit of the series

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

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

**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.**

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

**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.**

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.

**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**.)

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.

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.

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.

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

Let and ,
where all the *d _{i} *and

a) *d _{i}*

b)* d _{k} = e_{k} *+ 1;

c)
*d _{i }= *0 for all

d)
*e _{i }= *9 for all

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 .