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Decimal representation of real numbers
Contents
How to think
about the decimal representation
Decimal
representation and geometric series
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… (infinite number of 3’s).
¨
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 "
¨
The mathematician may think of it as a precisely given number,
namely ,
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
Abstractmath.org always
takes the mathematician's point of view. If I refer to 3.14159, I mean the
rational number 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. Agitated objection.
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 inifinite
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
This inequality gives an estimate of the accuracy of this approximation:
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.
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
The main theorem
about infinite geometric series is that, for ,