A real number has a decimal representation. It gives the approximate location of the number on the real line.
¨ 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.
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,
takes the mathematician's point of view. If I refer to 3.14159, I mean the
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).
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 notation
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
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:
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.
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 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.
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.
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 di and ei are decimal digits, and suppose that for some integer the following four statements are all correct:
a) di = ei for ;
b) dk = ek + 1;
c) di = 0 for all i > k; and
d) ei = 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 .