Jon Skeet said:
1/3 is in the range, but can't be stored exactly. Which part of that
statement do you disagree with?
Just as an example: Consider the SQL NUMERIC(2, 1) data type. 1/3 is
*absolutely not* a member of the set of valid values for this type. The
complete set of valid values for this type is:
{ -9.9, -9.8, -9.7, -9.6, -9.5, -9.4, -9.3, -9.2, -9.1, -9.0, -8.9, -8.8, -8.7,
-8.6, -8.5, -8.4, -8.3, -8.2, -8.1, -8.0, -7.9, -7.8, -7.7, -7.6, -7.5, -7.4,
-7.3, -7.2, -7.1, -7.0, -6.9, -6.8, -6.7, -6.6, -6.5, -6.4, -6.3, -6.2, -6.1,
-6.0, -5.9, -5.8, -5.7, -5.6, -5.5, -5.4, -5.3, -5.2, -5.1, -5.0, -4.9, -4.8,
-4.7, -4.6, -4.5, -4.4, -4.3, -4.2, -4.1, -4.0, -3.9, -3.8, -3.7, -3.6, -3.5,
-3.4, -3.3, -3.2, -3.1, -3.0, -2.9, -2.8, -2.7, -2.6, -2.5, -2.4, -2.3, -2.2,
-2.1, -2.0, -1.9, -1.8, -1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1.0, -0.9,
-0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0.0, 0.1, 0.2, 0.3, 0.4,
0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,
2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4,
3.5, 3.6, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9,
5.0, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 6.0, 6.1, 6.2, 6.3, 6.4,
6.5, 6.6, 6.7, 6.8, 6.9, 7.0, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9,
8.0, 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 9.0, 9.1, 9.2, 9.3, 9.4,
9.5, 9.6, 9.7, 9.8, 9.9 }
Likewise with NUMERIC (10, 2), DECIMAL (24, 18), or any other valid
precision and scale settings for the NUMERIC and DECIMAL types.
The fact that you cannot accurately store a value that is not a member of
the set of valid values for a type is of no significance. Since by
definition 1/3 is not a member of the set of valid values for the DECIMAL or
NUMERIC types, why do you find it surprising that 1/3 cannot be stored
exactly using these types?
They're the same thing. Converting 0.1 base 3 (i.e. 1/3) into a decimal
*is* a conversion from one base to another, and the Decimal and Numeric
types *do not* store that number exactly.
No, they are two different arguments. As explained, 0.1 is a non-repeating
decimal, whereas 1/3 is an infinitely repeating decimal which cannot be
represented exactly in decimal form, period. The fact that an infinitely
repeating decimal cannot be represented *exactly* using a DECIMAL or NUMERIC
type should come as no surprise to you or anyone else for that matter. The
fact that a non-repeating decimal with a scale of 1 or 2 (just one or two
digits after the decimal point) cannot be stored *exactly* using FLOAT and
REAL data types comes as a surprise to many people.
I'm happy to disagree with any number of people, if what they're saying
doesn't make sense.
That's perfectly fine; however, as long as the standard calls INT, DECIMAL,
NUMERIC, etc., "exact" numeric data types and it calls FLOAT and REAL
"approximate" numeric data types, you may as well prepare yourself to
disagree with every SQL database programmer and designer in the world.
In that case the same can be said for Float. Every Float value is an
exact value - you can write down the *exact* binary string it
represents, and even convert it into an exact decimal representation.
Sounds great! Prove it: Store 28.08 as a FLOAT value and write doen the
*exact* binary string it represents. Then convert it into an exact decimal
representation. Here's something to get you started:
DECLARE @r FLOAT
SELECT @r = 28.08
SELECT @r
--Result in QA: 28.079999999999998
I don't think that makes any difference - when maximum precision is
used, the range of valid values is given in terms of a minimum and a
maximum, and not every value in that range is exactly representable. If
you're going to treat it as "every value that can be exactly
represented can be exactly represented" then as I say, the same is true
for Float.
Of course you don't think it makes any difference, or you would have quoted
the full quote instead of just the last part that appears to substantiate
your argument. The point is that when the precision and scale are turned
down the set of valid values changes for a given NUMERIC or DECIMAL data
type declaration. See the example above for NUMERIC (2, 1) for an example.
The range of valid values for NUMERIC (2, 1) is nowhere near -10^38 + 1 nor
+10^38 - 1. The minimum and maximum values in this case are -9.9 and +9.9,
and the set of valid values includes only decimal numbers that consist of
two digits where one of those digits is after the decimal point. Your
example of 1/3 does not fit in that set since it has more than 1 digit after
the decimal point (in fact, it has infinitely more digits after the decimal
point).
No, I'm recognising the mathematical reality that the existence (and
value) of a number doesn't depend on what base you represent it in.
The mathematical reality is that very few people through the course of human
history have needed to use Base 3. The reality is that NUMERIC and DECIMAL
types do not represent every single number "in existence", nor every single
number between the minimum and maximum values specified by the precision and
scale set by the user. The real reality is that the user defines the set of
*valid values* as well as implicit upper and lower bounds when they declare
a DECIMAL or NUMERIC type. The reality is that every single *valid value*
in this set can be stored *exactly*; that is, you get out of it exactly what
you put into it.
I wasn't talking about usefulness, I was talking about existence.
Suggesting that 1/3 isn't a number is silly, IMO.
1/3 is a number, and I for one did not say otherwise. Now let me tell you
what is really silly (IMO):
- Suggesting that because you can't store an exact representation of the
infinitely repeating fraction 1/3 in a NUMERIC or DECIMAL (or any data type
for that matter) makes it inexact is ludicrous. As mentioned above, 1/3 is
not defined as a *valid value* for any NUMERIC or DECIMAL precision and
scale.
- Saying that trying to store the infinitely repeating decimal number 1/3
and getting less than an infinitely repeating decimal number is *the same
thing* as storing the non-repeating decimal number 1/10 and getting
something other than 0.1 back as a result is absolutely ridiculous.
Maybe a couple of examples will help, but let's get away from numbers and
bases for a minute and talk about character data (so we don't have to worry
about my decimal bias, or your love for "Base 3"):
0. Consider the VARCHAR data type on SQL 2000. It maxes out at 8,000
characters. You absolutely cannot store an infinite number of 'A'
characters in a VARCHAR(8000); therefore it is not exact by your reasoning.
By your reasoning, the only "exact" representation would be one in which you
could store every possible combination of characters ever, in an infinitely
long character string; even though a character string with 9,000 characters
is not a member of the set of valid values for a VARCHAR(8000). Likewise
you cannot store an infinitely repeating decimal (like 1/3) which, by your
reasoning, makes the NUMERIC and DECIMAL data types "inexact" as well;
despite the fact that 1/3 is not a member of the set of *valid values*
defined for the type.
1. Consider an imaginary data type similar to VARCHAR. We'll call it
APPROXIMATE VARCHAR. The APPROXIMATE VARCHAR will store basically the
strings we pass it; however, it may lose some accuracy in the least
significant characters (right-hand side of the string) due to the way it is
stored. If you were to store 8000 'A' characters in an APPROXIMATE
VARCHAR(8000), but then select the value back you might get 7,999 'A'
characters and one 'B' character. This might be no big deal if you really
only ever need to use the first 10 characters anyway, which appears to be
IEEE's rational behind their representation of floating point numbers. The
analogy is, of course, storing 28.08 in a FLOAT and retrieving 28.07999...
or 28.080000002 or some such number that *approximates* the exact value you
stored originally.
By your reasoning, both of these situations are the same: i.e., trying to
store an infinite number of characters in a data type that has a
well-defined limitation on the number of characters that can be stored *is
the same thing* as storing an approximation of the character string you pass
in. This is pretty silly, IMO.
Regardless of the reason, 1/10 is not a repeating decimal in decimal
(yes,
I'm showing my "bias" for decimal again). It has a finite representation
in
decimal. 1/3 is a repeating decimal in decimal. Since it's decimal
representation repeats infinitely, it's impossible to store it exactly
anywhere. However, with regards to Decimal and Numeric data types in
SQL,
"0.33333" is not a member of the set of valid values for a DECIMAL (10,
2)
or a NUMERIC (10, 3) [for instance].
Indeed.
Glad we agree on something.
There's no reason why they shouldn't - why would it not be able to
guarantees tha the number is essentially reproducible, which is all I
can understand that we're talking about? The binary string "0.010101"
represents an exact number and no others.
SQL REAL and FLOAT types are based on the IEEE standard which guarantees
that it will store an approximation of a value you feed to it. Since we're
dealing with binary computers (as opposed to analog computers) we can expect
that once our exact value is converted to an approximation the internal
representation *of that approximation* will be exact (1's and 0's).
However, the exact value you feed it is still converted to an approximation,
as shown by the OP. NUMERIC and DECIMAL types do not suffer from this
approximation. They store only values from the set of *valid values*
defined for their precision and scale, and they store them exactly. In your
example of 1/3, storing it as a NUMERIC (2, 1) results in "0.3" being
stored, and that rounding occurs during casting, not during the storage
operation. Basically it is attempting to cast 1/3, which is not a member of
the valid set of numbers for NUMERIC(2, 1), to a valid member of that set.
It has nothing to do with the exactness with which "0.3" is actually stored;
whereas that is the entire issue with FLOAT and REAL.
Do you regard 28.08 as in the "range" for FLOAT? It seems unfair to
demand that it is while saying that 1/3 isn't in the "RANGE" for
NUMERIC, but that's what you've basically got to do in order to claim
that FLOAT is "approximate" but DECIMAL is "exact".
Life is not fair; however, your argument here is lacking.
FLOAT is a floating point type and 28.08 is a non-repeating decimal. The
fact that FLOAT cannot accurately store 28.08 exactly but DECIMAL and
NUMERIC can proves the point very well.
DECIMAL and NUMERIC are fixed-point types with a guarantee of storage
"exactness". As mentioned, you define the precision (total number of
digits) and scale (number of digits after the decimal point) for DECIMAL and
NUMERIC, and in the process you implicitly define the upper and lower
bounds, as well as the set of *valid values* for the DECIMAL and NUMERIC
types. The guarantee is that any value that is a member of the set of
*valid values* for those types will be stored exactly. If you want to store
1/3 as DECIMAL or NUMERIC you have to cast it (implicitly or explicitly) to
a member of the set of *valid values* defined for the precision and scale
you have chosen. 1/3 is not a *valid value* from the set of *valid values*
for any NUMERIC or DECIMAL type (since they all have a finite scale).
Consider a DECIMAL (5, 4). You can store only 4 digits after the decimal
point; therefore you cannot store an infinite number of 3's after the
decimal point. Therefore 1/3 is not in the set of valid values for that
type (see previous for example).
Personally I don't find your argument about not being able to store an exact
representation of an infinitely repeating decimal on machines with a finite
amount of memory to be all that compelling a reason to call DECIMAL and
NUMERIC data types "inexact".
