# What is the actual lowest possible positive REAL number

MSDN says that the range of REAL numbers is - 3.40E + 38 to -1.18E - 38, 0 and 1.18E - 38 to 3.40E + 38. Apparently the true lower limit is much lower. The following script populates a REAL column with 1.401298E-45:

``````CREATE TABLE a
(
r1 REAL NULL ,
r2 REAL NULL ,
r3 REAL NULL
) ;
GO
INSERT  INTO a
( r1, r2 )
VALUES  ( 1.18E-37, 10 ) ;
GO
DECLARE @i INT ;
SET @i = 1 ;

WHILE @i < 20
BEGIN ;

UPDATE  a
SET     r1 = r1 / r2 ;

SELECT  r1 ,
r2
FROM    a ;

SET @i = @i + 1 ;

END ;
GO
DROP TABLE a ;

r1            r2
------------- -------------
1.18E-38      10

(snip)

r1            r2
------------- -------------
1.401298E-45  10
``````

Can anyone tell me what is the actual lowest possible positive number?

• There's definitely some weird behaviour with this type. The smallest scalar literal I was able to cast and select was 1.1754944E-38, which came back as 1.175494E-38 (note the missing 4 at the end) -- if you try to cast 1.175494E-38 directly, you get zero back. – Jon Seigel Jul 3 '12 at 2:18

The minimum positive (subnormal) single-precision floating-point value is 2−149 ≈ 1.4 × 10−45. The minimum positive normal value is 2−126 ≈ 1.18 × 10−38 (reference).

``````DECLARE
@r1 real = POWER(2e0, -126),
@r2 real = POWER(2e0, -23)

SELECT
@r1,
@r2,
@r1 * @r2,
CONVERT(binary(4), @r1 * @r2);
``````

For double-precision, the minimum positive subnormal is:

``````DECLARE @r1 float = POWER(2e0, -1075);

SELECT @r1, CONVERT(binary(8), @r1);
``````