title: Can a IEEE 754 real number "cover" all integers within its range?
body: <p><em>The original question was edited (shortened) to focus on a problem of precision, not range.</em></p>
  
  <p>Single, or double precision, every representation of real number is limited to (-range,+range). Within this range lie some integer numbers (1, 2, 3, 4..., and so on; the same goes with negative numbers).</p>
  
  <p>Is there a guarantee that a IEEE 754 real number (float, double, etc) can "cover" <strong>all</strong> integers within its range? By "cover" I mean the real number will represent the integer number exactly, not as (for example) "5.000001". </p>
  
  <p><em>Just as reminder: <a href="http://www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html" rel="noreferrer">http://www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html</a> nice explanation of various number representation formats.</em></p>
  
  <p><strong>Update:</strong></p>
  
  <p>Because the question is for "can" I am also looking for the fact this cannot be done -- for it quoting a number is enough. For example "no it cannot be done, for example number 1748574 is not represented exactly by float number" (this number is taken out of thin air of course).</p>
  
  <p><strong>For curious reader</strong></p>
  
  <p>If you would like to play with IEEE 754 representation -- on-line calculator: <a href="http://www.ajdesigner.com/fl_ieee_754_word/ieee_32_bit_word.php" rel="noreferrer">http://www.ajdesigner.com/fl_ieee_754_word/ieee_32_bit_word.php</a></p>