title: Why do we ignore co-efficients in Big O notation?
body: <p>While searching for answers relating to "Big O" notation, I have seen many SO answers such as <a href="https://stackoverflow.com/questions/3255/big-o-how-do-you-calculate-approximate-it">this</a>, <a href="https://stackoverflow.com/questions/487258/plain-english-explanation-of-big-o/487278#487278">this</a>, or <a href="https://stackoverflow.com/questions/11032015/how-to-find-time-complexity-of-an-algorithm">this</a>, but still I have not clearly understood some points.</p>
  
  <p>Why do we ignore the co-efficients?</p>
  
  <p>For example <a href="https://stackoverflow.com/questions/11032015/how-to-find-time-complexity-of-an-algorithm">this answer</a> says that the final complexity of <code>2N + 2</code> is <code>O(N)</code>; we remove the leading co-efficient <code>2</code> and the final constant <code>2</code> as well.</p>
  
  <p>Removing the final constant of <code>2</code> perhaps understandable.  After all, <code>N</code> may be very large and so "forgetting" the final <code>2</code> may only change the grand total by a small percentage.</p>
  
  <p>However I cannot clearly understand how removing the <em>leading</em> co-efficient does not make difference.  If the leading <code>2</code> above became a <code>1</code> or a <code>3</code>, the percentage change to the grand total would be large.</p>
  
  <p>Similarly, apparently <code>2N^3 + 99N^2 + 500</code> is <code>O(N^3)</code>.  How do we ignore the <code>99N^2</code> along with the <code>500</code>?</p>