grows too fast to display all its values on the chart alsowe use the scientific notation for numberswhereae+ denotes the ceiling and floor functions one additional comment concerning the functions above is in order the value of logarithm is typically not an integeryet the running time of an algorithm is usually expressed by means of an integer quantitysuch as the number of operations performed thusthe analysis of an algorithm may sometimes involve the use of thefloor function and ceiling functionwhich are defined respectively as follows the largest integer less than or equal to the smallest integer greater than or equal to analysis of algorithms in classic storythe famous mathematician archimedes was asked to determine if golden crown commissioned by the king was indeed pure goldand not part silveras an informant had claimed archimedes discovered way to perform this analysis while stepping into (greekbath he noted that water spilled out of the bath in proportion to the amount of him that went in realizing the implications of this facthe immediately got out of the bath and ran naked through the city shouting"eurekaeureka!,for he had discovered an analysis tool (displacement)whichwhen combined with simple scalecould determine if the king' new crown was good or not that isarchimedes could dip the crown and an equal-weight amount of gold