Symbols used


We say that a divides b, or that b is a multiple of a, if there is an integer q such that b = qa. In this case, we write a\mid b and say that a is a divisor of b. If a does not divide b, we write a\nmid b.
e.g. 3\times 4=12, so 3\mid 12 and 3\nmid 13.
Knuth et al in Concrete Mathematics use m\backslash n and say “The notation `m|n’ is actually much more common than `m\n’ in current mathematics literature. But vertical lines are overused – for absolute values, set delimiters, conditional probabilities, etc. – and backward slashes are underused. Moreover, `m\n’ gives an impression that m is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.”

greatest common divisor

The greatest common divisor of two positive integers a and b, written (a,b), is the largest number that divides evenly (no remainder) into both. If a and b have no common factor, (a,b)=1.

factorial !

n!=n\times (n-1)\times (n-2)\dots \times 3 \times 2 \times 1
e.g. 5!=5\times 4\times 3\times 2 \times 1=120
0! is defined as 1

floor/ceiling functions

\lceil x \rceil means x rounded up to the next-highest integer.
\lfloor x \rfloor means x rounded down. e.g. \lfloor 4.76 \rfloor =4 and \lceil 4.76 \rceil =5


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