Floor, Ceiling and Fractional Functions

Number-Theory

Floor

The floor function also called the greatest integer function is defined as , it is the greatest integer such that .

Fractional Part

The fractional part of denoted by For positive this is simply the decimal part but for negative integers it’s the the fractional part. E.g. .

Ceiling

The ceiling function gives the integer larger than . It’s is defined as

Properties

Let and .

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Hermite’s Identity

Proof 1

1. The key fact is that
2. Therefore and .
3. So .
4. Note that and , together this gives
5. Using this we get that
6. Using this we get .

Proof 2 INSERT PROOF

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Floor Function of Rational Numbers

Let and be the remainder when is divided by then. . This follows from Euclid’s Division Lemma with .

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Floor Function and Divisors

Properties

1. The number of multiples of which are
3. Number of Divisors
4. Sum of Divisors

Explanations

1. The number of multiples of which are
   1. Let where the amount of multiples of .
   2. .
2. 1. This is a consequence of property 1. If then there will be one more multiple of than so you get .
   2. If then there will be the same amount of multiples of so you get .
3. Number of Divisors
   1. Form and table with rows and columns . The element is if is a multiple of . Count the total number of ‘s,
   2. Fix column . The number of ‘s in the th column is by property 1. So .
   3. Fix row . The number of ‘s in the th row is simply . So .
   4. Meaning .
4. Sum of Divisors
   1. Same construction as property 3. However instead of writing a write . And consider the sum of the whole table
   2. Fix column . The sum of the th column is
   3. Fix row . The sum of the th row is
   4. Meaning