Number domains

(Anna Košková, Petra Malešová)

NATURAL NUMBERS, INTEGERS

Natural Numbers

- remain undefined (in secondary school mathematics).

Notation: N... the set of all natural numbers

Examples: 1, 2, 3, ..., 105, 106, ...N

0, -1, -2, ..., -100, ... N

3,5; - 4/3; , …N

- N is closed under the operations addition and multiplication.

Integers

- remain undefined (in secondary school mathematics).

Notation: Z … the set of all integers

- integers are all natural numbers (1, 2, 3, …), their negatives (-1, -2, -3, … ) and thenumber 0. Therefore NZ.

- Z is closed under the operations addition, subtraction and multiplication and is not closed

under the operation division.

Number Position System

p – adic position system: base pN, p1

digits (characters): 0, 1, ..., (p–1)

(AnAn-1…A1A0)p = Anpn+An-1pn-1+…+A1p1+A0p0

digits An, An-1, …, A1, A0 {0, 1, …, (p-1)}

Base:Digits:

p = 10decadic0, 1, ..., 9

p = 2binary0, 1

p = 600, 1, ..., (58), (59)

p = 160, 1, ..., 8, 9, (10), (11), (12), (13), (14), (15)

- in the system with the base p=16 usually the character A is used for (10), B for (11), Cfor(12), D for (13), E for (14), F for (15).

Division

Definition of division: Let n,dZ be given numbers.

d| n (d is a divisor of n, n is divisible byd), if qZ: n=dq.

Observation: For each n,dZthere exists at most one qZ such that n = dq.

( This qZ is said to be the quotient. )

Definition of division with remainder: Let nZ,dN, r{0, 1, ..., (d-1)} be given numbers.

r is the remainder of n divided by d, if qZ: n = dq+ r.

Observation: For each nZ, dN there exists exactly one pair r{0, 1, ..., (d-1)}, qZ such

that n = dq+ r.

Notation:a ≡ b mod d ( a is congruent with b modulo d )

means that a, b have the same remainder divided by d

Observation:dN: d 0 mod d

aZ: a 0 mod dd | a

aZ: aa mod d

a,bZ: ab mod d  ba mod d

a,b,cZ: ( ab mod dbc mod d )ac mod d.

Prime Numbers, Composite Numbers

Definition: A natural number n is a prime number if there exist exactly two distinct natural divisors of n.

A natural number n is a composite number if there exist more than two distinct natural divisors of n.

Consequence: 1 is not a prime number, neither a composite number.

Theorem (fundamental theorem of arithmetic): Each natural number n1 has exactly one factorization to prime factors ( up to the order offactors).

Observation:

n is a composite number if and only if there exists a prime number p such that p|n.

d is a divisor of ( with prime factors p1p2<…<pk ) if and only if , , ..., such that .

Theorem: There exist infinitely many prime numbers.

Greatest Common Divisor, Least Common Multiple

Definition: Given a,bZ:

GCD(a,b) is the maximal dN such that d | ad | b.

LCM(a,b) is the minimal nN such that a|nb|n.

Observation:For each a,bN there exists exactly one GCD(a,b) and exactly one LCM(a,b).

If and

( with prime factors p1p2<…<pk ) then

GCD(a,b) = ,

LCM(a,b) = .

( Analogically for more than two numbers. )

Observation:For each dZ: if d | ad | b thend | GCD(a,b). ( GCD is not only greater than other common divisors, GCD is divisible by each common divisor. )

For each nZ: if a | nb | n then LCM(a,b)| n. ( LCM is not only less than other natural common multiples, LCM is a divisor of each common multiple. )

Theorem: For each a,bN:

GCD(a,b)  LCM(a,b) = ab.

if ab then GCD(a,b) = GCD(a,b-a).

( Moreover: if D(x,y) = {dZ; d | xd | y } denotes the set of all common divisors of x, y thenD(a,b) = D(a,b-a). Therefore the pair a, b-a has exactly the same common divisors as the pair a, b. )

Definition:a,bZ are relatively prime numbers if there exists no dN–{1} such that d|ad|b.

Observation: For each a,bN:

a,b are relatively primes a,b have no common prime factor  GCD(a,b) = 1.

RATIONAL NUMBERS, REAL NUMBERS

Rational Numbers

Definition:x is a rational number if mZ, nN such that x=.

Notation:Q … the set of all rational numbers

Examples: 0,801; ; , …Q

Observation:NZQ.

Properties of operations +,  in Q:

  1. Q is closed under the operations +, 
  2. commutative laws: a,b:a+b = b+a,ab = ba
  3. associative laws: a,b,c: a+(b+c) = (a+b)+c, a(bc) = (ab)c
  4. distributive law: a,b,c: a(b+c) = ab+ac
  5. there exists exactly one neutral element for the operation + and exactly one neutral element for the operation  in Q:a: a+0 =a, a1 = a
  6. for each aQ there exists exactly one inverse element (-a) for the operation + and exactly one inverse element for the operation  in Q(except of a=0):

aQ: a+(-a) = 0, a≠0a = 1

Real Numbers

- remain undefined in secondary school mathematics.

Main idea: to extend Q to a number domain so that the new number domain contains the measure (a number for length) of each geometric segment. ( Because there exists no rational number e.g. for the length of the hypotenuse of the right triangle with sides of length 1.)

- real axis: a geometric line with two distinct points 0,1.

- real number: each point of the real axis. ( This means that the concept of a real number corresponds more with the idea of “address” than with the idea of “amount”. )

Notation:R … the set of all real numbers

Examples:π, , , ...R

Hierarchy of number domains:N  Z  Q  A  R  C

(where Adenotes algebraic numbers,

C denotes complex numbers – not studied within this chapter )

- irrational numberx: each xR–Q (for example ,,eR–Q)

- transcendental number x: each xR–A ( for example ,eR–A, but A–Q )

Observation: =

=

etc.

are two distinct decadic forms of the same real number (moreover, this is the unique type of double decadic form of the same real number).

Theorem (decadic form of rationals): For eachxR: xQx has a periodic decadic form

(including finite forms being periodic with infinitely many zeros).

Theorem (density of rationals and irrationals in R): For each a,bR: if ab then

xQ,yR–Q such that axb, ayb.

Consequence: For each a,bR: if ab then there exist infinitely many xQ, yR–Qsuch that

axb, ayb.

Interval notation: For a,bR, ab:

Interval, definition / Picture / Example

Closed

/ a, b = {xR; axb} /
(includes end points) / 0, 10
Open / (a, b) = {xR; axb} /
(excludes end points) / (-1, 5)
Semi-Open / (a, b = {xR; axb} / / (-3, 1
a, b) = {xR; axb} / / -4, -1)
Infinite / a, ) = {xR; ax} / / 0, )
(a, ) = {xR; ax} / / (-3, )
(-, b = {xR; xb} / / (-, 0
(-, b) = {xR; xb} / / (-, 8)
(-, ) = R / / (-, )

Absolute value

Definition: Let aR. The absolute value of a is

a = afor a 0,

a = -afor a 0.

Consequence: For each aR:

a = 0 a = 0

a 0

-a = a

a = max{a,-a}

, , …

a = d(a,0) ... the distance between aand 0 on the real axis

Other properties: For each a,bR:

ab= ab

=

a–baba+b

a–b= d(a,b) ... the distance between aand b on the real axis

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