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; 5 , …  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
the number 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, p>1
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 = 10
decadic
0, 1, ..., 9
p=2
binary
0, 1
p = 60
0, 1, ..., (58), (59)
p = 16
0, 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),
C for (12), D for (13), E for (14), F for (15).
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Division
Definition of division: Let n,dZ be given numbers.
d | n ( d is a divisor of n, n is divisible by d ), if qZ: n=dq.
Observation: For each n,dZ there 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 of factors ).
Observation:
n is a composite number if and only if there exists a prime number p n such that p | n.
d is a divisor of n  p11  p 2 2  ...  p k k ( with prime factors p1<p2<…<pk ) if and only if
 1  0,1,..., 1 ,  2  0,1,...,  2 , ...,  k  0,1,...,  k  such that d  p11  p 2 2  ...  p k k .
Theorem: There exist infinitely many prime numbers.
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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 a  p11  p 2 2  ...  p k k and b  p11  p 2 2  ...  p k k
( with prime factors p1<p2<…<pk ) then
GCD(a,b) = p1min{1 , 1 }  p 2min{ 2 ,  2 }  ...  p kmin{ k ,  k } ,
LCM(a,b) = p1max{ 1 , 1 }  p 2max{  2 ,  2 }  ...  p kmax{  k ,  k } .
( Analogically for more than two numbers. )
Observation: For each dZ: if d | a  d | b then d | 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 then D(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
 153 1
; ,…Q
22
2
Observation: NZQ.
Examples: 0,801;
3
m
.
n
Properties of operations +,  in Q:
Q is closed under the operations +, 
commutative laws: a,b: a+b = b+a, ab = ba
associative laws: a,b,c: a+(b+c) = (a+b)+c, a(bc) = (ab)c
distributive law: a,b,c: a(b+c) = ab+ac
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
1
exactly one inverse element   for the operation  in Q ( except of a=0 ):
a
1
aQ: a+(-a) = 0, a≠0  a   = 1
a
1.
2.
3.
4.
5.
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:
2 , π,
5
40 ,
3
5
, ...  R
Hierarchy of number domains: N  Z  Q  A  R  C
( where A denotes algebraic numbers,
C denotes complex numbers – not studied within this chapter )
- irrational number x: each xR–Q ( for example 2 ,,eR–Q )
- transcendental number x: each xR–A ( for example ,eR–A, but
2 A–Q )
Observation: 0, 9 = 1, 0
15,419 = 15,420
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 ).
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Theorem ( decadic form of rationals ): For each x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–Q such that
axb, ayb.
Interval notation: For a,bR, ab:
Interval, definition
Closed
a, b = {x  R; a  x  b}
Open
(a, b) = {x  R; a  x  b}
Picture
Example
(includes end points)
(excludes end points)
0, 10
(-1, 5)
Semi-Open (a, b = {x  R; a  x  b}
(-3, 1
a, b) = {x  R; a  x  b}
-4, -1)
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
(-, )
Infinite
Absolute value
Definition: Let aR. The absolute value of a is
a = a
for a  0,
a = -a for a  0.
Consequence: For each aR:
a = 0  a = 0
a  0
-a = a
a = max{a,-a}
a2  a ,
4
a4  a , …
a = d(a,0) ... the distance between a and 0 on the real axis
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Other properties: For each a,bR:
ab= ab
a
a
=
b
b
a–b  ab  a+b
a–b = d(a,b) ... the distance between a and b on the real axis
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