The opposite number of the natural number a is denoted by* –*a.

The set of all natural numbers, zero and the opposite of natural numbers is called the set of integers and is denoted by

**Z, ie.**

*Z = {…, – 3, -2, -1, 0, 1, 2, 3, … }*

Elements -1, -2, -3, … are negative integers, and natural numbers 1, 2, 3, … are positive integers. Zero is neither a positive nor a negative number.

Absolute value of an integer a, ** d**enoted by | a |, is the number itself if it is positive or zero, and is the opposite number if the given one is negative, ie.

#### 1. BASIC RULES FOR CALCULATION OF INTEGRAL NUMBERS

#### 1.1. COLLECTION

If *a, *b ∈* N*o, then:

1. – (*–* a) =* *a; – (+* *a) = *-a*

2. (+* a*) + (*+ *b) = + (*a* +* *b)

3. (-*a*) + (*–* b) = –* (a* +* *b)

4. (+* *a) + *(*– b) = *+ *(a *–* b), fo*r *a>* b*

5. (+* *a) + *(*– b) = *– *(b *–* a), fo*r* a <b

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#### 1.2. MULTIPLICATION

If *a, *b ∈* N*o, then:

1. (+* *a) · (+* *b) = + *(a*b)

2. (-*a*) · (-*b*) = + *(a*b)

3. (-*a*) · (+* b*) = – *(a*b)

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#### 1.3. SUBTRACTION AND DIVISION OF INTEGRAL NUMBERS

The difference between any two integers is an integer, ie.

(∀*a, *b ∈* *Z) *(a-*b) ∈* Z*

Subtraction of integers is defined by addition. ie

(∀*a, *b ∈ *Z*) *a-b = a + (- b)*

For given integers a and* *b, *t*he quotient a:* b *is not always

is determined i*n* Z, ie. is not an integer.

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#### 1.4. CONGRUMENTS

Let a*, b *be integers and m b*e* a natural number. A a*n*d b* *are said to be congruent by modulus m,* *if a an*d* b *h*ave the same resi*d*ues when dividing by m, ie. *a = mq₁ + r* and* b = mq₂ + *r. Symbolic congruence is denoted by the following

way:

**a ≡ b (mod m)**

Example. *42 ≡ 98 (mod 4)*

=>

42 = 4 · 10 + 2 and 98 = 4 ∙ 24 + 2

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#### 1.5. OPERATIONS. GROUPS

Let G be an empty set. An operation in Set G is a rule or rule * according to which each ordered pair of element*s x*, y* *∈ G is joined by exactly one element

*Z* ∈ *G*, i.e. *x * y = z*

Each non-empty set G, tog*e*ther with some of its operations * is called a groupoid and is written* (G, **)

Example 1.

(N, +); (N, ∙); (Z, -) are groupoids, and (N, -); (Z,:) are not groupoids.

For a groupoid (G, *) it is said that:

– is commutative if * is a commutative operation, ie. **х * у = у * **х, for any x

*, y ∈ G*

– is associative (or is a subgroup), if * is an associative operation, ie.

**z), for any x**

*(x **y)** z = x ***(y ***, у, z*∈

*G*.

-there is a neutral element, if there is an elem

*e*nt i

*s*∈ G such th

*at x * e = e * x = e*-there is an inverse element if for ev

*e*ry

*x*∈ G, there

*ex*ist

*s*x ‘∈ G, such

**= is (the el**

*that x * x’ = x ‘* x**e*ment x is called inverse).

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#### 1.6. GROUP

A groupoid (*G*, *) is a group, if it meets the conditions:

1 “(*G₁*) (*G*, *) is an associative groupoid, ie. semi-group

; 2 “*(G*₂) *(*G, *) has a neutral element;

3″ (*G₃*): Each element in (G,* **) has an inverse element;

4 “(G*₄)*: If the groupoid *(*G, *) is commutative, then (G*,* *) is said to be a commutative group or an Abelian group.

– Celi broevi –