Integers

The opposite number of the natural number a is denoted bya.
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, denoted 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 ∈ No, then:

1. – ( a) = a; – (+ a) = -a
2. (+ a) + (+ b) = + (a + b)
3. (-a) + ( b) = – (a + b)
4. (+ a) + (– b) = + (a b), for a> b
5. (+ a) + (– b) = (b a), for a <b

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1.2. MULTIPLICATION

If a, b ∈ No, then:

1. (+ a) · (+ b) = + (ab)
2. (-a) · (-b) = + (ab)
3. (-a) · (+ b) = – (ab)

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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, the quotient a: b is not always
is determined in Z, ie. is not an integer.

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1.4. CONGRUMENTS

Let a, b be integers and m be a natural number. A and b are said to be congruent by modulus m, if a and b have the same residues 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 elements x, y ∈ G is joined by exactly one element
ZG, i.e. x * y = z
Each non-empty set G, together 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. (x * y) * z = x * (y * z), for any x, у, zG.
-there is a neutral element, if there is an element is ∈ G such that x * e = e * x = e
-there is an inverse element if for every x ∈ G, there exists x ‘∈ G, such that x * x’ = x ‘* x = is (the element 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 “(G₂) (G, *) has a neutral element;
3″ (G₃): Each element in (G, *) has an inverse element;
4 “(G₄): If the groupoid (G, *) is commutative, then (G, *) is said to be a commutative group or an Abelian group.

– Celi broevi –

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