Integers
On a thermometer, temperatures above 0 are designated as positive (+) temperatures, while temperatures below 0 are labeled as negative (-) temperatures. Similarly, when measuring geographic features, sea level is established as 0, where elevations are indicated by positive (+) numbers and depths by negative (-) numbers. This demonstrates that numbers less than zero are necessary to quantify certain real-world measurements.
Positive Integers
Integers that are strictly greater than zero are called positive integers. These numbers continue infinitely in the pattern +1, +2, +3, … The set of positive integers is denoted by Z+ and is defined as: Z+ = {1, 2, 3, 4, 5, …}
Negative Integers
Integers that are strictly less than zero are called negative integers. These numbers continue infinitely in the pattern −1, −2, −3, … The set of negative integers is denoted by Z– and is defined as: Z– = {−1, −2, −3, −4, …}
The Set of Integers
The union of the positive integers, negative integers, and zero forms the set of integers. This set is denoted by Z and is expressed as: Z = Z+ ∪ {0} ∪ Z– Written in roster notation, the set of integers is:
Z = {…, −3, −2, −1, 0, 1, 2, 3, …}
Relationship Between Natural Numbers and Integers
Every element in the set of natural numbers is also an element of the set of integers. Therefore, the set of natural numbers is a subset of the set of integers. This relationship is written as: N ⊆ Z Consequently, every natural number is an integer.
Alternative Set-Theoretic Definition of Integers
Integers can also be rigorously constructed using ordered pairs of natural numbers:
An integer can be defined as an equivalence class of ordered pairs of natural numbers, where the ordered pair (a, b) corresponds to the evaluation:
(a, b) = a – b.
Let a be a non-zero natural number;
(a, 0) = a – 0 = a;
(0, a) = 0 – a = -a.
For any natural number r, an ordered pair of the form (r, r) represents the integer zero (0):
(r, r) = r – r = 0.
Examples:
- (7, 3) = (5, 1) = (4, 0) = 4 − 0 = +4
- (3, 9) = (1, 7) = (0, 6) = 0 − 6 = −6
- (5, 5) = (1, 1) = (0, 0) = 0 − 0 = 0
Example
Given that \(a \cdot b = 6\) and \(b \cdot c = 15\), where \(a\), \(b\), and \(c\) are integers, let us find the minimum possible value of the product \(a \cdot b \cdot c\).
Since \(a\), \(b\), and \(c\) must be integers,
\(a \cdot b = 6 \implies 1 \cdot 6 = (-1) \cdot (-6) = 2 \cdot 3 = (-2) \cdot (-3)\)
and
\(b \cdot c = 15 \implies 1 \cdot 15 = (-1) \cdot (-15) = 3 \cdot 5 = (-3) \cdot (-5)\)
To minimize the product \(a \cdot b \cdot c\), we must analyze the possible combinations for \(a\), \(b\), and \(c\). The values for \((a, b, c)\) can be selected as \((-6, -1, -15)\) or \((2, -3, -5)\). Evaluating these combinations shows that the minimum possible value of the product \(a \cdot b \cdot c\) is:
\((-6) \cdot (-1) \cdot (-15) = -90\).