Integers
Integers are the set of numbers that includes all natural numbers (0, 1, 2, 3, ...) along with their negative counterparts (-1, -2, -3, ...). This set, denoted by the symbol \(\mathbb{Z}\), encompasses all whole numbers without decimal or fractional parts. It can be expressed as: $$ \mathbb{Z} = \{ ..., -3, -2, -1, 0, 1, 2, 3, ... \} $$
Integers form one of the foundational number systems in mathematics.
Formally, the set of integers is represented by the letter Z (from the German word Zahlen, meaning "numbers") and can be written as:
$$ Z = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \} $$
In this set, zero acts as the central point: numbers to its right are positive, and those to its left are negative, all of which are integers.
- Positive Numbers: These are all integers greater than zero (1, 2, 3, …). The “+” sign is usually omitted when writing positive numbers, so 3 and +3 are treated as the same value.
- Negative Numbers: These represent the opposite of positive numbers and are always less than zero (-1, -2, -3, …). Negative numbers are commonly used in practical situations to indicate things like temperatures below zero or financial debt.
- Zero: Zero is the only integer that is neither positive nor negative. It is the additive identity, meaning that adding zero to any integer leaves the integer unchanged.
This structure creates a balanced system in which each positive number has a corresponding negative.
For instance, the opposite of 5 is -5, and vice versa. Similarly, the opposite of 6 is -6, and so on.
Example: Temperatures below zero are expressed as negative integers, while those above zero are positive. For instance, -5 °C represents a temperature five degrees below freezing. Another example involving integers is a person’s bank account balance: a balance of +50 euros shows a credit, while -50 euros indicates a debt of the same amount.
Key Characteristics of Integers
The set of integers has the following characteristics:
- Infinite Set: The set of integers has no end. It extends infinitely in both the positive and negative directions, with no upper or lower bounds.
- Order: Integers are arranged in a continuous, infinite sequence, increasing positively and decreasing negatively, with zero at the center.
Imagine a number line with zero in the center, where positive numbers stretch to the right and negative numbers extend to the left. This setup makes it easier to understand how integers can represent both positive and negative amounts—similar to a bank balance that could be in credit (positive) or overdrawn (negative).
Put simply, given two integers, a and b, you can always determine that:
- a = b if they occupy the same point on the line;
- a < b if a is to the left of b on the line;
- a > b if a is to the right of b on the line.
- Discrete: The set of integers is discrete because there are no integers between any two consecutive values.
- Closure Under Operations: The set of integers is closed under addition, subtraction, and multiplication. This means the sum, difference, or product of any two integers is always an integer. For example: $$ 3 + 2 = 5 $$ $$ 3 - 2 = 1 $$ $$ 3 \cdot 2 = 6 $$
The set is not closed under division. For instance, dividing 3 by 2 does not yield an integer but a rational number (1.5). $$ 3 : 2 = 1.5 $$
- Opposites: Each integer has an opposite (or additive inverse) within the set. For instance, the opposite of 3 is -3, and adding a number to its opposite always results in zero. $$ 3+(-3)=0 $$
- Absolute Value: Every integer has an absolute value, which represents its distance from zero on the number line. For example, the absolute value of -3 is 3, and the absolute value of +3 is also 3. The absolute value is always positive or zero. $$ | 3 | = |-3 | = 3 $$
Operations with Integers
Although integers may seem simple, working with them requires attention to specific rules. Here are some of the most useful:
- Addition and Subtraction: When adding two integers with the same sign, add their absolute values and retain the same sign in the result. $$ 4 + 3 = +(|4|+|3|)=7 $$ $$ -4 + (-3) =-(|4|+|3|) = -7 $$ When adding integers with opposite signs, find the difference between their absolute values and retain the sign of the number with the larger absolute value: $$ 5 + (-3) =+( |5|-|3|)= 2 $$ $$ -5 + 3 = -(|5|-|3|)= -2 $$
- Multiplication and Division: The product or quotient of two integers is calculated by multiplying or dividing their absolute values, with the result positive if the numbers have the same sign and negative if they have opposite signs. $$ 4 \cdot 3 = 12 $$ $$ -4 \cdot (-3) = 12 $$ $$ 4 \cdot (-3) = -12 $$
Integers are a cornerstone of mathematics and have countless applications across daily life and scientific disciplines.
Understanding their properties and the rules for working with integers provides a solid foundation for tackling even more advanced mathematical concepts.
