Introduction to Integers

Definition and Notation of Integers What Are Integers? An integer is any whole number that can be positive, negative, or zero. Integers represent quantities without fractional or decimal parts. The set of all integers is denoted by the symbol $\mathbb{Z}$ and can be written as: $$\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$$ Integers extend the familiar counting numbers $\{1, 2, 3, \ldots\}$ by including zero and all negative whole numbers. This extension is important because it allows us to represent both quantities (like a debt of 5 dollars) and positions relative to a reference point. On a number line, each integer occupies a distinct point with equal spacing between consecutive integers. Zero serves as the reference point, with positive integers to the right and negative integers to the left. Fundamental Properties of Integers Closure Properties A crucial feature of the integers is that certain operations always produce another integer. We say the integers are closed under these operations. Closure under addition: Adding any two integers always yields another integer. For example, $5 + (-3) = 2$ and $(-4) + (-2) = -6$. Closure under subtraction: Subtracting any integer from another always yields an integer. For instance, $7 - 10 = -3$ and $(-5) - 3 = -8$. Note that subtraction can be viewed as adding the additive inverse, so this follows from closure under addition. Closure under multiplication: Multiplying any two integers always yields an integer. Examples include $4 \times 6 = 24$ and $(-3) \times 5 = -15$. Non-closure under division: Division of integers does not always produce an integer. For example, $7 \div 3 = 2.\overline{3}$, which is not an integer. This is why we must be careful when dividing integers—a remainder may remain. The Additive Identity and Additive Inverses Zero is the additive identity because adding zero to any integer leaves that integer unchanged: $n + 0 = n$ for all integers $n$. Every integer has an additive inverse. For any integer $n$, its additive inverse is $-n$, and they satisfy: $$n + (-n) = 0$$ For example, the additive inverse of 7 is $-7$ (since $7 + (-7) = 0$), and the additive inverse of $-3$ is $3$ (since $-3 + 3 = 0$). Commutativity and Associativity Addition is commutative: The order in which we add integers doesn't matter: $$a + b = b + a$$ For example, $5 + (-2) = 3$ and $(-2) + 5 = 3$. Multiplication is commutative: The order of multiplication doesn't affect the result: $$a \times b = b \times a$$ For example, $4 \times (-3) = -12$ and $(-3) \times 4 = -12$. Addition is associative: When adding three or more integers, we can group them in any way: $$(a + b) + c = a + (b + c)$$ For example, $(2 + 5) + (-3) = 7 + (-3) = 4$ and $2 + (5 + (-3)) = 2 + 2 = 4$. Multiplication is associative: Similarly, we can group factors in any way: $$(a \times b) \times c = a \times (b \times c)$$ For example, $(2 \times 3) \times (-4) = 6 \times (-4) = -24$ and $2 \times (3 \times (-4)) = 2 \times (-12) = -24$. Arithmetic Operations with Integers Addition and Subtraction When adding integers, we combine their values while carefully tracking their signs. A useful way to think about this: moving right on the number line for positive numbers and left for negative numbers. When subtracting an integer, recall that this is equivalent to adding its additive inverse. So $a - b = a + (-b)$. Multiplication and the Sign Rule To multiply integers, multiply their absolute values (discussed below), then apply the sign rule: Positive × Positive = Positive (e.g., $3 \times 4 = 12$) Negative × Negative = Positive (e.g., $(-3) \times (-4) = 12$) Positive × Negative = Negative (e.g., $3 \times (-4) = -12$) Negative × Positive = Negative (e.g., $(-3) \times 4 = -12$) Division and Absolute Value When dividing integers, perform the division and note that a remainder may result. For example, $17 \div 5 = 3$ remainder $2$. The absolute value of an integer $n$, denoted $|n|$, measures its distance from zero on the number line, ignoring sign. Formally: If $n \geq 0$, then $|n| = n$ If $n < 0$, then $|n| = -n$ For example, $|7| = 7$ and $|-7| = 7$. Absolute value is useful when applying the sign rule, since we compute $|a| \times |b|$ first, then determine the sign. Divisibility, Factors, and Multiples The "Divides" Relationship An integer $d$ divides an integer $n$ (written $d \mid n$) when $n \div d$ yields an integer result with no remainder. In other words, $d$ divides $n$ if there exists an integer $k$ such that $n = d \times k$. For example, $3$ divides $12$ because $12 \div 3 = 4$, an integer. However, $3$ does not divide $10$ because $10 \div 3 = 3.\overline{3}$, which is not an integer. Factors An integer $f$ is a factor of an integer $n$ if $f$ divides $n$ without remainder. Equivalently, $f$ is a factor of $n$ if $n = f \times k$ for some integer $k$. For example, the factors of $12$ are $\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\}$ because each of these divides $12$ evenly. Note that both positive and negative factors count, and both $1$ and the number itself are always factors. Multiples An integer $m$ is a multiple of an integer $k$ if $k$ divides $m$ without remainder. Equivalently, $m$ is a multiple of $k$ if $m = k \times j$ for some integer $j$. For example, the multiples of $5$ are $\{0, \pm 5, \pm 10, \pm 15, \ldots\}$ because each can be written as $5 \times j$ for some integer $j$. Every integer is a multiple of $1$, and zero is a multiple of every integer. Key Relationship Notice that "factor" and "multiple" are related concepts: if $f$ is a factor of $n$, then $n$ is a multiple of $f$. For instance, since $3$ is a factor of $12$, then $12$ is a multiple of $3$. <extrainfo> Integers and Ring Theory In abstract algebra, the integers form what is called a ring—a mathematical structure with two operations (addition and multiplication) that satisfy certain properties. A ring must have: An additive identity (zero) Every element must have an additive inverse Addition and multiplication must be associative Multiplication must be distributive over addition The integers $\mathbb{Z}$ satisfy all these conditions, making them the prototype example of a ring. This abstract framework allows mathematicians to study other sets that behave similarly to the integers, even though they might contain very different types of objects. </extrainfo>