Difference between revisions of "The Integers"

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<li><strong>Proof of a)</strong> If <span class="math-inline"><math>a \mid b</math></span> and <span class="math-inline"><math>b \mid c</math></span> then there exists <span class="math-inline"><math>q_1, q_2 \in \mathbb{Z}</math></span> such that:</li>
 
<li><strong>Proof of a)</strong> If <span class="math-inline"><math>a \mid b</math></span> and <span class="math-inline"><math>b \mid c</math></span> then there exists <span class="math-inline"><math>q_1, q_2 \in \mathbb{Z}</math></span> such that:</li>
 
</ul>
 
</ul>
<div style="text-align: center;"><math>\begin{align} \quad aq_1 = b \quad \mathrm{and} \quad bq_2 = c. \end{align}<math></div>
+
<div style="text-align: center;"><math>\begin{align} \quad aq_1 = b \quad \text{and} \quad bq_2 = c. \end{align}</math></div>
 
<ul>
 
<ul>
 
<li>Substituting the first equation into the second gives us:</li>
 
<li>Substituting the first equation into the second gives us:</li>
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<ul>
 
<ul>
 
<li><strong>Proof of c)</strong> From (b) if we set <span class="math-inline"><math>x = 1</math></span> and <span class="math-inline"><math>y = 1</math></span> we get that <span class="math-inline"><math>a \mid (b + c)</math></span> and if we set <span class="math-inline"><math>x = 1</math></span> and <span class="math-inline"><math>y = -1</math></span> we get that <span class="math-inline"><math>a \mid (b - c)</math></span> as desired. <span class="math-inline"><math>\blacksquare</math></span></li>
 
<li><strong>Proof of c)</strong> From (b) if we set <span class="math-inline"><math>x = 1</math></span> and <span class="math-inline"><math>y = 1</math></span> we get that <span class="math-inline"><math>a \mid (b + c)</math></span> and if we set <span class="math-inline"><math>x = 1</math></span> and <span class="math-inline"><math>y = -1</math></span> we get that <span class="math-inline"><math>a \mid (b - c)</math></span> as desired. <span class="math-inline"><math>\blacksquare</math></span></li>
 +
</ul>
 +
 +
 +
==The Division Algorithm==
 +
<p>One rather important aspect of the divisibility of integers is that if <span class="math-inline"><math>a, b \in \mathbb{Z}</math></span> then <span class="math-inline"><math>a</math></span> can be written as the product of some quotient <span class="math-inline"><math>q</math></span> with <span class="math-inline"><math>b</math></span> plus a remainder <span class="math-inline"><math>r</math></span>. For example, if <span class="math-inline"><math>a = 11</math></span> and <span class="math-inline"><math>b = 3</math></span>, then <span class="math-inline"><math>a = 3(b) + 2</math></span> where <span class="math-inline"><math>q = 3</math></span> and <span class="math-inline"><math>r = 2</math></span>.</p>
 +
<p>The following theorem known as the Division Algorithm shows us that for any pair of integers <span class="math-inline"><math>a, b \in \mathbb{Z}</math></span> where <span class="math-inline"><math>b > 0</math></span> that the form <span class="math-inline"><math>a = bq + r</math></span> exists and is unique.</p>
 +
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
<td><strong>Theorem 1 (The Division Algorithm):</strong> Let <span class="math-inline"><math>a, b \in \mathbb{Z}</math></span> with <span class="math-inline"><math>b > 0</math></span>. Then there exists unique <span class="math-inline"><math>q, r \in \mathbb{Z}</math></span> where <span class="math-inline"><math>0 \leq r < b</math></span> and such that <span class="math-inline"><math>a = bq + r</math></span>.</td>
 +
</blockquote>
 +
<p><em>The value of <span class="math-inline"><math>r</math></span> is often called the <strong>Remainder</strong> when <span class="math-inline"><math>a</math></span> is divided by <span class="math-inline"><math>b</math></span>. That is, <span class="math-inline"><math>a</math></span> is equal to some multiple <span class="math-inline"><math>q</math></span>, known as the <strong>Quotient</strong> of <span class="math-inline"><math>b</math></span>, plus a remainder <span class="math-inline"><math>r</math></span> for which <span class="math-inline"><math>0 \leq r < b</math></span>.</em></p>
 +
<ul>
 +
<li><strong>Proof:</strong> Let <span class="math-inline"><math>a, b \in \mathbb{Z}</math></span> with <span class="math-inline"><math>b > 0</math></span> and consider the following set <span class="math-inline"><math>A</math></span>:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad A = \{ a - bq : q \in \mathbb{N} \} = \{ a, a - b, a - 2b, ... \}. \end{align}</math></div>
 +
<ul>
 +
<li>Consider <span class="math-inline"><math>B \subset A</math></span> of nonnegative elements in <span class="math-inline"><math>A</math></span>. If <span class="math-inline"><math>a > 0</math></span> then <span class="math-inline"><math>a \in B</math></span> and if <span class="math-inline"><math>a < 0</math></span> then <span class="math-inline"><math>a - ba = a(1 - b) \in B</math></span>, so <span class="math-inline"><math>B</math></span> is a nonempty set. Moreover, <span class="math-inline"><math>B</math></span> is a nonempty subset of integers and so by the Well-Ordering principle there exists a least element of <span class="math-inline"><math>B</math></span>, say <span class="math-inline"><math>r \in B</math></span> is this least element. Since <span class="math-inline"><math>r \in B</math></span> we have that <span class="math-inline"><math>r = a - bn</math></span> for some <span class="math-inline"><math>n \in \mathbb{N}</math></span> and so:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad a = bq + r \end{align}</math></div>
 +
<ul>
 +
<li>So <span class="math-inline"><math>a</math></span> if of the form we desire. Now since <span class="math-inline"><math>r \in B</math></span> and every element in <span class="math-inline"><math>B</math></span> is nonnegative then we have that <span class="math-inline"><math>0 \leq r</math></span>. Suppose that <span class="math-inline"><math>r \geq b</math></span>. Then <span class="math-inline"><math>r = b + p</math></span> where <span class="math-inline"><math>p \geq 0</math></span> and so:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad a = bq + r = bq + (b + p) = b(q + 1) + p \end{align}</math></div>
 +
<ul>
 +
<li>Therefore <span class="math-inline"><math>p = a - b(q + 1) \geq 0</math></span> and so <span class="math-inline"><math>p \in B</math></span>. Since <span class="math-inline"><math>b > 0</math></span> and <span class="math-inline"><math>q+1 > q</math></span> we have that <span class="math-inline"><math>0 \leq p < r</math></span>. But this implies that <span class="math-inline"><math>r</math></span> is not the least element of <span class="math-inline"><math>B</math></span> which is a contradiction, so our assumption that <span class="math-inline"><math>r \geq b</math></span> is false, and so <span class="math-inline"><math>0 \leq r < b</math></span>.</li>
 +
</ul>
 +
<ul>
 +
<li>We lastly show that <span class="math-inline"><math>q</math></span> and <span class="math-inline"><math>r</math></span> are unique for <span class="math-inline"><math>a = bq + r</math></span> and <span class="math-inline"><math>0 \leq r < b</math></span>. Suppose that <span class="math-inline"><math>a = bq_1 + r_1</math></span> and <span class="math-inline"><math>a = bq_2 + r_2</math></span> where <span class="math-inline"><math>0 \leq r_1, r_2 < b</math></span>. Then by subtracting these two equations we get:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad 0 = bq_1 - bq_2 + r_1 - r_2 \\ \quad 0 =b(q_1 - q_2) + r_1 - r_2 \\ \quad r_2 - r_1 = b(q_1 - q_2) \end{align}</math></div>
 +
<ul>
 +
<li>Therefore <span class="math-inline"><math>r_2 - r_1</math></span> is a multiple of <span class="math-inline"><math>b</math></span>. Since <span class="math-inline"><math>0 \leq r_1 < b</math></span> and <span class="math-inline"><math>0 \leq r_2 < b</math></span> we must have that <span class="math-inline"><math>-b < r_2 - r_1 < b</math></span>. The only multiple of <span class="math-inline"><math>b</math></span> such that <span class="math-inline"><math>-b < r_2 - r_1 < b</math></span> is <span class="math-inline"><math>0</math></span>, so:</li>
 +
</ul>
 +
<div style="text-align: center;"><math>\begin{align} \quad r_2 - r_1 = 0 \\ \quad r_1 = r_2 \end{align}</math></div>
 +
<ul>
 +
<li>Since <span class="math-inline"><math>r_1 = r_2 = r</math></span>, we have that <span class="math-inline"><math>0 = b(q_1 - q_2)</math></span>. Since <span class="math-inline"><math>b > 0</math></span> this implies that <span class="math-inline"><math>q_1 - q_2 = 0</math></span> and so <span class="math-inline"><math>q_1 = q_2</math></span>. Therefore <span class="math-inline"><math>a</math></span> can be uniquely expressed in the form <span class="math-inline"><math>a = bq + r</math></span> where <span class="math-inline"><math>0 \leq r < b</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 
</ul>
 
</ul>
  

Latest revision as of 15:48, 16 November 2021

An integer is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not.

The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold letter "Z".

is a subset of the set of all rational numbers , which in turn is a subset of the real numbers . Like the natural numbers, is countably infinite.

The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

Symbol

The symbol can be annotated to denote various sets, with varying usage amongst different authors: , or for the positive integers, or for non-negative integers, and for non-zero integers. Some authors use for non-zero integers, while others use it for non-negative integers, or for {–1, 1}. Additionally, is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p-adic integers.

Algebraic properties

Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integers are shown in blue and negative integers in red.

Like the natural numbers, is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, 0), , unlike the natural numbers, is also closed under subtraction.

The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring .

is not closed under division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).

The following table lists some of the basic properties of addition and multiplication for any integers a, b and c:

Properties of addition and multiplication on integers
Addition Multiplication
Closure: a + b    is an integer a × b    is an integer
Associativity: a + (b + c) (a + b) + c a × (b × c) (a × b) × c
Commutativity: a + b b + a a × b b × a
Existence of an identity element: a + 0 a a × 1 a
Existence of inverse elements: a + (−a) 0 The only invertible integers (called units) are −1 and 1.
Distributivity: a × (b + c) (a × b) + (a × c)    and    (a + b) × c (a × c) + (b × c)
No zero divisors: If a × b 0, then a 0 or b 0 (or both)

The first five properties listed above for addition say that , under addition, is an abelian group. It is also a cyclic group, since every non-zero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to .

The first four properties listed above for multiplication say that under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that under multiplication is not a group.

All the rules from the above property table (except for the last), when taken together, say that together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in  for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings.

The lack of zero divisors in the integers (last property in the table) means that the commutative ring  is an integral domain.

The lack of multiplicative inverses, which is equivalent to the fact that is not closed under division, means that is not a field. The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes as its subring.

Although ordinary division is not defined on , the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers a and b with b ≠ 0, there exist unique integers q and r such that a q × b + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The integer q is called the quotient and r is called the remainder of the division of a by b. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions.

The above says that is a Euclidean domain. This implies that is a principal ideal domain, and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.

Order-theoretic properties

is a totally ordered set without upper or lower bound. The ordering of is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... An integer is positive if it is greater than zero, and negative if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:

  1. if a < b and c < d, then a + c < b + d
  2. if a < b and 0 < c, then ac < bc.

Thus it follows that together with the above ordering is an ordered ring.

The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered. This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.

Construction

Representation of equivalence classes for the numbers −5 to 5
Red points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line.

In elementary school teaching, integers are often intuitively defined as the (positive) natural numbers, zero, and the negations of the natural numbers. However, this style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey the various laws of arithmetic. Therefore, in modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b).

The intuition is that (a,b) stands for the result of subtracting b from a. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:

precisely when

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using [(a,b)] to denote the equivalence class having (a,b) as a member, one has:

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

Hence subtraction can be defined as the addition of the additive inverse:

The standard ordering on the integers is given by:

if and only if

It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.

Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (i.e., the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted n (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 0.

Thus, [(a,b)] is denoted by

If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.

This notation recovers the familiar representation of the integers as {..., −2, −1, 0, 1, 2, ...} .

Some examples are:

In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines. Integers are represented as algebraic terms built using a few basic operations (e.g., zero, succ, pred) and, possibly, using natural numbers, which are assumed to be already constructed (using, say, the Peano approach).

There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2) and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms.

The technique for the construction of integers presented above in this section corresponds to the particular case where there is a single basic operation pair that takes as arguments two natural numbers and , and returns an integer (equal to ). This operation is not free since the integer 0 can be written pair(0,0), or pair(1,1), or pair(2,2), etc. This technique of construction is used by the proof assistant Isabelle; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.


Integer Divisibility

Definition: Let . Then is said to be Divisible by , or, is said to Divide written if there exists a such that . The number is said to be a Divisor of and is said to be a Multiple of . If there does not exist an such that then we say that does not divide , or, is not divisible by denoted .

For example, the number divides since for we have that , so we write . However, the number does not divide any odd number for , so in this case we have that , i.e., for all integers .

We will now look at some rather elementary proofs regarding the divisibility of integers.

Theorem 1: Let .

a) If and then .
b) If and then for all , .

c) If and then and .

For the following proofs we utilize the fact that the set of integers is closed under standard addition and standard multiplication - that is, if then and .

  • Proof of a) If and then there exists such that:
  • Substituting the first equation into the second gives us:
  • Since we have that their product . Therefore .
  • Proof of b) If and then there exists such that:

(3)

  • Multiply the first equation by and the second equation by to get:
  • We now add both of these equations:
  • Since we have that and so .
  • Proof of c) From (b) if we set and we get that and if we set and we get that as desired.


The Division Algorithm

One rather important aspect of the divisibility of integers is that if then can be written as the product of some quotient with plus a remainder . For example, if and , then where and .

The following theorem known as the Division Algorithm shows us that for any pair of integers where that the form exists and is unique.

Theorem 1 (The Division Algorithm): Let with . Then there exists unique where and such that .

The value of is often called the Remainder when is divided by . That is, is equal to some multiple , known as the Quotient of , plus a remainder for which .

  • Proof: Let with and consider the following set :
  • Consider of nonnegative elements in . If then and if then , so is a nonempty set. Moreover, is a nonempty subset of integers and so by the Well-Ordering principle there exists a least element of , say is this least element. Since we have that for some and so:
  • So if of the form we desire. Now since and every element in is nonnegative then we have that . Suppose that . Then where and so:
  • Therefore and so . Since and we have that . But this implies that is not the least element of which is a contradiction, so our assumption that is false, and so .
  • We lastly show that and are unique for and . Suppose that and where . Then by subtracting these two equations we get:
  • Therefore is a multiple of . Since and we must have that . The only multiple of such that is , so:
  • Since , we have that . Since this implies that and so . Therefore can be uniquely expressed in the form where .


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