The Basic Number Systems
Natural numbers, denoted by N, are the first numbers we learn, including 1, 2, 3, and so on We become familiar with basic operations like addition and multiplication, which are binary operations that combine two natural numbers to produce another natural number These operations exhibit important properties such as commutativity, associativity, and the distributive law While subtraction and division are considered inverse operations of addition and multiplication, they are not classified as basic operations When working exclusively with natural numbers, we face limitations with subtraction and division, as these operations do not always yield natural numbers; for instance, 3−5 and 3÷5 do not result in natural numbers.
To perform subtraction, we include the number 0 and the negative counterparts of natural numbers, forming the set of integers Z = {…, -3, -2, -1, 0, 1, 2, 3, …} The number 0 serves as the identity element for addition, meaning that for any integer a, the equation a + 0 = a holds true Additionally, the negative of any integer a acts as its additive inverse, resulting in the equation a + (-a) = 0.
Exploring the concepts of zero and negative numbers in addition raises intriguing questions, such as why multiplying any number by zero results in zero and why the product of two negative numbers yields a positive outcome More broadly, what is the true significance of multiplying by a negative number? Is it merely an extension of repeated addition? These questions warrant further examination.
In the set of integers Z, the operations of addition and multiplication retain the same properties as in the natural numbers N Notably, Z includes an identity element and inverses for addition, enabling the subtraction of any integer from another while always resulting in another integer.
Division can only be performed on specific pairs of integers, and while the number 1 serves as the identity element for multiplication in the set of integers (Z), most integers lack multiplicative inverses To enable division, we need to introduce fractions, leading to the creation of the set of rational numbers (Q) with defined operations.
Definition 1.1.1 The set Q of rational numbers is the set of all quotients of integers (i.e., fractions),
, and we define i a b = a 0 b 0 if and only if ab 0 0 , ii a b + c d = ad+bc bd , iii a b ãc d = ac bd.
Note that since a = a 1 for an integer a, every integer is also a rational number and we have
Z⊆Q The operations of addition and multiplication in Q still satisfy all of the properties as in the set of integers Z.
If \( q \neq 0 \) is a rational number expressed as \( q = \frac{a}{b} \) with \( a \neq 0 \), then \( r = \frac{a}{b} \) is also rational The equation \( q \times r = \frac{a}{b} \times \frac{b}{a} = 1 \) shows that \( \frac{b}{a} \) serves as the multiplicative inverse of \( \frac{a}{b} \) Therefore, for any non-zero rational number \( q = \frac{a}{b} \), its multiplicative inverse, or reciprocal, is \( q^{-1} = \frac{b}{a} \).
The construction of the set of real numbers, R, typically involves advanced analytical tools; however, for our discussion, a simpler approach using decimal expansions will be adequate We will begin by reviewing the fundamentals of decimal expansions and then explore how they apply to rational numbers.
A positive integer m can always be written in its decimal form and expressed as a sum of multiples of non-negative powers of 10: m=nknk−1 n2n1n0=nkã10 k +nk−1ã10 k−1 +ã ã ã+n2ã10 2 +n1ã10 1 +n0ã10 0
Similarly, a positive numberr < 1 with a terminating decimal expansion can be written as a sum of multiples of negative powers of 10: r= 0.d1d2 dk−1dk=d1ã10 −1 +d2ã10 −2 +ã ã ã+dk−1ã10 −(k−1) +dkã10 −k
If the decimal expansion does not terminate, then r is an “infinite sum” of multiples of negative powers of 10: r = 0.d 1 d 2 d 3 =d 1 ã10 −1 +d 2 ã10 −2 +d 3 ã10 −3 +ã ã ã.
A real number can be expressed as a finite sum of non-negative powers of 10, combined with a potentially infinite sum of negative powers of 10, where the coefficients in this representation correspond to the digits in its decimal expansion.
Rational numbers can be defined by their decimal expansions, which either terminate or repeat For instance, the fraction 3/8 equals 0.375, while 9/7 results in the repeating decimal 1.285714285714 To confirm this characterization, we need to demonstrate that any rational number \( \frac{a}{b} \) has a decimal expansion that either terminates or repeats Additionally, it is essential to establish that every decimal expansion that terminates or repeats corresponds to a rational number \( \frac{a}{b} \).
Every rational number \( \frac{a}{b} \) has a decimal expansion that is either terminating or repeating For our analysis, we can assume that \( \frac{a}{b} \) is positive and that \( a < b \) The decimal representation of \( \frac{a}{b} \) is derived through the process of long division of \( a \) by \( b \).
To perform long division with decimals, we start by placing a decimal point to the right of the dividend and adding sufficient zeros to exceed the dividend's value Next, we divide this new number, placing the quotient above the division sign and determining a remainder that satisfies the condition 0 ≤ r < b This process continues by dividing the remainder by the divisor, as illustrated in the example of 3 ÷ 17.
The boldface numbers in the example are the remainders.
If a remainder of 0 is obtained at some stage, then the decimal expansion terminates, as for 3 8 :
If the remainder is never zero, each remainder \( r \) satisfies the condition \( 16r < 6b - 1 \) Consequently, there are only a finite number of possible remainders, which means that eventually, a remainder must repeat Once a remainder repeats, the sequence of quotients and remainders will also repeat indefinitely, resulting in a repeating decimal expansion, as illustrated by the example of \( 54 \div 7 \).
Notice that 16 is the first remainder to repeat in this example and the corresponding quotient, 2, is the start of the repeating decimal.
Exploration: Under what conditions on a fractiona/bin lowest terms will the decimal expansion be terminating? Investigate this question by searching in number theory texts or Internet sources.
Every terminating or repeating decimal expansion represents a rational number For simplicity, we can consider these decimal numbers as positive and less than one A terminating decimal with k decimal places, expressed as d1d2 dk, can be converted into a fraction by using the digits d1, d2, , dk.
A repeating decimal's period is defined as the length of its shortest repeating sequence of digits For instance, the decimal 0.454545 has a period of 2, while 0.1234563456 has a period of 4.
A repeating decimal of period k, say
A rational number R can be expressed in the form R = 0.d1d2 dj r1r2 rk To convert R into a fraction, we first multiply it by 10^k, shifting the decimal point k places to the right This results in 10^k R, which aligns the digits of R and 10^k R after a certain decimal point due to the periodic nature of R with a period of k.
10 k R−R= (10 k −1)R will be a terminating decimal, hence equal to some fraction T Therefore (10 k −1)R =T and
Example: Write the repeating decimalR= 0.12345345 .= 0.12 345 as a fraction.
The period of R is 3, so we calculate 10 3 R= 1000R:
Complex Numbers
In this section we introduce the arithmetic and geometry of complex numbers.
The set of complex numbers, denoted as C, consists of symbols in the form of a + bi, where a and b are real numbers Two complex numbers a + bi and c + di are equal if and only if a = c and b = d The addition of complex numbers is defined as (a + bi) + (c + di) = (a + c) + (b + d)i, while their multiplication is given by (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
Note that the definition of multiplication implies (with a = c = 0, b = d = 1) that i 2 = −1. Therefore we think ofias the square root of−1 Of course (−i) 2 =−1 as well, so we definei=√
−1, the principal square root of −1 If cis a positive real number, we also define √
−c=√ cãi. With the convention that i 2 = −1, multiplication of complex numbers follows the usual rules for multiplying binomials:
(a+bi)ã(c+di) +adi+bci+bdi 2 = (ac−bd) + (ad+bc)i.
Example: Ifw= 3 + 5iand z= 2−7i, we have w+z= (3 + 5i) + (2−7i) = (3 + 2) + (5−7)i= 5−2i and wãz= (3 + 5i)ã(2−7i) = [3(2)−5(−7)] + [3(−7) + 5(2)]i= 41−11i.
Any real number \( a \) can be expressed as \( a + 0i \), allowing us to treat it as a complex number Therefore, the set of real numbers \( \mathbb{R} \) is a subset of the set of complex numbers \( \mathbb{C} \) This establishes a clear relationship between these two mathematical sets.
In the complex number system C, the elements 0 (0 + 0i) and 1 (1 + 0i) serve as the identity elements for addition and multiplication, respectively For any complex number z represented as a + bi, its additive inverse is denoted as -z, which equals -a - bi These properties can be easily verified through direct calculations.
To understand multiplicative inverses and division in complex numbers, we need to establish some definitions A complex number, represented as z = a + bi, consists of a real part and an imaginary part Specifically, the real part of z is denoted as Re(z) = a, while the imaginary part is represented as Im(z) = b.
(Note that the imaginary part of zis band NOTbi.)
Definition 1.2.3 Ifz=a+biis a complex number, the(complex) conjugateofz isz=a−bi. Examples:
The following properties of conjugates are easy to verify using the definitions.
Proposition 1.2.4 If z and w are complex numbers, then i z+w=z+w ii zãw=zãw.
Proof (i) Letz=a+biand w=c+di, so that z=a−biand w=c−di We have z+w = (a+bi) + (c+di)
= (a−bi) + (c−di) by Definition 1.2.1 (ii),
(ii) The proof of (ii) is similar and is left as an exercise (See Exercise 1.2.11.) Proposition 1.2.5 If z=a+bi is a complex number, then i z+z= 2a= 2Re(z) ii z−z= 2bi= 2Im(z)ãi.
Proof (i) Letz=a+biso that z=a−biand Re(z) =a We have z+z = (a+bi) + (a−bi)
= 2a, and so z+z= 2a= 2Re(z) as claimed.
(ii) The proof of (ii) is similar and is left as an exercise (See Exercise 1.2.12.)
The proof of the next result is a computation similar to the previous proofs and is left as an exercise (See Exercise 1.2.13.)
Proposition 1.2.6 If z =a+bi is a complex number, then zãz =a 2 +b 2 , a non-negative real number.
In the study of complex numbers, we examine multiplicative inverses and division For a non-zero complex number \( z = a + bi \), at least one of \( a \) or \( b \) must be non-zero, ensuring that \( a^2 + b^2 \) is also a non-zero real number Consequently, it follows that \( \frac{1}{z} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i \), demonstrating how to compute the inverse of a complex number effectively.
Proposition 1.2.7 Ifz=a+biis a non-zero complex number, then there is a complex numberz −1 such that zz −1 = 1 In particular, z −1 = z zz = z a 2 +b 2 = a a 2 +b 2 − b a 2 +b 2 i.
To express the quotient of two complex numbers, \( \frac{a+bi}{c+di} \), in standard form, multiply both the numerator and denominator by the conjugate of the denominator, resulting in a real number in the denominator This method resembles the process of rationalizing a denominator that includes a root, as demonstrated by the equation \( \frac{a+bi}{c+di} = \frac{(a+bi)(c−di)}{c^2 + d^2} \).
(c+di)(c−di) = (ac+bd) + (bc−ad)i c 2 +d 2 = ac+bd c 2 +d 2 +bc−ad c 2 +d 2 i.
It is best to learn the procedure demonstrated here, and not to memorize this final formula.
1 Ifz= 2 + 3i, then the multiplicative inverse or reciprocal of z is z −1 = 1
2 Ifw=−1 + 3iand z= 2−5i, the the quotient w/z is w z = −1 + 3i
Complex numbers are visually represented as points in the complex plane, using the standard xy-coordinate system where the x-axis denotes the real part and the y-axis represents the imaginary part For a complex number z = a + bi, its graphical representation corresponds to the coordinates (a, b) Consequently, real numbers, such as a = a + 0i, are depicted as points on the real axis at (a, 0), aligning the real number line with the real axis in this geometric framework.
In complex number theory, the reflection of a complex number \( z \) across the real axis is denoted as \( \overline{z} \) The distance of a complex number \( z = a + bi \) from the origin is calculated using the formula \( \sqrt{a^2 + b^2} \) This concept parallels the absolute value of real numbers, which represents the distance from the number to the origin on the number line, leading to a formal definition for complex numbers.
Definition 1.2.8 If z = a+bi is a complex number, the absolute value or modulus of z, denoted |z|, is the distance from z to the origin in the complex plane Thus
Note that this also says zz=|z| 2
In the geometric interpretation of complex numbers, let z = a + bi and w = c + di represent points P = (a, b) and Q = (c, d) in the complex plane, with O = (0, 0) as the origin This framework allows us to visualize the addition and multiplication of complex numbers through their corresponding points in the plane.
Q, and O are not all on the same line Then the point S = (a+c, b+d) representing the sum z+w= (a+c) + (b+d)iis the endpoint of the diagonal of the parallelogram with the line segments
In the context of vector addition within the vector space R², the sides OP and OQ are depicted, as shown in Figure 1.2 This illustration serves to highlight the geometric representation of vector addition.
Figure 1.2: Addition of Complex Numbers
In geometric terms, the sum \( z + w \) can be visualized by plotting points \( P \), \( Q \), and the origin \( O \) on a straight line When both points \( P \) and \( Q \) are positioned on the same side of the origin, their sum \( z + w \) will also lie on that side, resulting in a straightforward linear addition Conversely, if \( P \) and \( Q \) are located on opposite sides of the origin, their sum may fall either at the origin or on the side of the point with the greater magnitude, illustrating the concept of vector addition in a more complex manner Experimenting with specific examples can further clarify these scenarios and their geometric implications.
The additive inverse of a complex number z = a + bi, represented as −z = −a − bi, serves as its reflection in the origin By utilizing geometric descriptions of addition and additive inverses, along with the principle that w − z = w + (−z), we can interpret the subtraction of complex numbers geometrically, as demonstrated in Figure 1.3.
Figure 1.3: Subtraction of Complex Numbers
To geometrically describe the multiplication of complex numbers, we utilize trigonometry and the polar form of complex numbers A point P = (a, b) in the plane is uniquely identified by its distance r from the origin O and the angle θ formed with the positive x-axis along the segment OP These values (r, θ) are referred to as the polar coordinates of point P.
A complex number z can be expressed in polar form, where its distance from the origin, known as the modulus, is calculated as r = |z| = √(a² + b²) Understanding this representation is essential for further applications in complex analysis.
Definition 1.2.9 The angle θ between the positive real axis and the line segment from the origin to the complex number z is the argument of z, denotedargz.
Note that the angle arg z is not unique, adding any integer multiple of 360 ◦ will yield another argument The polar form of z=a+biis illustrated in Figure 1.4.
The polar form of complex numbers establishes a relationship between their standard and polar representations through the equations a = r cos θ, b = r sin θ, and r = |z| = √(a² + b²), with tan θ = b/a Consequently, a complex number z can be expressed in polar form as z = r(cos θ + i sin θ).
Note thatr =|z|is a non-negative real number and cosθ+isinθ is a complex number of modulus pcos 2 θ+ sin 2 θ= 1, hence lies on the unit circle centered at the origin.
1 Find the polar form of the complex numberz= 3 + 3√
3 Since z is in the first quadrant and tanθ=√
3, we haveθ= 60 ◦ Therefore, the polar form ofz is z= 6 (cos 60 ◦ +isin 60 ◦ ).
2 Find the polar form of the complex numberw=−2√
3 The reference angle is θ 0 = arctan(1/√
3) = 30 ◦ , and sincew is in the second quadrant, we have θ= 180 ◦ −30 ◦ = 150 ◦ Therefore, the polar form of wis w= 4 (cos 150 ◦ +isin 150 ◦ ).
The next result says that in order to multiply two complex numbers, we multiply their moduli and addtheir arguments.
Theorem 1.2.10 If z 1 =r 1 (cosθ 1 +isinθ 1 ) and z 2 =r 2 (cosθ 2 +isinθ 2 ), then z1z2 =r1r2[cos(θ1+θ2) +isin(θ1+θ2)].
Proof Suppose we have two complex numbersz1 =r1(cosθ1+isinθ1) andz2=r2(cosθ2+isinθ2). Their product is z1z2 = r1(cosθ1+isinθ1)ãr2(cosθ2+isinθ2)
= r 1 r 2 [(cosθ 1 cosθ 2 −sinθ 1 sinθ 2 ) +i(cosθ 1 sinθ 2 + sinθ 1 cosθ 2 )].
By the angle sum formulas for sine and cosine (see Appendix A), we have cosθ 1 cosθ 2 −sinθ 1 sinθ 2 = cos(θ 1 +θ 2 ) and cosθ 1 sinθ 2 + sinθ 1 cosθ 2 = sin(θ 1 +θ 2 ).
Substituting yields z 1 z 2 =r 1 r 2 [cos(θ 1 +θ 2 ) +isin(θ 1 +θ 2 )], as claimed Using this theorem and mathematical induction (see §2.1), we obtain the following corollary. Corollary 1.2.11 If z=r(cosθ+isinθ) andn is a positive integer, then z n =r n (cosnθ+isinnθ).
In the case where r= 1, Corollary 1.2.11 becomes
Algebraic Properties of Number Systems
1.3 Algebraic Properties of Number Systems
In this section, we discuss various algebraic properties that our number systems share Before reading further, consider the following questions.
1 What properties of addition or multiplication of natural numbers are used in the following equations or calculations? Write out the properties carefully Whyare they true?
2 Besides those demonstrated above, do you know any other “basic” properties of addition or multiplication of natural numbers?
3 Does subtraction or division make sense inN?
When examining the set of integers Z, it is essential to determine whether the properties applicable to natural numbers N also hold true In this context, we can define zero and negative numbers in terms of addition by recognizing that adding zero to any integer yields the same integer, while adding a negative number effectively subtracts from the original integer Additionally, subtraction can be understood as the addition of a negative number, allowing us to maintain consistency in mathematical operations across both sets of numbers.
5 With 0 and−ndefined via addition, use the known properties of addition and multiplication to show:
(c) (−1)ã(−1) = 1, or more generally, (−a)ã(−b) =aãb for all a, b∈Z.
6 With the definitions discussed above, we can define subtraction in terms of addition in Z. Does subtraction satisfy all of the same properties as addition? If not, which ones fail?
7 In order to be able to subtract, we extended from N to Z What set is needed in order to allow us to divide?
8 Are all of the properties of addition and multiplication forZalso valid forQ? What additional properties does Qhave thatZ does not?
9 How can we define division in Q in terms of multiplication? Does division satisfy all of the same properties as multiplication? If not, which ones fail?
10 Do addition and multiplication in Rand Csatisfy the same properties as inQ?
The questions posed relate to the fundamental properties of addition and multiplication within the discussed number systems Below is a summary of the properties that a set S, on which addition and multiplication are defined, may or may not satisfy.
Definition 1.3.1 (Algebraic Properties) Let S be a set on which addition(+) and multiplica- tion (ã) are defined We define the following (potential) properties.
The properties of addition include several key principles: Closure under addition states that for any elements a and b in a set S, their sum a + b also belongs to S The Associative Law of Addition indicates that the grouping of numbers does not affect the sum, as shown by the equation a + (b + c) = (a + b) + c for all elements a, b, and c in S According to the Commutative Law of Addition, the order of addition does not matter, meaning a + b is equal to b + a for all a and b in S The Additive Identity property asserts that there exists an element 0 in S such that adding it to any element a yields the same element, as expressed by a + 0 = 0 + a for all a in S Lastly, the concept of Additive Inverses states that for every element a in S, there exists an element -a in S such that their sum equals the additive identity, resulting in a + (-a) = (-a) + a = 0.
The properties of multiplication in a set S include closure, which states that the product of any two elements a and b in S is also in S The associative law indicates that for any elements a, b, and c in S, the equation a × (b × c) equals (a × b) × c The commutative law asserts that for all elements a and b in S, the product a × b is equal to b × a Additionally, the multiplicative identity property reveals that there exists an element 1 in S such that multiplying any element a by 1 yields a, affirming that a × 1 = 1 × a = a Lastly, the concept of multiplicative inverses states that for every non-zero element a in S, there exists an element a⁻¹ in S such that the product of a and its inverse equals 1, meaning a × a⁻¹ = a⁻¹ × a = 1.
Property Relating Addition and Multiplication: xi Distributive Laws: cã(a+b) =cãa+cãb for all a,b, and c in S,
(a+b)ãc=aãc+bãc for all a,b, and c in S.
The two distributive laws, known as the left and right distributive laws, govern the distribution of multiplication over addition In a set S where property (viii) is satisfied and multiplication is commutative, only one of these distributive laws is required, as the other can be derived from the commutativity of multiplication.
The set Nof natural numbers satisfies all of these properties except (iv), (v), and (x) The set
The set W, defined as W = N ∪ {0}, includes the additive identity 0, fulfilling the requirement of property (iv) However, W lacks additive inverses for all its elements except for 0, which means it does not satisfy property (v).
The set of integers, denoted as Z, includes natural numbers, zero, and negative integers Integers exhibit all the properties of natural numbers, with zero serving as the additive identity, as demonstrated by the equation a + 0 = 0 + a for any integer a Each integer a has a corresponding negative integer -a, fulfilling the condition a + (-a) = (-a) + a = 0, which establishes -a as the additive inverse of a Additionally, the number 1 acts as the multiplicative identity in Z, since 1 × a = a × 1 for every integer a However, for any integer a other than 1 or -1, there is no integer b that satisfies the equation a × b = b × a = 1, indicating that Z does not satisfy property (x).
1.3 ALGEBRAIC PROPERTIES OF NUMBER SYSTEMS 19
The set Q of rational numbers includes all integers and their quotients, or fractions By applying Definition 1.1.1 and leveraging established integer properties, it can be demonstrated that Q meets properties (i)–(ix) and (xi) of Definition 1.3.1 For any non-zero rational number a/b, where a ≠ 0, the reciprocal b/a is also a rational number, confirming that a/b multiplied by its reciprocal equals 1 Therefore, a/b serves as the multiplicative inverse of itself, further validating that Q satisfies property (x).
1 Assuming the known properties of Z, show that multiplication inQis commutative.
Proof Let a b and c d be any rational numbers, so a, b, c, d∈Z We then use properties ofZin Definition 1.3.1 to show that a b ã d c = d c ã a b We have a b ã c d = ac bd by Definition 1.1.1 (iii),
= ca db by 1.3.1 (viii), commutativity of multiplication inZ,
= c dãa b by Definition 1.1.1 (iii), and so multiplication inQis commutative
2 Assuming the known properties of Z, show that addition inQis associative.
Proof Let a b , d c , and f e be any rational numbers, soa, b, c, d, e, f ∈Z By definition of addition inQ(Definition 1.1.1 (ii)), we have a b + c d
+ e f = ad+bc bd + e f = (ad+bc)f + (bd)e
= a b +cf +de df = a(df) +b(cf+de) b(df)
We use properties ofZ in Definition 1.3.1 to show that these expressions are equal We have a b + c d
= [(ad)f+ (bc)f] +b(de) b(df) by 1.3.1 (xi), (vii) in Z,
= a(df) + [b(cf) +b(de)] b(df) by 1.3.1 (ii), (vii) inZ,
= a(df) +b(cf+de) b(df) by 1.3.1 (xi) inZ,
Therefore ( a b + c d ) + f e = a b + ( c d + e f ) and addition inQis associative
The set of real numbers, denoted as R, adheres to the properties outlined in Definition 1.3.1 through standard addition and multiplication By applying the definitions of addition and multiplication from Definition 1.2.1, along with established properties of real numbers, it can be demonstrated that the complex numbers, represented as C, fulfill most properties of Definition 1.3.1, specifically properties (i)–(ix) and (xi) Furthermore, Proposition 1.2.7 confirms that C also meets property (x) of Definition 1.3.1 Consequently, the sets of rational numbers (Q), real numbers (R), and complex numbers (C) collectively satisfy all the properties specified in Definition 1.3.1.
Example: Assuming the known properties of R, show that multiplication inCis commutative.
Proof Leta+biandc+dibe any complex numbers, soa, b, c, d∈R We use properties ofR in Definition 1.3.1 to show that (a+bi)ã(c+di) = (c+di)ã(a+bi) We have
(a+bi)ã(c+di) = (ac−bd) + (ad+bc)i by Definition 1.2.1 (iii),
= (ca−db) + (da+cb)i by 1.3.1 (viii) inR,
= (ca−db) + (cb+da)i by 1.3.1 (iii) in R,
= (c+di)ã(a+bi) by Definition 1.2.1 (iii), and so multiplication inQis commutative
A set S, equipped with defined operations of addition (+) and multiplication (ã), is classified as a ring if it meets specific properties outlined in Definition 1.3.1 Furthermore, if S fulfills these properties and additionally satisfies property (viii), it is identified as a commutative ring Additionally, when S is a ring that also meets property (ix), it is referred to as a ring with identity or a ring with 1 Lastly, a set S that complies with all properties (i)–(xi) of Definition 1.3.1 is recognized as a field.
ThusNis NOT a ring because properties (iv) and (v) are not satisfied By the discussion above, we have the following result.
Theorem 1.3.3 The set of integers Z is a commutative ring with 1, and Q, R, and Care fields.
When examining mathematical sets such as functions, polynomials, and matrices, it's essential to determine whether they qualify as rings or fields based on the definitions of addition and multiplication A ring is a set equipped with two operations that satisfy certain properties, while a field is a more restrictive structure that requires the existence of multiplicative inverses for non-zero elements By analyzing these sets, one can identify their algebraic structures and understand their roles in broader mathematical contexts.
In the rings Z, Q, and R, the elements 0, -a, and 1 fulfill the properties (iv), (v), and (ix), respectively Additionally, in the rings Q or R, the element a - 1 = 1/a meets property (x) In any general ring S, these elements are defined based on the corresponding properties.
In mathematics, the additive identity element, denoted as "0," is defined as any element of a set S that fulfills a specific property, regardless of its appearance The additive inverse, represented as "−a," is the element that, when added to "a," results in the additive identity "0." Both "0" and "−a" are fundamentally defined through the operation of addition It's important to note that in most rings, including the complex numbers (C), the notions of "positive" or "negative" elements do not exist.
In a ring S, subtraction is defined through addition For any elements a and b in S, the operation a - b is expressed as a + (-b), where -b represents the additive inverse of b.
1.3 ALGEBRAIC PROPERTIES OF NUMBER SYSTEMS 21
In a set S that adheres to the commutative property of multiplication, as well as the definitions of the multiplicative identity "1" and the multiplicative inverse "a −1", we can also define division in terms of multiplication Specifically, for any elements a and b in S, where b is not equal to zero, the expression a ÷ b or a/b is defined as the product of a and the multiplicative inverse of b, denoted as a × b −1.
In the rational numbers Q, for example, if q = a b and r = d c 6= 0, then we have r −1 = d c , the reciprocal of r, andqữr =qãr −1 , hence a b ÷ c d = a b ãc d
= a b ãd c. This explains the rule for dividing fractions we all learned in school.
Division is not usually defined in a ring that is not commutative Since the elements b −1 ãaand aãb −1 may not be equal, the expression aữb would be ambiguous.
Sets and Equivalence Relations
To establish a more rigorous framework for our number systems, it is essential to introduce the concept of an equivalence relation within a set We will begin by revisiting key terminology and notation from set theory, adopting a somewhat informal approach to the subject.
A set A is defined as a collection of objects, known as its elements When an object a belongs to set A, it is denoted as a ∈ A, while an object b that does not belong to set A is represented as b /∈ A.
A set of particular importance is the set containing nothing.
Definition 1.4.2 The empty set is the (unique) set with no elements, denoted∅.
In set theory, two sets A and B are considered equal, denoted as A = B, if they contain the same elements Additionally, if every element of set A is also found in set B, then A is referred to as a subset of B, written as A ⊆ B Furthermore, if A is a subset of B but the two sets are not equal, we classify A as a proper subset of B, denoted A ⊂ B.
In set theory, if A is a subset of B (A ⊆ B), it indicates that every element a in A is also an element of B Conversely, if A is properly contained in B (A ⊂ B), it means that all elements of A are in B, but there exists at least one element b in B that is not in A Additionally, it is important to note that the empty set (∅) is a subset of any set A.
It's important to differentiate between the symbols ∈ and ⊆ in set theory The symbol ∈ denotes membership of an element within a set, while ⊆ indicates that one set is a subset of another For instance, if A = {1, 2, 3}, we can say that 1 ∈ A, meaning 1 is an element of A, and {1} ⊆ A, indicating that {1} is a subset of A However, using 1 ⊆ A or {1} ∈ A is incorrect.
The following result provides the most common method for showing that two sets are equal. Proposition 1.4.4 Two sets A and B are equal if and only if A⊆B andB ⊆A.
The proposition says that we can show A = B by showing that if a ∈ A then a ∈ B and if b∈B thenb∈A.
We next consider some methods for constructing new sets from old.
Definition 1.4.5 Let A, B, A1, A2, , An be sets. i The union of A and B is the set A∪B ={x|x∈A orx∈B} More generally, the union of A1, A2, , An is the set
Ai ={x|x∈Ai for somei}. ii Theintersectionof A and B is the setA∩B={x|x∈Aandx∈B} More generally, the intersection of A 1 , A 2 , , A n is the set
The term "or" is utilized in an inclusive manner, indicating that if x belongs to set A or set B, it signifies that x is an element of at least one of these sets, and it may also belong to both sets simultaneously.
Two sets, A and B, are considered disjoint if their intersection is empty, denoted as A∩B = ∅ A set A can be described as a disjoint union of subsets A1, A2, , An when it can be expressed as the union of these subsets, such that A = A1∪A2∪ ∪An, with the condition that each pair of subsets, Ai and Aj, are disjoint for all i ≠ j In this context, the subsets A1, A2, , An are said to form a partition of the set A.
Definition 1.4.7 Let A andB be sets The Cartesian productof A and B is the set
A×B ={(a, b)|a∈A, b∈B} consisting of all ordered pairs (a, b), where a∈A and b∈B.
The elements of A×B areordered pairs, and two elements (a 1 , b 1 ) and (a 2 , b 2 ) are equal if and only ifa1=a2 andb1 =b2 Thus ifA={1,2}=B for example, then (1,2) and (2,1) are different elements of A×B.
We are now prepared to define equivalence relations.
Definition 1.4.8 A relation R on a set A is a subset ofA×A.
In mathematics, a relation is typically defined as a set, but we often simplify this concept by referring to elements of set A as being "related" to one another Practically, we express this relationship using the notation aRb, indicating that element a is related to element b if the pair (a, b) belongs to the relation R.
It's important to understand that the term "related" does not imply a two-way connection The definition allows for the possibility that if element a is related to element b, it does not necessarily mean that b is related to a Therefore, while (a, b) may be part of the relation R, the reverse pair (b, a) might not exist within that same relation.
We will be primarily interested in a particular type of relation called an equivalence relation.
Definition 1.4.9 An equivalence relation on a set A is a relation ∼ satisfying, for all a, b, and c in A: i a∼a (reflexive property). ii If a∼bthen b∼a(symmetric property). iii If a∼band b∼c, then a∼c (transitive property).
An equivalence relation ∼ on a set A indicates that two elements, a and b, are equivalent if a ∼ b This concept generalizes the idea of equality, meaning that while a and b may not be equal, they are considered "the same" concerning a specific property It's important to note that equality itself is also classified as an equivalence relation.
In the set of real numbers R, we can establish an equivalence relation denoted by a ∼ b if the absolute values of a and b are equal (|a| = |b|) This relation is verified to be an equivalence relation, as it satisfies the necessary properties For instance, the numbers 3 and -3 are considered equivalent under this relation, as they share the same distance from the origin, despite not being equal.
In the context of equivalence relations, it is beneficial to treat all elements of a set that are equivalent to a specific element as identical We designate a name for the collection of these equivalent elements.
Definition 1.4.10 If ∼ is an equivalence relation on a set A and a is an element of A, the equivalence class of ais the set [a] ={b∈A|a∼b}.
The equivalence class of a is just the set of all elements ofA that are equivalent to a In the example above, [a] ={a,−a} Thus [3] ={3,−3}= [−3] The fact that 3 ∼ −3 and [3] = [−3] is not a coincidence.
Proposition 1.4.11 Let ∼ be an equivalence relation on a set A For a and b in A, [a] = [b] if and only if a∼b.
The proposition states that if two elements of set A are equivalent, their corresponding equivalence classes will be identical According to the defining properties of an equivalence relation, every element within a specific equivalence class must be equivalent to one another.
Every element of set A belongs to an equivalence class, denoted as [a] Consequently, A can be expressed as the union of these equivalence classes Furthermore, it is important to note that this union is disjoint, meaning the equivalence classes partition set A into distinct, non-overlapping subsets.
Proposition 1.4.12 Let ∼ be an equivalence relation on a set A If a and b are elements of A, then either [a] = [b] or [a]∩[b] =∅. §1.4 Exercises
In the following exercises,A,B, and C are sets.
1 Show that if A⊆C and B ⊆C, thenA∪B ⊆C.
2 Show that if C⊆A and C⊆B, thenC⊆A∩B.
7 Verify that the relation ∼ on R, defined by a∼ b if and only if |a|= |b|, is an equivalence relation.
8 Verify that the relation ∼ on R, defined by a ∼ b if and only if a−b is rational, is an equivalence relation.
9 Verify that the relation∼on the set of fractions{ a b |a, b∈Z, b6= 0}, defined by a b ∼ c d if and only if ad, is an equivalence relation.
10 Show that if ∼is an equivalence relation on a setA and a∼b, then [a] = [b].
11 Show that if∼is an equivalence relation on a setA andaand bare elements of Asuch that[a]∩[b] is not empty, thena∼b (This proves Proposition 1.4.12.)
Formal Constructions of Number Systems
While many people today are familiar with various number systems, complex numbers remain a challenge for some Historically, concepts like zero, negative numbers, irrational numbers, and complex numbers have posed significant difficulties compared to natural numbers To truly understand and accept the existence of these mathematical entities, it's essential to have tangible examples that demonstrate their expected behaviors.
This section offers a formal and systematic development of various number systems We will start by constructing the field Q of rational numbers based on the ring Z of integers Next, we will derive the integers Z from the natural numbers N, followed by the construction of the complex numbers C from the real numbers R.
The natural numbers can be modeled through a set of axioms known as the Peano axioms; however, this abstract construction will not be discussed in this article Additionally, a construction of the real numbers is omitted, as it involves analytical tools that exceed the scope of this text.
Construction of the Rational Numbers
The construction of rational numbers, while logically derived from integers, is more relatable to our understanding as fractions The key differences lie in notation and the formalization of equivalent fractions We will explore the construction of rational numbers, assuming all properties of the integer ring, defining rational numbers as equivalence classes of ordered pairs of integers.
This construction represents a specific instance of creating the "field of fractions of an integral domain," a concept explored in ring theory Additionally, it enables the derivation of other fields from rings that resemble the integers, including polynomial rings.
Definition 1.5.1 Let Q={(a, b)|a, b∈Z, b6= 0}and define a relation∼onQ by(a, b)∼(a 0 , b 0 ) if and only if ab 0 0
Proposition 1.5.2 The relation ∼defined on Q is an equivalence relation.
Definition 1.5.3 For (a, b) in Q, let [(a, b)] be the equivalence class of (a, b) under the relation
∼ Define Q to be the set of equivalence classes of elements of Q, that is,
To comprehend the construction of the rational numbers Q, consider the element (a, b) as the fraction a/b For two pairs (a, b) and (a₀, b₀) in Q, they are equivalent, denoted as (a, b) ∼ (a₀, b₀), if and only if the cross products ab₀ and a₀b are equal This implies that a/b equals a₀/b₀, establishing that the equivalence relation ∼ mirrors the standard equivalence of fractions.
The element [(a, b)] in the set of rational numbers Q represents the fraction a/b, which can also be expressed as any equivalent fraction Additionally, the set of integers Z can be seen as a subset of Q by associating each integer z with the rational number represented by the element [(z, 1)] in Q.
1.5 FORMAL CONSTRUCTIONS OF NUMBER SYSTEMS 29
The concept of equality among fractions is formally represented by treating elements of the set of rational numbers, Q, as equivalence classes For instance, the rational number 0.666 is denoted by the single equivalence class [(2,3)], rather than by its various equivalent fractions such as 2/3, 4/6, or 6/9.
Keep in mind that the elements of Q are equivalence classes, hence are sets For example,
For (a, b) in Q, [(a, b)] = [(ac, bc)] for every non-zero integerc, just as a b = ac bc
Next, we must define addition and multiplication on Q so that Q behaves like the rational numbersQ Compare the following definition to Definition 1.1.1.
Definition 1.5.4 Forx= [(a, b)]andy= [(c, d)]inQ, we define additionx+yand multiplication xãy as follows: i x+y= [(a, b)] + [(c, d)] = [(ad+bc, bd)] ii xãy= [(a, b)]ã[(c, d)] = [(ac, bd)].
The operations of addition and multiplication involving x and y are influenced by the specific representatives (a, b) and (c, d) selected for the equivalence classes To ensure the validity of these definitions, it is crucial to establish that these operations remain consistent regardless of the chosen representatives.
Proposition 1.5.5 If [(a, b)] = [(a 0 , b 0 )] and [(c, d)] = [(c 0 , d 0 )] in Q, then
That is, addition and multiplication inQ arewell-defined.
The set of rational numbers, Q, is closed under both addition and multiplication, as defined by the operations within Q and the closure properties of integers, Z Direct calculations can be performed to verify this closure under the specified operations.
Proposition 1.5.6 The following hold in Q. i The element 0= [(0,1)] is the additive identity element ofQ. ii If x= [(a, b)]is an element of Q, then the additive inverse of xis −x= [(−a, b)].
The expression [(a, b)] equals zero if and only if the pair (a, b) is equivalent to (0, 1), which implies that a raised to the power of 1 is equal to b raised to the power of 0, leading to the conclusion that a must be zero Conversely, if [(a, b)] is not equal to zero, then a cannot be zero, and it follows that [(b, a)] is also a member of the set Q This principle is applied in the subsequent result.
Proposition 1.5.7 The following hold in Q. i The element 1= [(1,1)] is the multiplicative identity element of Q. ii If x= [(a, b)]6=0 is an element ofQ, then the multiplicative inverse ofx is x −1 = [(b, a)].
In the field of rational numbers (Q), both addition and multiplication are demonstrated to be commutative and associative, adhering to the distributive law These properties can be established using the definitions of Q, along with the known characteristics of integers (Z) The proofs for these properties closely resemble those found in Exercises 1.3.2–1.3.5, leading to the formulation of a significant theorem regarding the arithmetic operations within Q.
Theorem 1.5.8 With Q, addition, and multiplication defined as above, Q satisfies properties (i)–(xi) of Definition 1.3.1 That is, Q is a field.
In abstract algebra, the rational number represented by the fraction \( \frac{a}{b} \) (or any equivalent fraction) identifies the element \([(a, b)]\) of the set of rational numbers \(Q\), demonstrating that \(Q\) is algebraically equivalent to itself This relationship implies that the two fields are isomorphic.
The construction of integers from natural numbers involves treating ordered pairs of natural numbers as equivalence classes This approach builds upon the properties established for natural numbers, ensuring a coherent mathematical framework for understanding integers.
Definition 1.5.9 Let Z={(a, b)|a, b∈N} and define a relation∼onZ by (a, b)∼(a 0 , b 0 ) if and only ifa+b 0 =b+a 0
Proposition 1.5.10 The relation∼ defined onZ is an equivalence relation.
Definition 1.5.11 For(a, b)inZ, let[(a, b)]be the equivalence class of(a, b)under the relation∼. Define Z to be the set of equivalence classes of elements of Z, that is,
The definitions of Z and Q exhibit notable similarities, particularly in their operations The key distinction lies in the operation used: while Q employs multiplication in its definition of ∼ (i.e., ab 0 = ba 0), Z utilizes addition (i.e., a+b 0 = b+a 0) This difference is expected, as Q is derived from Z to introduce multiplicative inverses, whereas Z is constructed from N to achieve additive inverses.
Principle of Mathematical Induction
Consider the following questions before reading further.
1 What is “inductive” reasoning? Can it be used to prove mathematical statements?
2 Verify the following statements for as many cases as you can:
(a) 2 2 n + 1 is prime for every integer n>0.
(b) Ifn is any even integer greater than 4, thennis the sum of two odd primes.
3 Are the statements above true? How many cases would we need to check in order to prove they are true? How many cases to prove false?
A mathematical proof employs deductive reasoning, where new statements are logically derived from established definitions or theorems When the hypotheses are valid and the reasoning is sound, the conclusion drawn must also be true.
Inductive reasoning aims to draw general conclusions from specific observations, making it useful for forming conjectures, but it cannot conclusively prove mathematical statements After identifying specific cases and hypothesizing a conjecture, a deductive approach based on established results is essential for validation, as many statements may hold true in limited instances but fail in broader contexts Notably, Fermat numbers, defined as Fn = 2^(2^n) + 1 for integers n > 0, were first observed by Fermat in the 1600s, highlighting the importance of rigorous proof in mathematics.
F0 = 3, F1 = 5, F2= 17, F3 = 257, F4 = 65537, are prime, and conjectured thatF n is prime for all n>0, but was unable to prove this conjecture. The conjecture was not true, as
The Fermat number F5 equals 4,294,967,297, which can be factored into (641)(6,700,417), indicating it is not prime To date, no additional prime Fermat numbers have been discovered The complete prime factorizations for Fermat numbers Fn, where 5 ≤ n ≤ 11, are known.
5 6 n6 32 are known to be composite Thus the observation that Fn is prime for 0 6n 6 4 is rather misleading.
The Goldbach Conjecture posits that every even integer greater than 4 can be expressed as the sum of two odd prime numbers Extensive verification has confirmed this for all even integers up to 4 × 10^14, with unpublished claims suggesting it holds true for values up to 3 × 10^17 Despite the substantial evidence supporting the conjecture, it has yet to be proven, leaving open the possibility that there may exist an even integer greater than 4 that cannot be represented as the sum of two odd primes.
The Principle of Mathematical Induction is a powerful technique used to prove statements concerning natural numbers or integers Unlike inductive reasoning, this method relies on a specific property of natural numbers, which we will assume for our proofs.
Theorem 2.1.1 (Well-ordering Principle) If S is a non-empty set of natural numbers, then
The natural numbers possess a unique property where any non-empty subset that is bounded below contains a smallest element This characteristic does not extend to the positive real numbers or positive rational numbers However, it can be generalized to the set of integers, affirming that any non-empty set of integers with a lower bound also has a minimum element.
The Principle of Mathematical Induction states that if a set S of natural numbers contains the number 1 and is such that for any number k in S, the successor k + 1 is also included in S, then S encompasses all natural numbers This means that every natural number n belongs to the set S.
To demonstrate the theorem, we employ a proof by contradiction based on the Well-ordering Principle We begin by accepting the hypotheses of the theorem while assuming that the conclusion is incorrect This approach leads us to a contradiction, thereby validating the theorem.
Assuming conditions (i) and (ii) are satisfied, if the set S does not encompass all natural numbers, then the set T, which consists of natural numbers not included in S, must be a non-empty subset Consequently, according to the Well-ordering Principle, the set T will have a smallest element, denoted as m.
The smallest natural number not included in the set S is denoted as m Since 1 is part of S, it follows that m must be greater than 1, making k equal to m−1 a natural number Given that k is less than m, it must be included in S According to the second condition, k + 1, which equals m, should also be in S, leading to a contradiction Therefore, our initial assumption that S does not encompass all natural numbers must be incorrect, thereby proving the theorem.
We usually do not formally consider a set S of natural numbers when using mathematical induction The following is an alternative version.
Theorem 2.1.3 (Principle of Mathematical Induction (alternative version))LetP(n) be a statement about natural numbers If i P(1)is true, and ii whenever P(k) is true, P(k+ 1) is also true, thenP(n) is true for alln>1.
The equivalence of the two versions of the Principle of Mathematical Induction becomes evident when we define S as the set of all natural numbers n for which the statement P(n) holds true.
2 Statement (i) is called thebase stepof the induction The assumption in (ii) that k∈S, or
P(k) is true, is called theinductive hypothesis, and the proof of (ii) is called theinductive step.
Induction does not require starting at 1; instead, it can begin with any integer n0 in the set S If we establish that P(n0) is true, we can conclude that P(n) holds for all integers n greater than n0 This flexibility allows S to encompass a range of integers, including negative numbers and zero.
1 Use induction to show that 1 3 + 2 3 + 3 3 +ã ã ã+n 3 = n 2 (n+ 1) 2
Proof (i) We first show that the statement is true for n = 1 When n = 1, the formula becomes 1 3 = 1 2 (1+1) 4 2 = 4 4 = 1, and so the statement is true.
(ii) Next, we assume the statement is true for n=k; that is,
4 (∗) and show that this implies the statement is true forn=k+ 1; that is,
Hence if the statement is true for n=k, then the statement is true for n=k+ 1.
Since (i) and (ii) hold, the statement is true for all n>1 by the Principle of Mathematical
2 Use induction to show that 1
Proof (i) We first show that the statement is true for n = 1 When n = 1, the formula becomes 1ã2 1 = 1+1 1 , and so the statement is true.
(ii) Next, we assume the statement is true for n=k; that is,
3ã4 +ã ã ã+ 1 k(k+ 1) = k k+ 1 (∗) and show that this implies the statement is true forn=k+ 1; that is,
= k+ 1 k+ 2. Hence if the statement is true for n=k, then the statement is true for n=k+ 1.
Since (i) and (ii) hold, the statement is true for all n>1 by the Principle of Mathematical
3 Prove de Moivre’s Theorem (Corollary 1.2.11 with r= 1); i.e., show that
(cosθ+isinθ) n = cosnθ+isinnθ for all n>1.
Proof (i) We first show that the statement is true for n = 1 When n = 1, the statement becomes (cosθ+isinθ) 1 = cos 1θ+isin 1θ or equivalently, cosθ+isinθ = cosθ+isinθ, which is clearly true.
(ii) Next, we assume the statement is true for n=k; that is,
(cosθ+isinθ) k = coskθ+isinkθ (∗) and show that this implies the statement is true forn=k+ 1; that is,
(cosθ+isinθ) k+1 = (cosθ+isinθ) k ã(cosθ+isinθ)
= (coskθ+isinkθ)ã(cosθ+isinθ) by (∗),
= cos[(k+ 1)θ] +isin[(k+ 1)θ], and therefore (cosθ+isinθ) k+1 = cos[(k+ 1)θ] +isin[(k+ 1)θ] Hence if the statement is true for n=k, then the statement is true for n=k+ 1.
Since (i) and (ii) hold, the statement is true for all n>1 by the Principle of Mathematical
4 Use induction to show thatn!>2 n forn>4 (Recall thatn! =n(n−1)(n−2)ã ã ã3ã2ã1.)
Proof (i) We first show that the statement is true for n = 4 When n = 4, the inequality becomes 4! > 2 4 , and since 4! = 24 and 2 4 = 16, the statement is true (Note that the statement is actually false for n= 1,2,3.)
(ii) Next, we assume the statement is true for n=k>4; that is, k!>2 k (∗) and show that this implies the statement is true forn=k+ 1; that is,
= 2 k+1 , and therefore (k+ 1)!>2 k+1 Hence if the statement is true for n=k, then the statement is true forn=k+ 1.
Since (i) and (ii) hold, the statement is true for all n>4 by the Principle of Mathematical
Induction §2.1 Exercises Prove the following statements using the Principle of Mathematical Induction.
8 d dxx n =nx n−1 forn>1, assuming only d dxx= 1 and the product rule for differentiation.
9 Ifx is a positive real number, then (1 +x) n >1 +nxfor all natural numbersn>1.
10 Ifx and y are real numbers, then (xy) n =x n y n for all natural numbersn>1.
11 4 n −1 is divisible by 3 for alln>0.
12 A set withn elements has exactly 2 n subsets for n>0.
Divisibility of Integers
Definition 2.2.1 Let a and b be integers, with a 6= 0 We say a divides b, and write a | b, if b=na for some integer n If adoes not divideb, we write a-b.
In mathematics, we say that a divides b, denoted as a|b, which indicates that b is a multiple of a or that b is divisible by a It's essential to understand that this notation does not represent a numerical value or the fraction a/b.
If a divides b, then the rational number a/b is an integer, as b can be expressed as n times a, where n is an integer, leading to a/b = n When dealing with integers, multiplication is generally more effective than division, making the above definition preferable for proving divisibility results.
We will need to be able to use several important basic properties of divisibility These are listed among the (potential) properties below Consider the following problems before reading further.
In addressing class preparation problems, evaluate whether each statement is TRUE or FALSE For every true statement, provide a proof to support its validity Conversely, for each false statement, present an example of integers that demonstrate its inaccuracy.
5 Ifaand b are positive anda|b, thena6b.
More generally, ifaand bare any non-zero integers anda|b, then|a|6|b|.
Statements 1, 3, 8, 13, 14, and 16 above are FALSE, the other statements are true Statement 11 is just a special case of statement 12 (let d=a) We restate the remaining true properties in the following theorem.
Theorem 2.2.2 outlines key divisibility properties for integers a, b, and c Firstly, if a is not zero, then it divides zero (a|0), and it also divides itself (a|a) Additionally, every integer is divisible by 1 (1|a) If a and b are non-zero integers such that a divides b (a|b), then the absolute value of a does not divide the absolute value of b (|a| ≠ |b|) Furthermore, if a divides 1, then a must be either 1 or -1 (a = ±1) If a divides b and b divides a, it follows that b equals either a or its negative (b = ±a) The property also states that if a divides b and b divides c, then a divides c (a|c) Moreover, if a divides b and d divides c, then the product of a and d divides the product of b and c (ad|bc) Lastly, if a divides both b and c, then a divides their sum (a|b+c).
Proof Property (i) follows from the definition and the fact that 0 = 0ãa Properties (ii) and (iii) both follow from a= 1ãa.
For (iv), note that if a|b, then b=na for some integer n, and so |b|=|n| ã |a| Sincea and b are non-zero,nis also non-zero and|n|>1 becausenis an integer Thus|b|=|n| ã |a|>1ã |a|=|a| and |b|>|a|as claimed.
For (v), we have by (iv) that if a|1, then |a|61 and a6= 0 Sincea is an integer, this leaves only the possibility that |a| = 1 and so a = ±1 Similarly, if a |b and b | a, then |a| 6 |b| and
|b|6|a|, hence|a|=|b| Again, sinceaand bare integers, this implies b=±aand (vi) holds.
To demonstrate that if \( a \) divides \( b \) and \( b \) divides \( c \), then \( a \) also divides \( c \), we start by expressing \( b \) as \( b = na \) and \( c \) as \( c = mb \) for some integers \( m \) and \( n \) Substituting \( b \) into the equation for \( c \), we get \( c = m(na) = (mn)a \) Since the integers are closed under multiplication, \( mn \) is also an integer, leading us to conclude that \( c = (mn)a \), which by definition confirms that \( a \) divides \( c \).
If a divides b and d divides c, we can express b as na and c as md for some integers m and n Consequently, the product bc can be rewritten as (na)(md), which simplifies to (nm)(ad) due to the properties of multiplication in the integers Since the integers are closed under multiplication, nm is also an integer Therefore, the equation bc = (nm)(ad) demonstrates that ad divides bc, confirming the validity of statement (viii).
If \( a \) divides both \( b \) and \( c \), then there exist integers \( m \) and \( n \) such that \( b = ma \) and \( c = na \) Consequently, by the distributive law in \( \mathbb{Z} \), we have \( b + c = ma + na = (m+n)a \) Since \( \mathbb{Z} \) is closed under addition, \( m+n \) is also an integer Therefore, \( b+c = (m+n)a \) confirms that \( a \) divides \( b+c \), thus satisfying the definition and proving the statement.
The following theorem is very useful and important Its proof is nearly identical to the proof of Theorem 2.2.2 (ix) above and is left as an exercise (see Exercise 2.2.7).
Theorem 2.2.3 (Combination Theorem) If a|b and a|c, then a|bx+cy for all integers x and y.
Theorem 2.2.2 (ix) is really just a special case of the Combination Theorem, with x =y = 1.
According to the Combination Theorem, by substituting x = 1 and y = −1, we can conclude that if a divides both b and c, then a also divides the difference b − c This insight gives rise to a valuable corollary, which readers are encouraged to prove as an exercise (refer to Exercise 2.2.9).
In other words, if a divides a sum of two integers and divides one of the two integers, then a must divide the other.
Divisibility is unaffected by the sign of integers; if \( a \) divides \( b \), then \( -a \) also divides \( b \), and similarly, \( a \) divides \( -b \) and \( -a \) divides \( -b \) Consequently, divisors of an integer appear in pairs, consisting of a positive divisor and its negative counterpart Therefore, we typically focus on positive divisors of non-negative integers, while noting that most results we establish are applicable to negative integers as well.
2 Use Definition 2.2.1 to prove that ifa|b and cis an integer, thena|bc.
3 Use Definition 2.2.1 to prove that ifa|c and b|c, thenab|c 2
4 Find integersa,b, and csuch that a|bc, buta-b and a-c.
5 Find integersa,b, and csuch that a|c andb|c, butab-c.
6 Show that if c6= 0 and ac|bc, then a|b.
7 Prove the Combination Theorem (Theorem 2.2.3):
Ifa|band a|c, thena|bx+cy for all integers xand y.
8 Find integersa,b, and csuch that a|b+c, buta-b and a-c.
9 Prove that if a|r+sand a|r, thena|s.
10 Use induction to prove that 4|5 n −1 for all n>1.
11 Use induction to prove that 5|6 n −1 for all n>1.
12 Show that if ais a fixed positive integer, thena|(a+ 1) n −1 for alln>1.