# CHAPTER 5: MODULAR ARITHMETIC

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called modular arithmetic. This will lead us to the system Z m where m is a positive integer. Unlike other number systems, Z m is finite: it has m elements. We will define an addition and multiplication operation on Z m, and show that they have many of the same properties that one finds in other number systems. In fact, we will prove that Z m has enough arithmetic properties to be a commutative ring. We will determine which elements of Z m have multiplicative inverses. These will be called the units of Z m. The set of units forms an abelian group. When m = p is a prime, we will show further that every non-zero element of Z p has a multiplicative inverse. This will show that Z p is a field. We will sometimes write F p for Z p to indicate that we are indicating a field with p elements. We will consider fields in general. We will end with a discussion of negative exponents for units, and show that such exponentiation satisfies the usual properties. 2. Congruence modulo m Let m be a fixed positive integer. We call m the modulus. Definition 1. If a, b Z are such that Rem(a, m) = Rem(b, m), then we say that a and b are congruent modulo m and write a b mod m or a m b. Remark 1. Recall that whenever the word if is used to define a new concept, it really means, from a logical point of view, if and only if. The above gives an example of this. The definition stipulates that Rem(a, m) = Rem(b, m) a b mod m a m b. Remark 2. The abbreviation mod m is short for modulo m, which in Latin means using modulus m. Theorem 1. Fix m. Then m is an equivalence relation on Z. Exercise 1. Prove the above. Date: March 21,

2 2 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN A very common use of congruences is to assert a r mod m where r is the remainder Rem(a, m). This is supported in the following theorem. For example, one would commonly say 7 2 mod 5. However, this is not the only valid use of congruences. One could also say that 7 17 mod 5, or 7 7 mod 5, or even 7 13 mod 5. Theorem 2. If a, m Z with m > 0 then a Rem(a, m) mod m. Lemma 3. If 0 c < m then Rem(c, m) = c. Proof of Lemma. Since c = 0 m+c and 0 c < m, the Quotient-Remainder Theorem (Ch. 4) forces c to be the remainder Rem(c, m). Proof of Theorem. Let r = Rem(a, m). Since 0 r < m we have r = Rem(r, m) by the above lemma. Thus Rem(a, m) = Rem(r, m). By definition of congruence, a m r. Since congruence is reflexive, an equality can always be converted to a congruence. The following says that for small integers, a congruence can be converted to an equality. Theorem 4. Suppose a, b, m N. Suppose also that 0 a < m and 0 b < m. Then a m b a = b. Proof. Assume a m b. Thus Rem(a, m) = Rem(b, m) by Definition 1. By Lemma 3, Rem(a, m) = a and Rem(b, m) = b. Thus a = b. The other direction follows from the fact that m is reflexive (congruence is an equivalence relation). Corollary 5. Suppose a, m Z with m > 0. Then there is exactly one b {0,..., m 1} such that a b mod m. Exercise 2. Justify the above corollary. Hint: use Theorems 2 and 4. The following is another characterization of congruence. It is often chosen as the definition of congruence in number theory books. Theorem 6. Let a, b Z. Let m be a positive integer. Then a m b m (a b). Proof. Suppose a b modulo m. Then a and b have the same remainder (but perhaps different quotients). So we have a = qm + r and b = q m + r for some q, q Z. Thus This implies that m (a b). a b = (qm + r) (q m + r) = (q q )m.

3 CHAPTER 5: MODULAR ARITHMETIC 3 Suppose m (a b). So a b = cm for some c Z. Thus a = b + cm. Apply the Quotient Remainder Theorem (Ch. 4) to b giving us b = qm + r with 0 r < m. Thus a = b + cm = (qm + r) + cm = (q + c)m + r. Since 0 r < m, this implies that the quotient and remainder for a divided by m are q + c and r respectively. In particular, a and b have the same remainder r. Thus a b mod m. Exercise 3. Use the above theorem to show that a b modulo 1 is always true (for all a, b Z). 3. Modular Arithmetic The first rule of modular arithmetic allows you to add a constant to both sides of a congruence. Theorem 7. Let a, b, c, m Z with m > 0. a b mod m = a + c b + c mod m. Proof. By Theorem 6, m (a b). But (a + c) (b + c) = a + c + ( b) + ( c) = a b. So m ( (a + c) (b + c) ). The conclusion follows from Theorem 6. Using the above twice gives the following Theorem 8. Let a, b, a, b, m Z with m > 0. a m a and b m b = a + b m a + b. Proof. By Theorem 7, we can add a to both sides of b m b : a + b a + b mod m By Theorem 7 again, we can add b to both sides of a m a : Now use transitivity. a + b a + b mod m Informal Exercise 4. Illustrate Theorems 8 and 10 with several examples. Not only can you add constants to congruences, you can multiply constants to congruences. Theorem 9. Let a, b, c, m Z with m > 0. a m b = ac m bc. Exercise 5. Use Theorem 6 to prove the above. Theorem 10. Let a, b, a, b, m Z with m > 0. a m a and b m b = ab m a b. Exercise 6. Prove the above. Hint: see Theorem 8 for ideas.

4 4 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN Exercise 7. Let a, b, m, n Z where n 0 and m > 0. Prove, by induction, that if a m b, then a n m b n. Informal Exercise 8. Use congruences to show that adding 52 hours to a clock is the same as adding 4 to a clock. Informal Exercise 9. Ignoring the effect of leap years, consecutive birthdays differ by 365 days. Suppose this is so, where the first of the consecutive birthdays occurs on a Friday. Use congruences to show that the second of the consecutive birthdays must be on a Saturday. Exercise 10. Suppose that a, m Z with m > 0. Show that you can freely add or subtract m in a congruence: a a + m a m mod m. For instance, if you are given a = 6 and m = 8, you could add 8 and conclude 6 2 modulo 8. Exercise 11. Suppose that a, k, m Z with m > 0. Suppose d m where d > 0. Show that if a m b then a d b. Hint: use Theorem Application to finding remainders In this section we give quick ways to find remainders when we divide by various small integers. The technique is based on writing a number in base 10, but it generalizes easily to other bases. Exercise 12. Show that and for all n N. 10 n 1 n 1 mod 9 10 n 1 mod 3. Informal Exercise 13. Let s be the sum of the digits of a number n N written in base 10. Show that and n s mod 9 n s mod 3. Informal Exercise 14. Use the sum of the digits method to find Rem(3783, 9) and Rem(12345, 3). What is the closest number to 45, 991 that is divisible by 9? Informal Exercise 15. Derive your own procedure for finding Rem(n, 7) where n N has up to three digits. Hint: work with 10 k modulo 7 for k = 0, 1, 2. Use this procedure to find Rem(249, 7) and Rem(723, 7).

5 CHAPTER 5: MODULAR ARITHMETIC 5 Exercise 16. Prove (without induction) that B n 0 mod B for all positive integers B and n. Hint: write n as m + 1. Informal Exercise 17. Show that if n is a natural number, and m is the last digit of n written in base B, then n m mod B. What is a quick way to find the remainder of n N when you divide by 10? what is Rem( , 10)? What is the remainder of [ ] 16 when you divide by 16? Exercise 18. Show that 10 n ( 1) n mod 11. Hint: use Exercise 7. Informal Exercise 19. Let n N, and write n in base 10 as k n = d i 10 i. Show that n i=0 k ( 1) i d i mod 11. i=0 Use this to find the remainder of 156, 347 when dividing by 11. Informal Exercise 20. Observe that Show that to find Rem(n, 4), you just need to replace n N with the number formed from the last two digits of n. Show that if d 1 and d 0 are the last two digits, then n 2d 1 +d 0 mod 4. Informal Exercise 21. How many digits do you need to consider when calculating Rem(n, 8)? Explain why. Informal Exercise 22. How many digits do you need to consider when calculating Rem(n, 5) or Rem(n, 2)? Explain why. Informal Exercise 23. Find the remainder of 337 when dividing by 2, 3, 4, 5, 7, 9, 10, and 11 using the techniques in this section. Informal Exercise 24. Give short cuts for finding the remainder of the number [ ] 8 when dividing by 7, 8 or Even and Odd By Corollary 5, if a Z then exactly one of the following can occur: a 0 mod 2 or a 1 mod 2. Definition 2. If a 0 mod 2 then a is called an even integer. If a 1 mod 2 then a is called an odd integers. Theorem 11. An even integer plus an even integer is even. An odd integer plus an odd integer is even. An even integer plus an odd integer is odd. An even integer times an even integer is even. An odd integer times an odd integer is odd. An even integer times an odd integer is even.

6 6 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN Proof. We consider the case of two odd integers. The other cases are left to the reader. If a, b Z are odd then by Theorem 8 a + b mod 2, so a + b is even. Also, by Theorem 10, so ab is odd. ab mod 2 Exercise 25. Prove the other cases in the above theorem. 6. The finite ring Z m. Since m is an equivalence relation, we can consider the equivalence classes under this relation. The set of equivalence classes Z m = {[a] m a Z} will be shown to be a ring (after we define a suitable + and ). At first one might think that this ring Z m is infinite since, for each a Z, we can form the equivalence class [a]. However, due to the properties of m, the number of elements in Z m is just m. Definition 3. Fix a positive integer m, and consider the equivalence relation m defined above. If a Z, then let [a] denote the equivalence class containing a under this relation. In other words, [a] = {x Z x m a}. Define Z m = { [a] a Z }. We call Z m the set of integers modulo m. We often write a for [a]. We also write [a] m when we want to be clear about the modulus. Informal Exercise 26. Describe the set [5] if m = 1. Show that [5] = [ 1] in this case. Informal Exercise 27. Describe the set [5] if m = 2. Show that [5] consists of the odd integers. Informal Exercise 28. Describe the set [5] if m = 3. Show that [5] = [2]. Informal Exercise 29. Describe the sets [0], [1], and [2] if m = 3. Theorem 12. Let m be a positive integer and a, b Z. Then a m b [a] = [b]. Proof. This is a general fact about equivalence classes (from set theory). Corollary 13. Let m be a positive integer and a, b Z. Then [a] m = [b] m Rem(a, m) = Rem(b, m) m (a b) Proof. These conditions are all equivalent to a m b by earlier results. The following shows that when working in Z m we can always limit ourselves to [b] with 0 b < m.

7 CHAPTER 5: MODULAR ARITHMETIC 7 Theorem 14. Suppose [a] Z m where m is a positive integer. Then there is exactly one b {0,..., m 1} such that [a] = [b]. Proof. Combine Corollary 5 with Theorem 12. Corollary 15. Let m be a positive integer. bijection f : {0,..., m 1} Z m. The rule x [x] defines a Proof. We show that f is injective and surjective. Suppose f(x) = f(y) where x, y {0,..., m 1}. Then [x] = [y]. So x y modulo m by Theorem 12. Therefore, x = y by Theorem 4 Now we show f is surjective. Let [a] Z m be an arbitrary element. We must find something in the domain that maps to [a]. By Theorem 14 there is a unique b {0,..., m 1} such that [a] = [b]. Thus f(b) = [a]. So f is surjective. Corollary 16. Let m be a positive integer. The set Z m has m elements. Proof. The above corollary gives a bijection {0,..., m 1} Z m. However, {0,..., m 1} has m elements (Chapter 3). Thus Z m has m elements (Chapter 2). Now we consider addition and multiplication in Z m. Informal Exercise 30. Show that [3] + [7] = [1] in Z 9 using the following definition of addition (and Theorem 12). Hint: first show [3] + [7] = [10]. Definition 4. Let m be a positive integer. Suppose [a], [b] Z m. Then [a] + [b] is defined to be [a + b] where addition inside [ ] is as in Chapter 3. This defines a binary operation + : Z m Z m Z m. Since this definition involves equivalence classes, and since there are several ways to denote the same class, we need to show that the definition of addition is well-defined. This is done in the following lemma. Lemma 17. Let m be a positive integer. If [a] = [a ] and [b] = [b ] in Z m then [a] + [b] = [a ] + [b ]. Proof. By Theorem 12, a m a and b m b. Then a + b m a + b by Theorem 8. So, by Theorem 12, [a + b] = [a + b ]. Many of the properties of addition for Z also apply to Z m. For example, we prove the commutative law. Theorem 18. Let [a], [b] Z m where m is a positive integer. Then [a] + [b] = [b] + [a].

8 8 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN Proof. Observe [a] + [b] = [a + b] (Def. of addition in Z m ) = [b + a] (Comm. Law for + in Z: Ch. 3) = [b] + [a] (Def. of addition in Z m ). Exercise 31. Prove the associative law of addition for Z m. Now we turn our attention to multiplication. Informal Exercise 32. Show that [3] [7] = [3] in Z 9 using the following definition of multiplication (and Theorem 12 or Corollary 13). Definition 5. Let m be a positive integer. Suppose [a], [b] Z m. Then [a] [b] is defined to be [ab], where multiplication inside [ ] is as in Chapter 3. This defines a binary operation : Z m Z m Z m. As with the definition of addition on Z m, we need to show that this definition is well-defined. This is done in the following lemma. Lemma 19. Let m be a positive integer. If [a] = [a ] and [b] = [b ] in Z m then [a] [b] = [a ] [b ]. Proof. Combine Theorem 12 with Theorem 10. Exercise 33. Let [a], [b] Z m where m is a positive integer. Then show [a] [b] = [b] [a]. Now we come to the key theorem of the section. Theorem 20. Let m be a positive integer. Then Z m is a commutative ring with additive identity [0] and multiplicative identity [1]. The additive inverse of [a] is [ a]. Exercise 34. Prove the above theorem. Hint: some pieces have been done in earlier theorems and exercises. For example, if you want to show [0] is the additive identity, you only need to show [a]+[0] = [a] since [0]+[a] = [a]+[0] from an earlier theorem. Exercise 35. Show that the additive inverse of [a] Z m is [m a]. Show that 2 is the additive inverse of 3 in Z 5. Remark 3. Since the additive inverse of [a] is [ a] you can write [a] = [ a] where the first signifies inverse in Z m, and the second signifies inverse in Z.

9 CHAPTER 5: MODULAR ARITHMETIC 9 Remark 4. Using the notation a for [a] we can write the above definitions and results as a + b = a + b ab = a b a = a = m a. The additive identity is 0. The multiplicative identity is 1. Remark 5. In any ring, 0 customarily denotes the additive identity. So we can write 0 = [0] = 0 where the left 0 is the identity in Z m and the middle and right 0 is the identity in Z. Likewise, 1 can denote the multiplicative identity in any ring: 1 = [1] = 1 where the left 1 is the identity in Z m and the middle and right 1 is the identity in Z. Informal Exercise 36. Make addition and multiplication tables for Z m for m = 1, 2, 3, 4, 5, 6. Your answers should be in the form a where 0 a < m, but to save time you do not have to write bars over the answer: if you write 3, everyone will know that you actually mean 3. Hint: use the commutative law to save time. Exercise 37. Suppose m is a positive integer, and that m = ab where a and b are positive and less than m (in other words, suppose that m is composite). Show that Z m is not an integral domain. Later, we will show that Z p is an integral domain if p is a prime number. 7. Units in a ring Every element in a ring has an additive inverse, but only some elements have multiplicative inverses. Any element with a multiplicative inverse is called a unit. Recall that we assume all rings have a multiplicative identity. Definition 6. Let R be a ring with multiplicative identity 1. If a, b R are such that ab = ba = 1 then we say that a and b are multiplicative inverses. We write b = a 1 and a = b 1 to indicate that b is the inverse of a and a is the inverse of b. If R is commutative, we only need to check ab = 1. An element a R is called a unit if it has an inverse. The set of units is written R : R def = {u R u is a unit}. Observe that R is a subset of R. Warning. The above use of the superscript 1 is different than its use in iteration. Informal Exercise 38. What are the units of Z? In other words, what is Z?

10 10 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN Informal Exercise 39. Make a multiplication table for Z 9. Use it to find Z 9. List all the inverses of all the units. To save time, you do not have to use bars or brackets in the tables. Hint: remember how to find remainders modulo 9, and use the fact that multiplication is commutative. Informal Exercise 40. Are Z and Z 9 closed under addition? Exercise 41. Let a be an element of a ring R. Show that if a is a unit, then its multiplicative inverse is unique. Exercise 42. Show that 1 and 1 are units in any ring R. Show that if R is a ring with 0 1 then 0 is not a unit. (Most rings have 0 1. The trivial ring is an exception: it has only one element so all elements are equal). Exercise 43. Show that if u is a unit in a ring R then so is u 1, and that ( u 1 ) 1 = u. Here is the main theorem of this section. It tells us which elements of Z m are units. Theorem 21. Let a Z m where m is a positive integer. Then a is a unit if and only if a and m are relatively prime. Proof. Suppose that a is a unit. This means that there is a b Z such that ab 1 mod m. In other words, m divides ab 1. In particular, ab 1 = mc for some c Z. So ab cm = 1. Any common divisor of a and m must divide the linear combination ab cm = 1 (Section 6 of Chapter 4). Thus the only positive common divisor of a and m is 1. This means that a and m are relatively prime. Now suppose that a and m are relatively prime. Consider the function f : Z m Z m defined by the rule x ax. We first show that f is injective. Suppose f ( b ) = f ( c ). Then a b = a c. In other words, ab m ac. This means that m divides ab ac = a(b c). Clearly a also divides a(b c). Since a and m are relatively prime, we have am a(b c) (Section 7, Ch. 4). Thus m (b c). This means that b m c, so b = c. We conclude that f is injective. Since f is injective, and maps a finite set to itself, it must also be surjective (Chapter 2). Thus there is an element b such that f ( b ) = 1. By definition of f, we have a b = 1. So a is a unit. Informal Exercise 44. Use the above theorem to identify Z m for m = 1 to m = 12. Make multiplication tables for Z m for m = 1, 2, 3, 4, 5, 7, 8, 10, 12. (To save time, you do not have to use bars or brackets in the tables.) As the tables from the above exercise show, the set Z m is closed under multiplication: Lemma 22. If a, b R are units in a ring R, then ab is a unit. Furthermore, (ab) 1 = b 1 a 1. If R is commutative then (ab) 1 = a 1 b 1 Proof. (sketch) First show b 1 a 1 is the inverse of ab. So the inverse of ab exists. In other words, ab is a unit.

11 CHAPTER 5: MODULAR ARITHMETIC 11 This lemma implies that for R multiplication gives a binary operation R R R. Multiplication is associative since R is a ring, and R is a subset of R. Since 1 is a unit in any ring, there is an identity for this operation. Furthermore, if u R then clearly u 1 R (see Exercise 43). Thus every element of R has an inverse in R. Thus we get the following: Theorem 23. If R is a ring, then the units R form a group under multiplication. If R is a commutative ring, then R is an abelian group. 8. The field F p In this section we will see that every non-zero element of Z p is a unit when p is a prime. Commutative rings with this property are very important, and are called fields. Because Z p is a field we sometimes write F p for Z p. Since every field is an integral domain, Z p is also an integral domain. We saw above that Z m is not an integral domain if m is composite. Theorem 24. If p is a prime, then every non-zero element of Z p is a unit. Proof. Let a Z p be non-zero. Observe that a is not a multiple of p (otherwise a p 0, a contradiction). Since p a and since p is a prime, a and p are relatively prime (Chapter 4). By Theorem 21, a is a unit. Definition 7. A field is a commutative ring F such that (i) 0 1, and (ii) every non-zero element is a unit. Remark 6. The conditions (i) and (ii) in the above definition can be folded into one condition: x F is a unit if and only if x 0. Remark 7. Fields are extremely important in mathematics. The number systems Q, R, C are all fields. In a field you can make use of all four basic algebraic operations +,,,, with the only restriction that you cannot divide by zero since zero is not a unit. Theorem 25. If p is a prime then Z p is a field. Proof. Observe that (i) 0 1 since p > 1. In addition, (ii) every non-zero element of Z p is a unit by Theorem 24. Definition 8. If p is a prime, then F p as another name for Z p. The field F p is an example of a finite field. Remark 8. Every field is an integral domain, but not all integral domains are fields. We leave the verification of these facts to the reader.

12 12 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 9. Exponentiation In Chapter 1 we used the idea of repeated multiplication to define exponentiation. This idea can be extended to any ring R. For units, we can extend this idea and define exponentiation for negative exponents. Exponentiation in a general ring satisfies familiar rules such as a m+n = a m a n. We have seen similar rules in the context of iteration. In fact, we will use the rules for iteration to prove the analogous rules for exponentiation. In this section we will consider exponentiation for general elements in a ring. In the next section we will consider exponentiation of units. Definition 9 (Exponentiation in rings). Suppose R is a ring and a R. Let M a : R R be defined by the rule x xa. If n N then a n def = M n a (1). Our strategy for studying exponentiation is to find identities between M a. This approach is very similar to that used in Chapter 3 when we studied the translation function A a. In what follows, let R be a ring. Lemma 26. Let a R. Then M 0 a = id and M 1 a = M a. Proof. These are basic properties of iteration (Chapter 1). Corollary 27. Let a R. Then a 0 = 1 and a 1 = a. Proof. Apply M 0 a and M 1 a to 1. The above lemma gives the result. Lemma 28. Let a, b R. Then M a M b = M ba. Proof. For any x R, M a M b (x) = M a ( Mb (x) ) = M a (xb) = (xb)a = x(ba) = M ba (x). The conclusion follows. Lemma 29. Let a, b R. Suppose ab = ba (which holds, for example, if R is a commutative ring). Then M a and M b commute. Proof. This follows from the above lemma since M ab = M ba. Lemma 30. Consider the iteration Ma n where a R and n N. Then there is an element c R such that Ma n = M c. Proof. Fix a R. Let S be the set of n N such that the conclusion holds. Observe that 0 S a since M 0 a = id = M 1. Suppose that n S a. This means M n a = M b for some b R. Thus M n+1 a = M a M n a = M a M b = M ba. (Lemma 28) Thus n + 1 S a. By induction, S a = N. The result follows. The following is very useful for deriving identities. Lemma 31. If n N and a R then M n a = M a n.

13 CHAPTER 5: MODULAR ARITHMETIC 13 Proof. By Lemma 30, M n a = M c for some c R. Apply M n a and M c to 1: a n def = M n a (1) = M c (1) = 1 c = c. Thus c = a n. But, M n a = M c. So M n a = M a n Lemma 32. Let a, b R. If M a = M b then a = b. Proof. Suppose M a = M b. Then M a (1) = M b (1). But M a (1) = 1 a = a and M b (1) = 1 b = b We know from Chapter 2 that iteration satisfies the law f m+n = f m f n for m, n N. We use this fact in the following. Theorem 33. Let a R where R is a ring, and let m, n N. Then Proof. Observe that a m+n = a m a n. M a m+n = Ma m+n (Lemma 31) = Ma n+m (Comm. Law, Ch. 1) = Ma n Ma m (f m+n = f m f n, Ch. 2) = M a n M a m (Lemma 31) = M a m an (Lemma 28) Thus a m+n = a m a n by Lemma 32. Theorem 34. Consider 0 R. If n is a positive integer then 0 n = 0. Proof. Write n as m + 1. Then, using the previous theorem, 0 n = 0 m+1 = 0 m 0 1 = 0 m 0 = 0. We know from Chapter 2 that iteration of functions satisfies the law (f m ) n = f mn for all m, n N. We use this fact in the following. Theorem 35. Let a R and m, n N. Then (a m ) n = a mn. Proof. Start with M a m = Ma m (Lemma 31). Then (M a m) n = ( Ma m ) n (Iterate same funct.) = Ma mn (Ch. 2: (f m ) n = f mn ) Apply both sides to 1: (a m ) n def = (M a m) n (1) = M mn a (1) def = a mn.

14 14 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN We know from Chapter 3 that if f : S S and g : S S commute (for some set S) then we have (f g) n = f n g n for all n N. We use this fact in the following. Theorem 36. Let a, b R and n N. Suppose ab = ba (which is true, for example, if R is a commutative ring). Then (ab) n = a n b n. Proof. By Lemma 29, M a and M b commute. So ( Mab ) n = ( M b M a ) n (Lemma 28) Apply both sides to 1: = M n b M n a (Ch. 3: (f g) n = f n g n ) = M b n M a n (Lemma 31) = M a n bn (Lemma 28) (ab) n def = ( M ab ) n(1) = Ma n b n(1) = an b n. Exponentiation can also be thought of in terms of finite products: Theorem 37. Let a R where R is a ring. If n is a positive integer then n a n = where (a i ) i=1,...,n is the constant sequence defined by the rule a i = a for all i {1,..., n}. Proof. (Sketch) This can be proved by induction. The details are left to the reader. Remark 9. If a Z then the definition of a n given in this section agrees with the definition of Chapter 1 whenever a 0. To see this, compare the two definitions and observe that µ a (x) = M a (x) for all x 0. By induction, it follows that µ n a = M n a for all n N. In particular, µ n a(1) = M n a (1). i=1 a i 10. Exponentiation of Units If a is a unit in a ring R, then we can define a u for negative u Z. In what follows let R be a ring. Lemma 38. If a R is a unit then M a has an inverse function and (M a ) 1 = M a 1. Proof. We know M a M a 1 = M 1 by Lemma 28. But M 1 = id by definition of M a. Likewise M a 1 M a = id. The conclusion follows. Corollary 39. If a R is a unit then M a is a bijection, and M u a is defined for all u Z.

15 CHAPTER 5: MODULAR ARITHMETIC 15 Now we can define a u for all u Z, even for negative u. Definition 10. Suppose a R is a unit and u Z. Then a u def = M u a (1). Recall from Chapter 3 that if f : S S is the identity function, then f u = id for all u Z. This is used below. Theorem 40. Let 1 be the multiplicative identity of a ring R. If u Z then 1 u = 1. Proof. Observe that M 1 is the identity. Then M1 u Thus 1 u = M1 u (1) = id(1) = 1. Informal Exercise 45. Find a 2 for all non-zero a F 7. compare to a 4? The following generalizes Lemma 30. is the identity (Ch. 3). How do these Lemma 41. Consider the iteration M u a where a R and u Z. Then there is an element c R such that M u a = M c. Proof. If u 0 this follows from Lemma 41. Now assume u < 0, so u = n for some n N. Then Ma u = Ma n (Rewrite) = ( Ma 1 ) n (Ch. 3: f uv = (f u ) v ) = ( n M a 1) (Lemma 38) Thus M u a = M c where c = ( a 1) n. = M n a 1 (Rewrite) = M (a 1 ) n (Lemma 31) The following generalizes Lemma 31. Lemma 42. Let a R be a unit. If u Z then M u a = M a u. Proof. (Sketch) This is similar to the proof of Lemma 31, but adapted from natural numbers to integers. The following generalizes Theorem 33. Theorem 43. Suppose a R is a unit and that u, v Z. Then a u+v = a u a v. Proof. This is similar to the proof of Theorem 33, but adapted from natural numbers to integers. The following generalizes Theorem 35. Theorem 44. Suppose a R is a unit and that u, v Z. Then (a u ) v = a uv.

16 16 LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN Proof. This is similar to the proof of Theorem 35, but adapted from natural numbers to integers. The following generalizes Theorem 36. Theorem 45. Let a, b R be units, and let u Z. If ab = ba then (ab) u = a u b u. Proof. This is similar to the proof of Theorem 36, but adapted from natural numbers to integers. Theorem 46. If a R is a unit and u Z then a u is also a unit. Proof. By Theorem 43, a u a u = a 0 and a u a u = a 0. By Corollary 27, a 0 = 1. The result follows. Appendix: miscellaneous comments The theory of exponentiation can be developed for groups as well. There is both an additive and a multiplicative version.

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 3. Applications of Number Theory

3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGraw-Hill The McGraw-Hill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

### Finite Fields and Error-Correcting Codes

Lecture Notes in Mathematics Finite Fields and Error-Correcting Codes Karl-Gustav Andersson (Lund University) (version 1.013-16 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents

### 5.1 Commutative rings; Integral Domains

5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Lectures on Number Theory. Lars-Åke Lindahl

Lectures on Number Theory Lars-Åke Lindahl 2002 Contents 1 Divisibility 1 2 Prime Numbers 7 3 The Linear Diophantine Equation ax+by=c 12 4 Congruences 15 5 Linear Congruences 19 6 The Chinese Remainder

### MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### Number Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures

Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### COMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:

COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative

### 26 Ideals and Quotient Rings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

### Stupid Divisibility Tricks

Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### Discrete Mathematics, Chapter 4: Number Theory and Cryptography

Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### 4.4 Clock Arithmetic and Modular Systems

4.4 Clock Arithmetic and Modular Systems A mathematical system has 3 major properies. 1. It is a set of elements 2. It has one or more operations to combine these elements (ie. Multiplication, addition)

### 12 Greatest Common Divisors. The Euclidean Algorithm

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography

V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 3 Congruence Congruences are an important and useful tool for the study of divisibility. As we shall see, they are also critical

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### Notes on Algebraic Structures. Peter J. Cameron

Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Let s just do some examples to get the feel of congruence arithmetic.

Basic Congruence Arithmetic Let s just do some examples to get the feel of congruence arithmetic. Arithmetic Mod 7 Just write the multiplication table. 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### a = bq + r where 0 r < b.

Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental

### Fractions and Decimals

Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

### Chapter 3. if 2 a i then location: = i. Page 40

Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### Math 223 Abstract Algebra Lecture Notes

Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### Computer Algebra for Computer Engineers

p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set

FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### 4.5 Finite Mathematical Systems

4.5 Finite Mathematical Systems We will be looking at operations on finite sets of numbers. We will denote the operation by a generic symbol, like *. What is an operation? An operation is a rule for combining

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system

CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Chapter 10. Abstract algebra

Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### MATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003

MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is