Integers and Number Theory

Integers and Number Theory Summary

I. Basic concepts of integers

A. Definitions:

1. The set of Integers is the union of the set of whole numbers, W, with the negative integers, I = W I = {‑3,-2,-1, 0, 1, 2, 3, …} where the negative integers, I are defined as follows:

Negative Integers: I is { 1,2, 3, …, n,… } where n is that unique number defined for each integer n I such that n+( n) = 0 = ( n)+n. That is the set of negative numbers, n, are those numbers that enable every whole number, n, to return to zero (home base).

2. Def: Absolute Value: The absolute value of an integer x, denoted |x|, is the distance between x and zero (i.e. the number of steps needed to return to 0 on the number line). No direction is associated with the concept of absolute value.

For any two integers, a, b and c, whether positive, negative or zero:

Closure ±,* a±b exists and is a unique integer.
a*b exists and is a unique integer.

Addition: a+b = c such that c = ±|a+b|, depending whether (a+b)≥0 or (a+b)<0

Multiplication: a*b = c where c = ±(|a|*|b| from repeated (fast) addition of |b| for |a| many summands and the sign of |a|*|b|. The sign of c depends on the signs of a and b and follows the rules for products of signed integers below.

Subtraction: a‑b = n if and only if, b+n = a, (subtraction undoes the result of addition, so is referred to as the inverse operation of addition)

Division: (Inverse operation of multiplication, undoes multiplication). For any integers a and b≠0,
a b is the unique integer c, if it exists, such that a = bc.

Discussion of the Zero Property of Division:

Case i.) Dividing 0 by a nonzero number: You can divide zero by any nonzero whole number because there exists an answer, namely zero itself, to the division sentence, 0 b=[?] and the answer is unique. That is there is one and only one answer to the question, “How many non-zero b’s can you add up to get the result of having zero-many of any quantity?” For integers, a=0, b≠0 there exists a solution to the equation 0 b=c namely c=0 such that 0*b=0=b*0. Further, zero is the one and only such solution by the zero property of multiplication.

However, you cannot divide any integer, zero or nonzero, by zero.

Case ii.) Dividing an Integer by 0: There are two cases to consider, a≠0 and a=0.

a. Non-zero divided by zero: You cannot divide a non-zero number by zero because there does not exist a number to satisfy the definition of the division equation. For a≠0, a 0=[?] iff [?] *0=a≠0 but[?] *0=0 by the zero property of multiplication.. No matter how many times you add zero to itself, the sum is always zero.

b. Zero divided by zero: If a=0, then there is no unique answer to the expression 0 0=[?]. One can put any integer x in the box because x *0=0 by the zero property of multiplication. So there does not exist a unique solution.

Summary:

Definitions multiplication and div:

2. a b = c if and only if (there exists a unique integer c such that) c*b=a provided b≠0.

0 b = c if b ≠ 0 because there exists an integer, namely c=0 that satisfies the remaining part of the definition of division, c*b=0*b=0.

a 0 is undefined for a≠0 because no integer c exists to satisfy the definition equation, c*0=a .

0 0 is undefined, because there is no unique integer solution such that c*0=0.

The relationships among the 4 operations‑addition, subtraction, multiplication, and division‑are shown in the Figure below

one

many of

The mathematical order for performing the four basic operations, +,-,x, , is as follows:

Parentheses, Exponents, Multiplication then Division from left to right, Addition then Subtraction from left to right.

Remember:

PEMDAS: Please excuse my dear aunt Sally.

Be able to use mental addition, subtraction, multiplication, and division as given in the book and be able to estimate using all four operations.

Multiplication algorithms: provide Visual representation, heavily use Place‑value, can be performed using both nontraditional and traditional algorithms

Identity: +, *,     For all integers a,
a.     There is a unique integer 0 such that a+0 = a = 0+a                  Id =0.
b.     There is a unique integer 1 such that a*1 = a = 1*a                 Id =1.

Inverse: +, For every integer a,
a. There is an integer ( a) such that a+( a) = 0 = ( a)+a.

B. Properties of the above definitions: For all integers a, b and c whether positive, negative or zero:

1. Properties of Additive Inverses:

a. ( a) = a (the additive inverse of of a negative integer is the integer itself)

b. (a+b) = a+ b (the inverse of a sum of integers = sum of their inverses)

c. a-b = a+ b (to subtract, add the inverse of the number to be subtracted)

d. (a-b) = a+b (the inverse of a difference of two integers = sum of the inverse of the subtrahend (the number that is subtracted) and the minuend (the first integer)

2. Properties of Multiplicative Inverses:

a. (a*b) = ( a)*b = a*( b) (inverse of a product = product of any one factor and the inverse of the other factor)

b. a = ( 1)*a (the inverse of an integer = negative one times the integer )

c. ( a)*( b) = ab (the product of the inverses of two integers = product of the integers)

d . ( a)(b–c) = ( 1*a)*(b–c) = ( 1*a)*(b+ c)=…= ab+ ac . This says that negative integers distribute through subtraction)

e. a‑(b‑c) = a‑b+c (consequence of above)

3. Order of operations (Please Excuse My Dear Aunt Sally, PEMDAS): All parentheses or exponents are done first in any arithmetic expression. Then multiplications and divisions are done first in the order of their appearance from left to right. "Aunt Sally" follows after these operations are completed orlastly, do additions and subtractions in the order of their appearance from left to right.

II. The system of integers

A. The set of integers, I = (…, 3, 2, 1, 0, 1, 2, 3....) along with the operations of addition and multiplication, satisfy the following properties:

Property	+	x	-
Closure	Yes	Yes	Yes	No
Commutative	Yes	Yes	No	No
Associative	Yes	Yes	No	No
Identity	Yes, 0	Yes, 1	No, (a-0) = a ≠ (0-a)	No,(a 1) = a ≠ (1 a)
Inverse	Yes	No	n/a there is no Identity	n/a there is no Identity
Distributive property of multiplication over addition (and subtraction because a-b = a+ b) a(b ±c) = ab±ac (the lunch box rule for mult over add of whole numbers)

B. Zero multiplication property of integers:

a*0 = 0 = 0*a, for any integer a. (Anything multiplied by zero or vice versa is zero.)

C. Addition, subtraction property of equality:

For any integers a, b, and c, if a=b, then a±c = b±c. (Equals added to (subtracted from) equals are equal)

D. Multiplication property of equality:

For any integers a, b, and c, if a = b, then ac = bc. (Equals multiplied by equals are equal).

E. Substitution property:

Any number substituted for its equal its results are equal. (Equals substituted for equals are equal).

For example, substitute ( a) for a, of a* b for ab, etc.

F. Cancellation properties of equality:

1. For any integers a, b, and c, if a±c = b±c, then a = b. (Equals subtracted from or added to equals are equal)

2. For any integers a, b, and c, if c ≠ 0 and ac = bc, then a = b. (Equals divided by equals are equal)

G. For all integers a, b, and c:

1. (a+b)(c+d) = ac + ad + bc + bd (FOIL – result of double distributivity,1st (c+d) right distribute over (a+b), then a and b separately left distribute over (c+d))

2. (a±b) =a ±2ab+b (follows fm foil to (a±b)(c±d) where c=a and b=d.

3. (a+b)(a-b) = a -b (difference‑of‑squares formula)

III. Divisibility

A. Definition: If a and b are any integers, then b divides a, denoted by b|a, if and only if there is an integer c such that a = cb. If b does not divide a, we write, b a

B. The following are basic divisibility theorems for integers a, b, and d:

1. If d|a and k is any integer, then d|ka. (If d divides an integer a, then d divides any multiple of a.)

2. If d|a and d|b, then d|(a± b). (If d divides a and d divides b, then d divides the sum or difference of a and b. This generalizes to: If d divides all the terms of a sum, then d divides the sum.)

3. If d|a and d b, then d (a ± b). (If d fails to divide any term of a sum or difference, then d does not divide the sum or difference.)

C. Condensed list of Divisibility tests

a. An integer is divisible by 2, 5, or 10 if and only if its units digit is divisible by 2, 5, or 10, respectively.

b. An integer is divisible by 4 if and only if the last 2 digits of the integer represent a number divisible by 4.

c. An integer is divisible by 8 if and only if the last 3 digits of the integer represent a number divisible by 8.

d. An integer is divisible by 3 or by 9 if and only if the sum of its digits is divisible by 3 or 9, respectively.

e. An integer is divisible by 7. Seven divides an integer if and only it 7 divides the result after you double and subtract.

f. An integer is divisible by a composite number n if and only if it is divisible by the highest power of any prime in its unique factorization. (*Fundamental Theorem of Arithmetic (FTA): Every positive integer is the unique product of powers of a finite number of primes; uniqueness means independent of the order of the prime factors.)

For example, 6 divide an integer if and only if the integer is divisible by both of the relatively prime factors 2 and 3.

IV. Prime and composite numbers

A. Positive integers that have exactly two positive divisors, one and itself, are called prime integers. Integers greater than 1 and not prime are called composites.

B. Fundamental Theorem of Arithmetic: Every composite number has one and only one prime factorization, aside from variation in the order of the prime factors. (Every factorization of an integer is unique up to order.)

C. Criterion for determining if a given number n is prime:

If n is not divisible by any prime p for which p ≤ n, then n is prime. (be able to explain)

D. If the prime factorization of a number, n, is n = p₁^q1. P₂ ^q2. P₃ ^q3. . . . P_m^qm, then the number of divisors of n is [(q₁)(q₂)(q₃). . . . . (q)+ 1)].

V. Greatest common divisor (GCD) and least common multiple (LCM)
(also called LCF for the lowest common factor)

A. The greatest common divisor (GCD) of two or more natural numbers is the greatest divisor, or factor, that the numbers have in common. (is used for simplifying sums or differences of fractions)

B. Euclidean algorithm: If a and b are whole numbers and a ≥ b, then GCD(a, b) = GCD(b, r), where r is the remainder when a is divided by b. The procedure of finding the GCD of two numbers aand b by using the above result repeatedly is the Euclidean algorithm.

C. The least common multiple (LCM) of two or more natural numbers is the least positive multiple that the numbers have in common.

D. GCD(a, b)*LCM(a, b) = ab.

E. If GCD(a, b) = 1, then a and b are relatively prime.

*VI. Modular Arithmetic

A. For any integers a and b, a is congruent to b modulo m if and only if, a-b is a multiple of m, where m is a positive integer greater than 1, or m|(a-b).

Two integers are congruent modulo m if, and only if, their remainders upon division by m are the same.

Students are to be able to define, explain the concepts of, and give examples of all the different models of the four operations of addition, subtraction, multiplication and division of integers as covered in class and in Chapter 4.

Students are to:

- know the difference and relationships that hold between “negative” numbers that are associated with the concept of direction and "subtraction," the inverse operation of addition.

-know the equivalencies of the following integer additive inverses:
-(-a) = a, -(a) = (-1)*a, (–a)(–b) = ab, –ab = (–a)b = a(–b) and (a ± b) = a ± ( b).

-be able to use and explain the properties shown in the various models given for these equivalencies but need not have to write-out the formal proofs.

For example, by the Charged Field Model,
3- 4 = 3+0- 4 = 3+(4+ 4) - 4 = (3+4)+( 4- 4) = (3+4)+0 = 3+4 = 7.m