in the set z on integers, we define addition and multiplicatio

in the set z on integers, we define addition and multiplicatio

1. In the set Z on Integers, we define Addition and Multiplication Prepared by : Mariela A. Camba

2. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM a = [(m, n)] b = [(p, q)] a + b = [( m+p , n+q )] a.b = [(mp+nq, mq+np)] and these are well defined can be seen easily .With these definitions it is easy to establish the Ring properties. For any two integers

3. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM What is a Ring properties? A ring is a set equipped with two operations usually referred to as addition and multiplication that satisfy certain properties: there are additive and multiplicative identities and additive inverses, addition is commutative, and the operations are associative and distributive.

4. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM DEFINITE INTEGRAL 1 Closure Property: 1 a+b , a.b are integers. The closure property states that when a set of numbers is closed under an arithmetic operation, performing the operation on any two numbers in the set always results in a number belonging to the same set of numbers.

5. The closure property holds true for addition, subtraction, and multiplication of integers. It does not apply for the division of two integers. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Closure Property for Integers The set of integers is given by Closure property of integers under addition The closure property of addition of integers states that the sum of any two integers will always be an integer. If a and b are any two integers, will be an integer. Closure property of integers under multiplication The product of any two integers will be an integer. If a and b are any two integers, a b will also be an integer.

6. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM 2 Associative as the name implies, means grouping. The origin of the term associative is from the word “associate”. Basic mathematical operations that can be performed using the associate property are addition and multiplication. This is normally applicable to more than 2 numbers. Associative property: a+(b+c) = (a+b)+c a.( b.c )=( a.b ).c

7. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Associative Property for Addition The addition follows associative property i.e. regardless of how numbers are parenthesized the final sum of the numbers will be the same. Associative property of addition states that: Associative Property for Multiplication Rule for the associative property of multiplication is:

8. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Property of Zero 3 The additive property of zero simply says that adding zero to any number does not change the number. In other words, x plus zero will always equal x. a+0=0+a=a ( 0 is called the additive identity for the set of integers and it is unique)

9. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Distributive Property 4 The distributive Property States that when a factor is multiplied by the sum/addition of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition operation. This property can be stated symbolically as: A ( B+ C) = AB + AC

10. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Additive Inverse 5 An additive inverse of a number is defined as the value, which on adding with the original number results in zero value. It is the value we add to a number to yield zero. Suppose, a is the original number, then its additive inverse will be minus of a, such that; a+(-a) = a – a = 0

11. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM If a, b ∈ Z then the equation a+x = b has a unique solution in the set of integers. For the sake of completeness, we just state the following properties of integers; Sum and product of two positive integers is positive. Product of two negative integers is positive. Product of a positive and a negative integer is negative. Given two integers a,b one and only one of the following holds. ( i ) a-b is positive (ii) a-b is negative, (iii) a-b=0 We write a > b if a-b is positive ,a < b if a-b is negative

12. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM If a+c = b+c then a=b If a. c= b.c , c ≠ 0 then a=b If a+c > b+c then a > b If a.c > b.c , c > 0 then a >b. If a.c > b.c , c < 0 then b > a.∎

13. MATH. EDUC. 208 Theory of Numbers Structure of Integers

14. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Type equation here.   The correspondence [(x,x+n)] n,( n ∈ N) is such that it preserves addition and multiplication, in the following sense: (i) [(x, x+n 1 )]+[(p, p+n 2 )]= [(x, x+ n 1 + n 2 )] (ii) ([(x, x+n 1 )].[(p, p+n 2 )]=[(x, x+ n 1 .n 2 )]

15. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Type equation here.  

16. MATH. EDUC. 209 ADVANCE CALCULUS – INTERGRAL AS LIMIT OF SUM Type equation here.   Thank You!