2. Integers The set of whole numbers and their opposites Integers are positive and negative numbers.from infinity…, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, … till infinity
3. Each negative number is paired with a positive number the same distance from 0 on a number line. A Number Line 0 1 2 -3 -2 3 -1
4. Algebraic properties Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative). The following lists some of the basic properties of addition and multiplication for any integers a, b and c.
6. Theoretical Properties Z is a totally ordered set without upper or lower bound. The ordering of Z is given by: ... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: if a < b and c < d, then a + c < b + d if a < b and 0 < c, then ac < bc. It follows that Z together with the above ordering is an ordered ring. The integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition
7. Construction The integers can be formally constructed as the equivalence classes of ordered pairs of natural numbers (a, b). The intuition is that (a, b) stands for the result of subtracting b from a. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule: precisely when Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; denoting by [(a,b)] the equivalence class having (a,b) as a member, one has: The negation (or additive inverse) of an integer is obtained by reversing the order of the pair: Hence subtraction can be defined as the addition of the additive inverse:
9. If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as {... −3,−2,−1, 0, 1, 2, 3, ...}. Some examples are:
10. Cardinality The cardinality of the set of integers is equal to (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N. If N = {0, 1, 2, ...} then consider the function: { ... (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) ... } If N = {1,2,3,...} then consider the function: { ... (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) ... } If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality.
12. Addition of Integers When the signs of the numbers being added are the same, add their absolute values--The sign of the answer is the same as the sign of the numbers added together When the signs of the numbers being added are different, subtract their absolute values--The sign of the answer is the same as the number with the largest absolute value
13. Example:- Let there be two numbers x=-4 and y=7 x+y = 3 Let there be two numbers x=-4 and y=-7 x+y = -11
14. Subtraction of Integers Add the opposite and refer to the addition rules (change the subtraction to addition and change the sign of the number being subtracted) Example:- Let there be two numbers x=4 and y=7 x-y = -3 y-x = 3
15. Add the opposite and refer to the addition rules (change the subtraction to addition and change the sign of the number being subtracted) Example:- Let there be two numbers x=-4 and y=7 x-y = -11 y-x = 11
16. Multiplication of Integers When the sign of the numbers being multiplied are the same, multiply the numbers--The sign of the answer is positive When the sign of the numbers being multiplies are different, multiply the numbers--The sign of the answer is negative
17. When the sign of the numbers being multiplied are the same, multiply the numbers--The sign of the answer is positive When the sign of the numbers being multiplies are different, multiply the numbers--The sign of the answer is negative Example:- Let there be two numbers x=-4 and y=7 x*y = -11
18. Division of Integers Same as the rules for multiplication of Integers Example:- Let there be two numbers x=4 and y=2 x/y = 2
