BINARY OPERATION THE REAL NUMBER SYSTEM Rational Numbers

BINARY OPERATION

THE REAL NUMBER SYSTEM • Rational Numbers ( ) are numbers which can be written as a fraction. By definition, all numbers in fractional form are rational numbers. Rational numbers will include all terminating and all recurring decimals. Some examples of rational numbers are: • Natural Numbers • Whole Numbers • Integers • Irrational Numbers ( fraction. For eg. ) are numbers which cannot be written as a

OPERATIONS • Unary Operation: An operation that can be performed on one element or one number. • For example: 1. finding the square root of all elements 2. Squaring of all elements 3. Reciprocal of all elements • Binary Operation ( or ): An operation that involves two elements or numbers. For example: addition, subtraction, multiplication and division.

PROPERTIES OF BINARY OPERATION • Closure • For any two elements in a given set, if we operate on these two elements and we end up with an element in the same set, then the set is closed with respect to that operation. That is: for all a, b A, then a b A. Activity 1. Is the set of real numbers closed with respect to addition? 2. Is the set of real numbers closed with respect to multiplication? 3. Is the set of integers closed with respect to division?

COMMUTATIVITY • Let a, b A. A binary operation is said to be commutative on the nonempty set A if and only if a b = b a. Activity 1. Is addition commutative with respect to the set of real numbers? 2. Is subtraction commutative with respect to the set of real numbers? 3. Is multiplication commutative with respect to the set of real numbers? 4. Is division commutative with respect to the set of real numbers?

ASSOCIATIVITY • Let a, b, c A. A binary operation is said to be associative if and only if a ( b c) = (a b) c. We get the same result however the elements are paired. To decide whether an operation is associative, we need three elements from the set. Activity 1. Is addition associative with respect to the set of real numbers? 2. Is subtraction associative with respect to the set of integers? 3. Is multiplication associative with respect to the set of real numbers? 4. Is division associative with respect to the set of real numbers?

DISTRIBUTIVITY • Let a, b, c A. For any two operations over if and only if a (b c) = (a and , b) (a distributes c). b

INVERSE • Let be a binary operation on a non-empty se A. Let a, b, e a is the inverse of b and b is the inverse of a if and only if a b = b a = e where e is the identity element in A. NOTE: (i) An element operated on its inverse is equal to the identity. (ii) Since e e = e, the identity is its inverse. Any element which is its own inverse is called a self-inverse.

IDENTITY • Let be a binary operation on a non-empty set A. If there exists an element e A such that e a = a e = a, for all e A, then e is called the identity element in A. • For any real numbers , a + 0 = 0 + a = a. Hence, 0 is the identity for addition of the real numbers. • The identity for multiplication is 1, since , 1 x a = a x 1 = a. • NOTE: Whenever the identity element operates on an element in A, the result is the element in A.

ACTIVITY • The binary operation is defined on the set {a, b, c, d} as shown in the table below. a b c d a b c b a b c d c b c d a d c d a b (a) Is this operation commutative? (b) Name the identity element or explain why none exist? (c) For each element having an inverse, name the element and its inverse. (d) Show that (a b) c = a (b c).