07 properties of real numbers
TRANSCRIPT
PROPERTIES OF REAL NUMBERS
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
QUIZ Write each union or intersection as a
single interval.
1. 2. [2,5) (4,9]2. [-2,2] [2,6)3. [2,6) [2,8)4. [1,5] [2,9]5. (3,6) (0,+)
I. PROPERTIES OF REAL NUMBERSEquality Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Let a, b, c, d be real numbers.Axiom 1. Reflexive Property for equality, a = a. The axiom states that any real number is equal to itself.
I. PROPERTIES OF REAL NUMBERSEquality Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 2. Symmetric Property for equality. If a = b then b = a. As an illustration, if a+b=x then x=a+b.
I. PROPERTIES OF REAL NUMBERSEquality Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 3. Transitive Property for equality. If a = b then b = c then a=c.
I. PROPERTIES OF REAL NUMBERSEquality Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 4. Addition Property for equality (APE). If a = b then a+c = b+c.
I. PROPERTIES OF REAL NUMBERSEquality Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 5. Multiplication Property for equality (MPE). If a = b then axc = bxc.
I. PROPERTIES OF REAL NUMBERSEquality Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 6. Substitution or Replacement Property states that a quantity may be substituted by an equal quantity. (i.e. if x=a+b then [(a+b)+1]2=(x+1)2.
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
The set of real numbers together with the operations addition and multiplication satisfies the following axioms called field axioms:
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 1. Closure Axiom. (Addition) If a and b are real numbers, then a+b is also a real number. (Multiplication) If and b are real numbers, then ab is also a real number. (The sum and product of any two real numbers are real numbers)
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Closure Property. A set of numbers is said to be closed under the operation addition(multiplication) if the sum(product) of any two elements in the set not necessarily distinct is also in the set. (i.e. The set Q)
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 2. Associative Axiom. (Addition) For any a,b,c є R, (a+b)+c=a+(b+c). (Multiplication) For any a,b,c є R, (ab)c=a(bc).
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 3. Commutative Axiom. (Addition) For any a,b є R, a+b =b+a. (Multiplication) For any a,b є R, ab=ba.
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 4. Distributive Axiom. For any a,b,c є R, thena(b+c)=ab+ac (left distribution)(b+c)a=ba+ca (right distribution)
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 5. Identity Axiom. (Addition)There exists a real number 0 such that a+0=a, for any a є R. (Multiplication)There exists a real number 1 not equal to 0 such that ax1=a, for any a є R.
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 6. Inverse Axiom. (Addition)For any real number a, there exists a unique real number denoted by –a such that a+(-a)=0. (any real number has a unique additive inverse) If a + b = 0 the b = -a.
I. PROPERTIES OF REAL NUMBERSField Axioms.
UNIVERSITY OF THE PHILIPPINES MINDANAO College Algebra
Axiom 6. Inverse Axiom. (Multiplication)For any non-zero real number a, there exists a unique real number denoted by 1/a such that a(1/a)=1. (any real number has a unique multiplicative inverse) if ab=1,a≠0, then b=1/a.