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Real Numbers: Definition. Property. System
This document defines and describes the properties of real numbers. Real numbers form a set that includes rational numbers like integers, fractions, as well as irrational numbers. Real numbers have two basic operations, addition and multiplication, and follow important properties like closure, commutativity, associativity, distributivity, identity, and inverses. The real number system is hierarchical, with natural numbers at the bottom and real numbers at the top, encompassing rational and irrational numbers.
Real Numbers: Definition. Property. System
- 1. Real Numbers DEFINITION.PROPERTIES. SYSTEM.
- 2. Real Numbers A value that represents a quantity along a number line. It is a set of undefined elements a, b, c, etc. or symbollically R = { a, b, c, … } It has two basic operation: addition and multiplication.
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- 4. Properties of Equality •Closure Property •Distributive Property •Commutative Property •Identity Property •Additive Identity •Multiplicative Identity •Associative Property •Inverse Property •Additive Inverse •Multiplicative Inverse
- 5. Properties of Equality Closure Property For any real number a, b € R, a+b € R and ab € R. Example: 1+2=3 R = {1,2,3}; 1,2,3 € R.
- 6. Properties of Equality Commutative Property For any real number a, b € R, a+b = b+a and ab = ba. Example: 4 + 2 = 2 + 4 4 x 2 = 2 x 4
- 7. Properties of Equality Associative Property For any real number a, b, c € R, (a + b) + c = a + (b + c) (ab)c = a(bc). Example: (4 + 2) + 3 = 4 + (2 + 3) (4 x 2) 3 = 4 (2 x 3)
- 8. Properties of Equality Distributive Property For any real number a, b, c € R, a (b + c) = ab + ac. Example: 4 (5 + 3) = 4 (5) + 4 (3)
- 9. Properties of Equality Identity Property Additive Identity For any real number a € R, a+0=a. Example: 4+0=4 Zero is the identity element for addition or additive identity. Multiplicative Identity For any real number a € R, a . 1 =a. Example: 4 . 1 =4 One is the identity element for multiplication or multiplicative identity.
- 10. Properties of Equality Inverse Property Additive Inverse For any real number a, there is a real number called additive inverse of a denoted by “-a” such that a + (-a) = 1. Example: -(ab) + (ab) = 0 Multiplicative Inverse For any real number a, there is a real number called multiplicative inverse of a denoted by “1/a” such that a . 1/a = 1. Example: 4 . ¼ = 1
- 11. The Real NumberSystem Real Numbers Rational Numbers Integers Positive Natural Numbers Fractions Zero Negative Irrational Numbers
- 12. Thank you! Presentedby: Kenneth R. Wasawas BSED - 3