Question :

What are the real numbers and properties of real numbers?

Subject

Mathematics

Standard

Class 9

Views

2085

Asked By

Rajiv Sharma

Thomas
Answer / Solution

The properties of real numbers have total of 21 properties :

  1. Distributive property
  2. Commutative property of addition
  3. Commutative property of multiplication
  4. Associative property of addition
  5. Associative property of multiplication
  6. Additive identity property
  7. Multiplicative identity property
  8. Additive inverse property
  9. Multiplicative inverse property
  10. Zero property of multiplication
  11. Closure property of addition
  12. Closure property of multiplication
  13. Addition property of equality
  14. Subtraction property of equality
  15. Multiplication property of equality
  16. Division property of equality
  17. Substitution property
  18. Reflective (or identity) property of equality
  19. Symmetric property of equality
  20. Transitive property of equality
  21. Law of trichotomy
Qayanat
Answer / Solution

Every real number corresponds to a point on a number line and every point on the number line corresponds to a real number. Let see the numbers on a number line.

Properties of the Real Number System

The following properties of the real numbers govern the operations of addition and multiplication.

Closure Property: 

The sum of two real numbers is a real number. This property is called the closure property of addition. For all real numbers a and b:

a + b is a real number.

The product of two real numbers is a real number. This property is called the closure property of multiplication. For all real numbers a and b:

a • b is a real number.

Commutative Property: 

When we add some real numbers, we can change the order in which two numbers are added without changing the sum. This property is called the commutative property of addition. For all real numbers a and b:

a + b = b + a

When we multiply real numbers, we can change the order of the factors without changing the product. This property is called the commutative property of multiplication. For all real numbers a and b:

a • b = b • a

Associative Property

When three numbers are added, two are added first and then their sum is added to the third. The sum does not depend on which two numbers are added first. This property is called the associative property of addition. For all real numbers a, b, and c:

(a + b) + c = a + (b + c)

When three numbers are multiplied, two are multiplied first and then their product is multiplied by the third. The product does not depend on which two numbers are multiplied first. This property is called the associative property of multiplication. For all real numbers a, b, and c:

a × (b × c) = (a × b) × c

Identity Property: 

When 0 is added to any real number a, the sum is a. THerefore, a real number 0 is called the additive identity. For every real number a:

a + 0 = a and 0 + a = a

When 1 is multiplied by any real number a, the product is a. The real number 1 is called the multiplicative identity. For every real number a:

a • 1 = a and 1 • a = a

Inverse Property: 

Two real numbers are called additive inverses if their sum is the additive identity, 0. For every real number a, there exists a real number –a such that:

a + (–a) = 0

Two real numbers are called multiplicative inverses if their product is the multiplicative identity, 1. For every real number a, which is not equal to 0, there exists a real number 1 such that:

a × 1/a  = 1

Distributive Property

The distributive property combines the operations of multiplication and addition that means a multiplication distributes over addition. For all real numbers a, b, and c:

a (b + c) = ab + ac and

(a + b) c = ac + bc

The Multiplication Property of Zero

Zero has no multiplicative inverse. The multiplication property of zero defines multiplication by zero. For all real numbers a and b:

ab = 0 if and only if a = 0 or b = 0

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