What are the real numbers and properties of real numbers?

Mathematics

Class 9

2085

Rajiv Sharma

Thomas

Answer / Solution

Answer / Solution

The properties of real numbers have total of 21 properties :

- Distributive property
- Commutative property of addition
- Commutative property of multiplication
- Associative property of addition
- Associative property of multiplication
- Additive identity property
- Multiplicative identity property
- Additive inverse property
- Multiplicative inverse property
- Zero property of multiplication
- Closure property of addition
- Closure property of multiplication
- Addition property of equality
- Subtraction property of equality
- Multiplication property of equality
- Division property of equality
- Substitution property
- Reflective (or identity) property of equality
- Symmetric property of equality
- Transitive property of equality
- Law of trichotomy

Qayanat

Answer / Solution### Properties of the Real Number System

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.

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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