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# Properties of Real Numbers

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 Sub Topics Real numbers are used to measure the continuous values. A real number can be rational number or irrational number or it may be positive, negative or zero. It may also be algebraic or transcendental. Properties of Real Numbers that are used in various expressions and equations as follows: 1. The real numbers are dense in nature. Within two real numbers, lies an infinite other real numbers.2. The real numbers holds true for Commutative Property under addition and multiplication as follows: a + b = b + a a * b = b * a3. The real numbers holds true for Associative Property under addition and multiplication as follows: a + ( b + c ) = ( a + b ) + c a * ( b * c ) = ( a * b ) * c4. Distributive property holds true on real numbers as follows: a * (b + c) = (a * b) + (a * c) a + (b * c) = (a + b) * (a + c)5. The additive and multiplicative identity property for real number is as follows: a + 0 = a a * 1 = a6. Inverse property of real number is as follows: a + (- a ) = 0 a * $\frac{1}{a}$ = 17. Zero property of real number is as follows: a * 0 = 0

## Commutative Properties of Real Numbers

Commutative property is an important part of mathematics, which is used to solve the problems of algebra. In a proper definition, real numbers are those numbers, that are not imaginary numbers. The real numbers did not have a name before imaginary numbers. Thatswhy, they were known as a real numbers. Commutative properties holds true under two operations which are addition and multiplication.

a + b = b + a

In the above given equation, it shows the commutative property of real numbers by addition. The commutative property of addition states that altering the arrangement of two numbers that are added does not make any modification in the final result. i.e, we can sum up numbers in any order but it does not affect the final result. In the general sense, in an additive equation, you can change the arrangement of real numbers anywhere. But, final result does not get affected through this shift of numbers. Here are some examples through which we can easily describe the commutative property of addition.

Example: 4 + 5 = 5 + 4 = 9

Commutative Property of Multiplication:

a * b = b * a

In the above given equation, it shows the commutative property of real numbers by multiplication. The commutative property of multiplication states that altering the arrangement of two numbers that are added does not make any modification in the final result i.e, we can multiply numbers in any arrangement. It wont affect the concluding result. In the general sense, in an multiplicative equation you can change the arrangement of real numbers anywhere. But, final result does not get affected through this shift of numbers. Here are some examples through which we can easily describe the commutative property of multiplication.

Example: 4 * 5 = 5 * 4 = 20

The commutative property of addition states that altering the arrangement of two numbers that are added does not make any modification in the final result i.e, we can sum up numbers in any order. But, it does not affect the final result.

a + b = b + a

The commutative property of addition can be explained by some real life examples as follows:

### Solved Example

Question: Two friends Tom and Terry were going to the market. Tom had Rs. 25 and Terry had Rs. 50 with him. Now, if Tom took Rs. 50 more and Terry took Rs. 25 from his mother, then do both of them have equal money?
Solution:
Money Tom had = Rs. 25
Money Tom took from his mother = Rs. 50
Total money Tom had =  25 + 50 = Rs. 75

Money Terry had = Rs. 50
Money Terry took from his mother = Rs. 25
Total money Terry had =  50 + 25 = Rs. 75

We observe that in both the cases, we get the same result. Thus, they have the same amount of money.

This property of addition is called commutative property of addition. According to the commutative property of addition, the sum of any two given numbers remain the same, even if we change the order of their existence. So, a + b = b + a, where a and b are any numbers.

## Commutative Property of Multiplication

The commutative property of multiplication states that altering the arrangement of two numbers that are multiplied does not make any modification in the final result i.e, we can multiply numbers in any order but it does not affect the final result.

a * b = b * a

The commutative property of multiplication can be explained by some real life examples as follows:

### Solved Example

Question: Two friends Riya and Jia went for a part time job. At the end of the day, Riya had five Rs. 10 coins and Jia had ten Rs. 5 coins. How much did they both earn in a day?
Solution:
Total money Riya earned = 5 * 10 = Rs. 50
Total money Jia earned = 10 * 5 = Rs. 50

We observe that in both the cases, we get the same result. Thus, they have same amount of money.

This property of multiplication is called commutative property of multiplication. According to the commutative property of multiplication, the product of any two given numbers remain the same, even if we change the order of their existence. So, a * b = b * a, where a and b are any numbers.

## Associative Property of Real Numbers

Associative property is an important part of mathematics, which is used to solve the problems of algebra. In a proper definition, real numbers are those numbers that are not imaginary numbers. The real numbers did not have a name before imaginary numbers. Thatswhy, they were known as real numbers. Associative properties holds true under two operations which are addition and multiplication.

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

In the above given equation, it shows the associative property of real numbers by addition. The Associative Property of Addition states that altering the arrangement of brackets between the operators does not make any modification in the final result i.e, we can sum up numbers in any order but it does not affect the final result. Here are some examples through which we can easily describe the Associative property of addition.

Example: 3 + (4 + 5) = (3 + 4) + 5 = 12

Commutative Property of Multiplication:

a * (b * c) = (a * b) * c

In the above given equation, it shows the associative property of real numbers by multiplication. The associative property of multiplication states that altering the arrangement of brackets between the operators does not make any modification in the final result i.e, we can multiply numbers in any arrangement. It wont affect the concluding result. Here are some examples through which we can easily describe the associative property of multiplication.

Example: 3 * (4 * 5) = (3 * 4) * 5 = 60

The Associative property of addition states that altering the arrangement of brackets between the operators does not make any modification in the final result i.e, we can sum up numbers in any order. But, it does not affect the final result.

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

The associative property of addition can be explained by some real life examples as follows:

### Solved Example

Question: Ashley goes to a super market and buys a chocolate for Rs. 10, a diary for Rs. 15 and a pen for Rs. 5. How much does she want to give to the cashier of the super market?
Solution:
Mental calculation done by ashley:
15 + 5 = 20
10 + 20 = 30
i.e, 10 + (15 + 5) = 30

Calculation done by the cashier:
10 + 15 = 25
25 + 5 = 30
i.e, (10 + 15) + 5 = 30

We observe that in both the cases, we get the same result.

This property of addition is called associative property of addition. According to the associative property of addition, the sum of any three given numbers remain the same, even if we change the arrangement of brackets between the operators. So, a + (b + c) = (a + b) + c where a, b and c are any numbers.

## Associative Property of Multiplication

The associative property of multiplication states that altering the arrangement of brackets between the operators does not make any modification in the final result i.e, we can multiply numbers in any order. But, it does not affect the final result.

a * (b * c) = (a * b) * c

The associative property of multiplication can be explained by the following examples:

Examples :

2 * (3 * 4) = (2 * 3) * 4 = 24
4 * (2 * 6) = (4 * 2) * 6 = 48
1 * (6 * 3) = (1 * 6) * 3 = 18
3 * (7 * 5) = (3 * 7) * 5 = 105
2 * (5 * 8) = (2 * 5) * 8 = 80

We observe that in both the cases, we get the same result for all the above examples.

This property of multiplication is called associative property of multiplication. According to the associative property of multiplication, the product of any three given numbers remains the same, even if we change the arrangement of brackets between the operators. So, a * (b * c) = (a * b) * c where a, b and c are any numbers.

## Distribution Property of Real Numbers

Real numbers are used to measure the continuous values. A real number can be rational number or irrational number or it may be positive, negative or zero. It may also be algebraic or transcendental.

The distributive property is a property for binary operations with at least two operands. This property comes in case when any expression has both the addition and multiplication operations. This is also known as the distribution property of multiplication over addition operation. According to this property of multiplication over addition, if any number or term is multiplied by the terms covered by the parenthesis, then we need to multiply that term to all the terms inside the parenthesis.
a * (b + c) = (a * b) + (a * c)
The distributive property can be explained by some examples as follows:
1. 5 * (4 + 3) = (5 * 4) + (5 * 3) = 35
2. 2 * (3 + 6) = (2 * 3) + (2 * 6) = 18

## Density Property of Real Numbers

Real numbers are used to measure the continuous values. A real number can be rational number or irrational number or it may be positive, negative or zero. It may also be algebraic or transcendental.

According to the density property, the real numbers are infinite. Between two real numbers, lies an infinite real numbers. For example, between 3 and 4, lies $\frac{3}{2}$, $\frac{3}{4}$, $\frac{3}{5}$, $\frac{3}{7}$, $\frac{3}{8}$, $\frac{3}{11}$, ....... etc. This means that real numbers cannot be counted and there can exist uncountable number of real numbers in the number system.

To prove that the real numbers are dense, follow the steps given below:

1. Consider any two real numbers say $\frac{3}{4}$ and $\frac{3}{8}$.
2. Add them. $\frac{3}{4}$ + $\frac{3}{8}$ = $\frac{9}{8}$
3. Divide it by 2 and it gives the mid-value of $\frac{3}{4}$ and $\frac{3}{8}$. So, $\frac{9}{16}$ lies between $\frac{3}{4}$ and $\frac{3}{8}$.
4. Similarly, we can find the mid-value of $\frac{3}{4}$ and $\frac{9}{16}$ and this will continue.
Hence it is proved that the real numbers are dense.

## Identity Property of Real Numbers

Real numbers are those numbers that are not imaginary numbers. The real numbers did not have a name before imaginary numbers. Thatswhy, they were known as real numbers. Identity properties holds true under two operations which are addition and multiplication.

a + 0 = a

The above equation shows the identity property of addition. The identity property of addition for real numbers states that, there exist a real number 0 such that, if we add any real number to that number, the number remains unchanged.

Example: 4 + 0 = 4

Here, we observe that, when 0 is added to a real number 4, the result remains unchanged. So, we say that 0 is additive identity.

Identity Property of Multiplication:

a * 1 = a

The above equation shows the identity property of multiplication. The identity property of multiplication for real numbers states that, there exists a real number 0 such that if we multiply any real number to that number, the number remains unchanged.

Example: 4 * 1 = 4

Here, we observe that, when 0 is multiplied to a real number 4, the result remains unchanged. So, we say that 0 is the multiplicative identity.

The identity property of addition for real numbers states that, there exist a real number 0 such that, if we add any real number to that number, the number remains unchanged.

a + 0 = a

The identity property of addition can be explained by the following examples:
1. 4 + 0 = 4
2. 2.7 + 0 = 2.7
3. 11 + 0 = 11
4. $\frac{3}{4}$ + 0 = $\frac{3}{4}$

## Identity Property of Multiplication

4. $\frac{3}{4}$ * 1 = $\frac{3}{4}$