[Maths Class Notes] on Additive and Multiplicative Identity Pdf for Exam

On a daily basis, we keep dealing with the world of numbers throughout our lives, a lot of the characteristics of real numbers are being used for the operations of such addition, subtraction, multiplication, division, etc. One of the properties includes a specific operation on the numbers appearing in obtaining the same number. This feature of the numbers, the world is called Identity property. The equation is always true no matter what different values have been chosen.

In the system of mathematics, the identity is always an equation which is an equilibrium relation X= Y and Y = X, X and Y are the variables and X and Y provide the similar value as each other despite whatever values they are (most probable numbers) are replaced for the variables. In other terms, X = Y and vice-versa is an identity if X and Y define identical functions. This refers that identity is a balance between the functions that are very differently determined.

The Basic Property of Identity

These identities were divided into various properties with the characteristics of real numbers, whole numbers, natural numbers and so on. These properties are separately explained with addition, subtraction, division, multiplication. 

The Identities of Closure Property

The identities of closure properties have been detailed explained with suitable examples and formulas.

1. Closure Property of Addition

It is a total of 2 variables which can be a real number, whole numbers, natural numbers, and so on. A collection of numbers is said to be closed if for a particular mathematical method if the results are obtained when an operation is performed on any of the two numbers in the collection and it is a part of the set. If a set of numbers is closed for a particular set of operations then it is said to be the closure property for that particular set of operations.

P + Q = R

Example: P = 9 

Q = 6

P + Q = R

9 + 6 = 15

2. The Closure Property of Subtraction

The difference between two natural numbers may give an outcome in which a negative number and sometimes it is a positive number. Therefore, the natural numbers under the subtraction method of closure property are not closed.

P + (-Q) = R

Example: P = 9 

Q = 6

P + (-Q) = R

9 + (-6) = 9 – 6 = 3

When number are interchanged,

P = 6 

Q = 9

P + (-Q) = R

6 + (-9) = 6 – 9 = -3

3. The Closure Property of multiplication

The product of any two numbers outcome is neither positive number nor negative numbers. It can be an integer number or a natural number or a whole number.

P * Q = R

Example : P = 6 

Q = 9

P * Q = R

6 * 9 = 54

4. The Closure Property of Division

The quotient of any two numbers results in decimal and a fraction in which it can neither be a positive number nor a negative number. It can be an integer number or a natural number or a whole number.

P / Q = R

Example : P = 6 

Q = 9

P / Q = R

6 / 9 = 0.67 (in decimal) or 2 / 3 (in fraction).

The Identities of Associative Property

A collection of associated functions that change in the grouping occurs due to operands of number changes; the result is called associative property.

1. Associative Property of Addition

When there are three or more numbers that are added, the total is always equal, although of the grouping of the addends.

(K + L) + M = K + (L + M)

Example: K = 8

L = 9

M =4

(K + L) + M = K + (L + M)

(8 + 9) + 4 = 8 + (9 + 4)

17 + 4 = 8 + 13

21 = 21 

2. Associative Property of Subtraction

The Change in the grouping occurs due to operands in the subtraction of the number that changes the result. Hence, the integers, whole numbers, rational numbers under subtraction do not oppose associative property.

(K – L) – M ≠K – (L – M)

Example: K = 8

L = 9

M =4

(K – L) – M ≠ K – (L – M)

(8 – 9) – 4 ≠8 – (9 – 4)

1 – 4 ≠8 – 5 

-3 ≠ 4

3. Associative Property of Multiplication

There will be no change in the result or outcome when the changes occur in the operands in the multiplication of numbers.

K * (L * M) = (K * L) * M 

Example: K = 8

L = 9

M =4

K * (L * M) = (K * L) * M 

8 * (9 * 4) = (8 * 9) * 4

8 * 36 = 72 * 4

288 = 288

4. Associative Property of Division

The Change in the grouping occurs due to operands in the division of numbers that change the result. Hence, the integers, whole numbers, rational numbers under division do not oppose associative property.

(K / L) / M ≠ K / (L / M)

Example: K = 8

L = 9

M =4

(K / L) / M ≠ K / (L / M)

(8 / 9) / 4 ≠ 8 / (9 / 4)

0.89 / 4 ≠ 8 / 2.25

0.2225 ≠ 3.56

The Identities of Commutative Property

A collection of numbers is supposed to be commutative for a specific mathematical function; the result obtained when changing the order of the operands does not change the result.

1. Commutative Property of Addition

The change in the order of operands in addition to the numbers does not cause any change in the result.

K + L = L + K 

Example: K = 5

L = 3

K + L = L + K

5 + 3 = 3 + 5

8 = 8 

2. Commutative Property of Subtraction

The change in the order of operands in subtraction to the numbers do change in the result. Therefore commutative property does not satisfy integers, whole numbers and rational numbers for subtraction.

K – L ≠ L – K 

Example: K = 5

L = 3

K – L ≠ L – K 

5 – 3 ≠3 – 5

2 ≠ -2

3. Commutative Property of Multiplication

The change in the order of operands in multiplication to the numbers does not cause any change in the result.

K * L = L * K

Example: K = 5

L = 3

K * L = L * K

5 * 3 = 3 * 5

15 = 15

4. Commutative Property of Division

The change in the order of operands in the division to the numbers does change the result. Therefore commutative property does not satisfy integers, whole numbers, and rational numbers for the division.

K / L = L / K

Example: K = 5

L = 3

K / L = L / K

5 / 3 = 3 / 5

1.67 = 0.6

The Identities of Distributive Property

This property clearly indicates that it includes distributing. This property states that if there are three numbers a, b and c, then the a( b +c ) = ab + ac. This means that a is distributed among two other numbers which are b and c respectively. 

1. Distributive Property of Multiplication over Addition 

In this case, the first number is multiplied with the sum of two numbers. For example, if 7 is multiplied by the sum of 20 and 8. This is represented by 7 ( 20 + 8). 

According to the distributive property rule, 

a(b + c) = ab + ac

7 ( 20 + 8) = 7 x 20 + 7 x 8

7( 28) = 140 + 56 

196 = 196 

This particular example shows that when the first number is multiplied with the sum of the other two numbers then it will be equal to the sum of the first number multiplied with both the other left numbers. 

2. Distributive Property of Multiplication over subtraction

In this case, the first number is multiplied by the difference between the two numbers. For example, if 10 is multiplied with the difference of 20 and 8. This is represented by 10 ( 20 – 8). 

According to the distributive property rule, 

a(b + c) = ab + ac

10 ( 20 – 8) = 10 x 20 – 10 x 8

10( 12) = 200 -80 

120 = 120

This particular example shows that when the first number is multiplied with the difference of the other two numbers then it will be equal to the difference of the first number multiplied with both the other left numbers. 

a(b – c) = ab – ac

The Identities of Additive Identity Property

The total of any number is always zero and is always the original number. The additive identity of numbers is the names that are suggested as a property of numbers which is used when we carry out additional operations. The property declares that when a number of variables are added to zero it shows to give the same number. Zero is always called the identity element, which is also known as additive identity. If we add any number with zero, the resulting number will be a similar number.

S + 0 = S = 0 + S

Where S is a real numbers

Example: the real numbers value is 24

S = 24

S + 0 = 24 + 0 = 24

The real number is 0

The Identities of Multiplicative Identity Property

The Multiplicative identity of numbers, as the name suggests, is a property of numbers that is engaged when carrying out multiplication functions. The multiplicative identity property says that whenever a number is multiplied by the number 1 (one) it will give the same number as the product. the multiplicative identity is 1 (the number one). It is true if the number will only be multiplied by 1 itself. The multiplicative identity property is represented as:

R × 1 = R = 1 × R (the real number is R)

Example: the value of the real number is 15

R = 15

R × 1 = R = 1 × R

15 * 1 = 15 = 1 * 15

15 = 15.

Additive Identity Property of Addition

The total of any number with zero is always the original number.in other words, if any of the natural numbers are added to or with zero, the sum is always the natural number that was to be added.

X + 0 = X

Example 1: 9 + 0 = 9

Example 2: 100 + 0 = 100

Difference Between the Additive Identity and Multiplicative Identity

Practice Questions

Fill in the Blanks:

1. It is true if the number will only be ____________by 1 itself (Ans: multiplied)

2. The difference between two natural numbers may give an outcome in which a _______________and sometimes it is _______________. (Ans: a negative number, positive number)

3. Zero is always called the ____________ (Ans:identity element).

Tick the Correct Answers

1. This equation P + (-Q) = R is for which property?

1. Closure property of addition.

2. Closure property of subtraction.

3. Associative property of subtraction.

4. Associative property of addition.

(Ans: b. Closure property of subtraction)

2. ________________does not satisfy integers, whole numbers and rational numbers for division.

1. Commutative property of division.

2. Commutative property of addition.

3. Commutative property of multiplication.

4. None of these.

(Ans: a. Commutative property of division )

3. S + 0 = S = 0 + S which is the real number in this equation?

1. S

2. 0

3. Both S and 0

4. None of the above

(Ans: a. S)