# Additive and Multiplicative Identity

## Additive and Multiplicative Identity and Basic Properties of Identity

On a daily basis, we keep dealing with the world of numbers throughout our lives, a lot of the characteristics of real numbers are been in use 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 are 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 provides 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 the 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 been separately explained with addition, subtraction, division, multiplication.

• • Closure property

• • Associative property

• • Commutative property

• • Distributive property

• • Additive identity property

• The Identities of Closure property:

The identities of closure property are been detailly explained with suitable examples and formula.

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 obtained when an operation is been 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 operation then it is said to be the closure property for that particular set of operation.

P + Q = R
Example: P = 9
Q = 6
P + Q = R
9 + 6 = 11

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 naturals numbers under 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

The closure property of multiplication

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

P * Q = R
Example : P = 6
Q = 9
P * Q = R
6 * 9 = 54

The closure property of division

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

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

Associative property of subtraction

The Change in the grouping occurs due to operands in the subtraction of number 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

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

Associative property of division

The Change in the grouping occurs due to operands in the division of number changes the result. Hence, the integers, whole numbers, rational numbers under division does 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 is obtained when changing the order of the operands does not change the result.

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

Commutative property of subtraction

the change in the order of operands in subtraction to the numbers do change happen 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

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

Commutative property of Division

the change in the order of operands in the division to the numbers do change happen in 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:

Distributive property of addition

a(b + c) = ab + ac

Distributive property of subtraction

a(b - c) = ab – ac

The Identities of additive identity property:

The total of any number is always zero and which is always the original number. The additive identity of numbers are the
names which suggested is a property of numbers which is used when we carrying out additional operations. The property declares that when a number of variables are is added to zero it show 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 which is engaged when carrying out multiplication functions. 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 rep
resented 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.

Difference between the additive identity and multiplicative identity
 Additive identity multiplicative identity The total of any number is always 0(zero) and which is always the original number. Zero is always called the identity element whenever a number is multiplied by the number 1 (one) it will give the same number as the productthe multiplicative identity is 1 (the number one)

Answer the following questions

• 1. What is identity?

• 2. Explain about The Closure Property of subtraction?

• 3. Write the difference between the additive identity and multiplicative identity?

• 4. Write briefly about commutative property?

• 5. Write about the associative property of subtraction?

• 6. Write a short note on the distributive property?

• 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?

• a. Closure property of addition

• b. Closure property of subtraction

• c. Associative property of subtraction

• d. Associative property of addition

• e. (Ans: b. Closure property of subtraction )

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

• a. Commutative property of division

• b. Commutative property of addition

• c. Commutative property of multiplication

• d. None of these

• e. (Ans: a. Commutative property of division )

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

• a. S

• b. 0

• c. Both S and 0

• d. None of the above

• (Ans: a. S)

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 been added to or with zero, the sum is always the natural number which was to be added.
X + 0 = X
Example 1: 9 + 0 = 9
Example 2: 100 + 0 = 100