Binary Operation

The mathematical operation that involves two elements is known as a binary operation. There are four main binary operations: addition, subtraction, multiplication, and division. When performing operations on two numbers, such as x and y, they are combined to form another number like x + y, x - y, x × y, or x ÷ y. These operations are represented by symbols such as ‘+’, ‘-’, ‘×’, and ‘÷’.
 

Binary operations are operations that require two inputs. The types of binary operations are:

• Binary Addition
• Binary Subtraction
• Binary Multiplication
• Binary Division

Binary Addition

If we add two natural numbers like 3 and 4, we get another natural number, 7. The same rule applies to real numbers, like 3.5 + 4.1 = 7.6. Thus, addition is a binary operation that is closed on both natural and real numbers.

Binary Subtraction

In binary subtraction, if we subtract two real numbers like 5.5 - 2.2 = 3.3, the answer is still a real number. Subtraction is a binary operation, but it is not commutative, meaning the order of the numbers matters. For example, 4 - 7 = -3, which is not a natural number.

Binary Multiplication

When multiplying two natural numbers like 3 × 4 = 12, we get another natural number as a result. This also applies to real numbers. So, multiplication is a binary operation for both natural and real numbers.

Binary Division

If we divide two real numbers like 6/2 = 3, the result is also a real number. However, dividing two natural numbers does not always result in a natural number. Division by zero is undefined and therefore not allowed. 

A binary operation takes two elements from a set and combines them to give a result that is also in the same set.  The key properties of binary operations includes:

• Closure property
• Associativity property
• Commutativity property
• Identity element 
• Inverse element

Closure Property in Binary Operations

The closure property means that if we take any two numbers from a set and perform an operation, the result will also be in the same set. 
For example, 4 + 8 = 12, and 12 is a natural number. Therefore, addition is closed for natural numbers.

Associativity of Binary Operations

For the associative property, when three numbers are involved, the way we group them does not affect the result. 
Example: 
(1 + 2) + 3 = 3 + 3 = 6
1 + (2 + 3) = 1 + 5 = 6
So, addition is associative.
But subtraction is not associative, because:
(5 - 3) - 2 = 2 - 2 = 0
5 - (3 - 2) = 5 - 1 = 4

Commutativity of Binary Operations

A binary operation is commutative if changing the order of the numbers does not affect the result. 
Example: 
5 + 4 = 9
4 + 5 = 9
So, addition is commutative. 

Identity Element of Binary Operations

An identity element is a value that, when used in an operation with any number, leaves the number unchanged. 
Example: 
2 + 0 = 2
0 + 2 = 2
Here, 0 is the additive identity.
3 × 1 = 3
1 × 3 = 3
Here, 1 is the multiplicative identity.

Inverse Element of Binary Operations

When a number is combined with its inverse, the result is the identity element. 
Example:
5 + (-5) = 0, 0 is the additive identity. 
4 × ¼ = 1, so ¼ is the multiplicative inverse of 4.

Binary Operation Table

When we have a small set, we can use a table to show how a binary operation works on that set. If we have a set X that includes just three numbers:
X = {3, 4, 5}
We define a binary operation W as:
W(x, y) represents the greater of x and y, also written as max {x, y}
Now, the binary table looks like:

We can read the table by looking at the row for the first number and the column for the second number. The number where they meet is the result of the operation.
W(3, 3) = max (3, 3) = 3
W(3, 4) = max (3, 4) = 4
W(4, 5) = max (4, 5) = 5
W(5, 3) = max (5, 3) = 5
W(4, 4) = max (4, 4) = 4
So, the table shows the maximum of each pair of numbers in the set.

Binary operations are used in financial calculations, resource distribution, and time management, and are applied in many aspects of daily life. Some of the real-life applications include:

• Shopping and billing: If you go shopping and buy two items, we use binary operations to calculate the cost of the items. For example, you bought a pencil for $10 and a notebook for $30; the total cost is calculated using addition, which is a binary operation.
• Temperature change: Binary operations can be used to calculate the temperature difference, showing how much the weather has changed. If the temperature in the morning is 20 °C and drops by 7 °C by evening, the change is found using subtraction. 
• Seating arrangements: Binary operations are useful for setting up events, exams, or parties. If there are 5 rows of chairs and each row has 6 chairs, the total number of chairs is found using multiplication.