 # Binary Operations

## Introduction to Binary Operations

Binary operation is an operation that requires two inputs. These inputs are known as operands. The binary operation of addition, multiplication, subtraction and division takes place on two operands. Even when we  add any three binary numbers, we first add two numbers and  then the third number will be added to the result of the two numbers. Thus,  the mathematical operations which are done with the two numbers are known as binary operations.

What is a Binary Operation?

The binary operation conjoins any two elements of a set. The results of the operation of binary numbers belong to the same set. Let us take the set of numbers as X on which binary operations will be performed. Now, we will perform binary operations such as addition, subtraction, multiplication and division of two sets (a and b) from the set X. The result of the operation (a and b) will be the other element that will belong to the same set X.

Hence, the binary operation is stated as an operation which is performed on set X. This function is derived by * A * A. Thus, the binary operation * performed on operands a and b is symbolized as a*b.

Binary operation Examples

Let us understand the binary addition on natural numbers and real numbers. If we add two operands which are natural numbers such as x and y, the result of this operation will also be a natural number. Same rule holds for real numbers as well.

+: R + R → R is derived by (x, y) → x + y

+: N + N N is derived by (x, y) → x + y

Let us understand the binary multiplication on natural numbers and real numbers. If we multiply two operands which are natural numbers such as x and y, the result of this operation will also be a natural number. Same rule holds for real numbers as well.

+: R × R → R is derived by (x, y) → x × y

+: N × N N is derived by (x, y) → x × y

Let us understand the binary subtraction on natural numbers and real numbers. If we subtract two operands which are real numbers such as x and y, the result of this operation will also be a natural number. Same rule does not hold for natural numbers because if we take two numbers such as x and y and perform binary subtraction on it, then the result will not be in real numbers.

For example = 3-4 = -1 (-1 is not a real number)

Hence,

-: R - R → R is derived by (x, y) → x – y

Let us understand the binary division on natural numbers and real numbers. If we divide two operands which are real numbers such as x and y, the result of this operation will also be a natural number. Same rule does not hold for natural numbers because if we take two numbers such as x and y and perform binary division on it, then the result will not be in real numbers.

For example: 1 ÷ 0 = 0 (0 is not a real number)

Hence, ÷ : R ÷ R → R is derived by (x, y) → x ÷ y

Binary Operation Types

Binary operations such as binary addition, binary subtraction, binary multiplication and  binary division are calculated similarly as the arithmetic operations are calculated in numerals.These are four types of binary operations namely

• Binary Subtraction

• Binary Multiplication

• Binary Division.

The result obtained after adding two binary numbers is the binary number itself. Binary addition is the simplest method to add any of the binary numbers. It can be calculated easily if we know the following rules.

Rules

• 0 + 0 = 0

• 0 + 1 = 1

• 1 + 0 = 1

• 1 + 1 =10

Let us take any two binary numbers and add them.

Add : 10001 + 11101 = 101110

Binary Subtraction

The result obtained after subtracting two binary numbers is the binary number itself. Binary subtraction is also the simplest method to subtract  any of the binary numbers. It can be calculated easily if we know the following rules.

Rules

• 0 – 0 = 0

• 0 – 1 = 1 (with a borrow of 1)

• 1 – 0 = 1

• 1 – 1 = 0

Let us take any two binary numbers and subtract them.

Binary Multiplication

The binary multiplications are calculated similarly as the other arithmetics numerals are calculated. Let us take any two binary numbers and multiply them.It can be calculated easily if we know the following rules.

Rules

• 0 × 0 = 0

• 0 × 1 = 0

• 1 × 0 = 0

• 1 × 1 = 1

Example

1101 * 1010 = 10000010

Binary Division

The method of binary division is similar to the 10 decimal system other than the base 2 system. It can be calculated easily if we know the following rules.

• 1 ÷ 1=1

• 1 ÷ 0 =0

• 0 ÷ 1 = Meaningless

• 0 ÷ 0= Meaningless

Let us understand binary division with an example.

Solved Example

1. Is * defined on the set (1, 2, 3, 4, 5) by x * y= LCM of x and y a binary operation. Justify your number.

(Hint: Use the below table to solve the question)

 * 1 2 3 4 5 1 1 2 3 4 5 2 2 2 6 4 10 3 3 a 3 12 15 4 4 4 12 4 20 5 5 a 15 20 25

Solution:

Let A = {1, 2, 3, 4, 5}, and x * y = LCM of x and y

Let x= 2 and y =3

x *y = 2 * 3 = 6 ϵ A

Since 6 is not included in set (1, 2, 3, 4, 5,)

Hence* is not a binary operation.

2.      Consider a binary operation * on the set { 1,2,3,4,5) given by the below multiplication table

1. Computer (2 *3) * 4 and 2 * (3* 4)

2. Is * commutative

3. Compute ( 2*3) * (4*5)

(Hint: Use the below table to solve the question)

 * 1 2 3 4 5 1 1 1 1 1 1 2 1 2 1 2 1 3 1 1 3 1 1 4 1 2 1 4 1 5 1 1 1 1 5

Solution:

1.     2 * 3 = 1 and 3 * 4= 1

Now, (2 * 3) * 4 = 1 * 4 =1 and 2 * (3 * 4) = 2 * 1 =1

2.     2 * 3=1 and 3 * 4=1

2 * 3= 3 * 2 and other element of the given set

Hence, the operation is commutative

3.  (2 * 3) * (4*5) = 1* 1=1

3. Show that none of the operations given below is identity.

Solution: Let the identity be I

1 a* I = a-I a

2.  a* I = a2 -I2 a

3.  a* I = a + aI a

4.  a * I = (a – I) a

5.  a * I = aI/4 a

6.  a * I= aI a

Hence, none of the operation given above has identity

Fun Facts

• Binary operation is often represented as * on set is a method of combining a pair of elements in that set that result in another element of the set.

• Gottfried Wilhelm Leibniz discovered a binary numeral system and stated that it can be used in a primitive calculating machine

Quiz Time

1. The result obtained on binary multiplication of 1010 * 1100 is

1. 0001111

2. 0011111

3. 1111100

4. 1111000

1. If * and ° are two binary operations sated by x * y = x +y and x°y = xy, then

1. (x +y)° z= xy + xz

2. ° is distributive on *

3.  * is distributive on °

4. x° ( y * z)= xz + yz

3. The law  x + y = y + x is known as

1. Commutative law

2.  Associative law

3. Closure law

4.  Distributive law

1. What is the Binary number system?

The binary number system is a system of numbering that uses only 2 digits such as 0 and 1 to represent any number instead of using the digits 0 to 9 to represent any number. The base-2 system is examined as the positional notation with 2 as a radix. The binary number system is usually performed internally on almost all the latest computers devices and laptops due to its direct implementation in electronic circuits through logic gates. Every digit in the binary number system is referred to as a bit.

A single binary digit in the binary number system is known as “Bit”. The binary number given below has 7 bits.

1111000

2. What are the properties of binary operations?

Closure property: An operation * on a non-empty set A has closure property, if x ϵ A, y ϵ A → x * y ϵ A.

Additions are the binary operation performed on each of the set of natural numbers (N), Integer (Z), Rational number (Q), Real number (R) Complex number (C).

The addition performed on the set of all irrational numbers is not considered as binary operations.

• Multiplication is a binary operation that is included on each of the set of natural numbers (N), Integer (Z), Rational number (Q), Real number (R) Complex number (C).

The multiplication performed on the set of all irrational numbers is not considered as binary     operations.

• Subtract is not considered as a binary operation on the set of Natural number (N)

• Division is not considered as a binary operation on each of the set of natural numbers (N), Integer (Z), Rational number (Q), Real number (R) Complex number (C).

Exponential operation (a,b) ab is considered as a binary operation on the set of natural numbers (N) but not on the set of integers (Z).