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 Addition

• Binary Subtraction

• Binary Multiplication

• Binary Division.

Binary Addition

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)

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)

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)2  ≠ a

5. a * I = aI/4 ≠ a

6. a * I= aI2  ≠ 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

2. 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

Tips to Learn Binary Operations

The text might have given you an idea of binary operations, their types and also, some solved examples which would have helped you to get a better understanding of the same. 

Maths is a subject that needs a lot of practice, and hence, the students are advised to focus on practising questions of each type and master the concepts. 

Apart from the practice, students might also consider adding the below-mentioned tricks and tips to their schedules to ace in it. 

Read ahead to know them. 

• Find a Good Studying Spot

A good studying spot is one where you have the least number of distractions and can study well without the diversion of mind. It’s possible that whenever you sit to study, you get stuck in millions of thoughts but a peaceful study environment is a key to dedication and focus. Some factors that matter a lot are study lighting that is important for your eyes, a silent room essential for maintaining focus and rules such as keeping the door closed, or keeping phone and devices away so that you don’t break your focus. 

• Stay Away From Your Phone

Try using your phone at a limit as you get a number of notifications which are one of the biggest distractions. The best way to stay away from your phone is either keeping it on flight mode, silent or switched off at the time when you’re studying. Take regular breaks while studying and use your phone for just 5-10 minutes. All you need to understand is a simple rule that says study now and distraction afterwards. You need not just remove all the distractions permanently but just temporarily as your brain needs some time to relax and get back to its full speed.

• Take a Break and Take care of Yourself

It’s important to take breaks at regular intervals otherwise your body might feel drained. It’s not just about one’s body’s energy but also about the mind. Taking a break relaxes your mind too which is essential for maintaining focus throughout the study time. Physical and mental health are equally important. Some students under stress tend to study for long hours without any break which is very unhealthy. 

• Organize Lectures Notes

Organising and rewriting are the key points that will help you excel in your studies with ease. You need to organise all your lecture notes in order, revising them from time to time and rewriting them. Rewriting your notes may look stupid but it isn't, it is actually very helpful and also makes you remember all of it within no time.  It has proved to be the most brilliant study technique and hence, you shouldn't be missing this one. 

• Leave Time for the Last-minute Review

Last-minute review is only possible if you have well-organised notes. Here organised notes come into the play. Leaving time for the last-minute revisions is important. You need to give yourself a good night's sleep and that too stress-free. You should try to keep your mind calm, especially when you are close to your exams. Remember good sleep will always increase your retention power and also make things more clear. The key to it is setting a good timetable that helps you relax and also leave time for last-minute revisions so that nothing happens in an overburdened environment.