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Binary Number System is a number system that is used to represent various numbers using only two symbols “0” and “1”. The word binary is derived from the word “bi” which means two. Hence, this number system is called Binary Number System. Thus, the binary number system is a system that has only two symbols.

There are generally various types of number systems and among them the four major ones are,

• Binary Number System (Number system with Base 2)
• Octal Number System (Number system with Base 8)
• Decimal Number System (Number system with Base 10)
• Hexadecimal Number System (Number system with Base 16)

Here, we are only going to learn about Binary Number System. This number system is very useful for explaining tasks to the computer. In the Binary Number System, we have two states “0” and “1” and these two states are represented by two states of a transistor. If the current passes through the transistor then the computer reads “1” and if the current is absent from the transistor then it read “0”. Thus, alternating the current the computer reads the binary number system. Each digit in the binary number system is called a “bit”. 

In this article, we will learn about the Binary Number System, the Conversion of the Binary Number System, the Binary Table, the Operation of Binary Numbers, Examples, and others in detail.

Binary Number System is the number system in which we use two digits “0” and “1” to perform all the necessary operations. In the Binary Number System, we have a base of 2. The base of the Binary Number System is also called the radix of the number system .

In a binary number system, we represent the number as,

In the above example, a binary number is given in which the base is 2. In a binary number system, each digit is called the “bit”. In the above example, there are 5 digits.

Binary Number Table

Binary to decimal conversion.

A binary number is converted into a decimal number by multiplying each digit of the binary number by the power of either 1 or 0 to the corresponding power of 2. Let us consider that a binary number has n digits, B = a n-1 …a 3 a 2 a 1 a 0 . Now, the corresponding decimal number is given as

D = (a n-1 × 2 n-1 ) +…+(a 3 × 2 3 ) + (a 2 × 2 2 ) + (a 1 × 2 1 ) + (a 0 × 2 0 )

Let us go through an example to understand the concept better.

Example: Convert (10011) 2 to a decimal number.

The given binary number is (10011) 2 . (10011) 2 = (1 × 2 4 ) + (0 × 2 3 ) + (0 × 2 2 ) + (1 × 2 1 ) + (1 × 2 0 ) = 16 + 0 + 0 + 2 + 1 = (19) 10 Hence, the binary number (10011) 2 is expressed as (19) 10 .

Decimal to Binary Conversion

A decimal number is converted into a binary number by dividing the given decimal number by 2 continuously until we get the quotient as 1, and we write the numbers from downwards to upwards.

Example: Convert (28) 10 into a binary number.

  Hence, (28) 10 is expressed as (11100) 2 .

Arithmetic Operations on Binary Numbers

We can easily perform various operations on Binary Numbers. Various arithmetic operations on the Binary number include,

Binary Addition

Binary multiplication, binary division.

Now let’s learn about the same in detail.

The result of the addition of two binary numbers is also a binary number. To obtain the result of the addition of two binary numbers, we have to add the digit of the binary numbers by digit. The table added below shows the rule of binary addition.

The result of the subtraction of two binary numbers is also a binary number. To obtain the result of the subtraction of two binary numbers, we have to subtract the digit of the binary numbers by digit. The table added below shows the rule of binary subtraction.

The multiplication process of binary numbers is similar to the multiplication of decimal numbers. The rules for multiplying any two binary numbers are given in the table,

The division method for binary numbers is similar to that of the decimal number division method. Let us go through an example to understand the concept better.

Example: Divide (101101) 2 by (110) 2

1’s and 2’s Complement of a Binary Number

• 1’s Complement of a Binary Number is obtained by inverting the digits of the binary number.

Example: Find the 1’s complement of (10011) 2 .

Given Binary Number is (10011) 2 Now, to find its 1’s complement, we have to invert the digits of the given number. Thus, 1’s complement of (10011) 2 is (01100) 2

• 2’s Complement of a Binary Number is obtained by inverting the digits of the binary number and then by adding 1 to the least significant bit.

Example: Find the 2’s complement of (1011) 2 .

Given Binary Number is (1011) 2 To find the 2’s complement, first find its 1’s complement, i.e., (0100) 2 Now, by adding 1 to the least significant bit, we get (0101) 2 Hence, the 2’s complement of (1011) 2 is (0101) 2

Uses of Binary Number System

Binary Number Systems are used for various purposes and the most important use of the binary number system is,

• Binary Number System is used in all Digital Electronics for performing various operations.
• Programming Languages uses  Binary Number System for encoding and decoding data.
• Binary Number System is used in Data Sciences for various purposes, etc.

Binary Formula Difference Between Decimal and Binary Number Systems Prime Number

Binary Number System Example

Example 1: Convert Decimal Number (98) 10 into Binary.

Solution: 

Thus, Binary Number for (98) 10 is equal to (1100010) 2

Example 2: Convert Binary Number (1010101) 2 to Decimal Number.

Given Binary Number, (1011101) 2 = (1 × 2 0 ) + (0 × 2 1 ) + (1 × 2 2 ) + (0 × 2 3 ) + (1 × 2 4 ) + (0 × 2 5 ) + (1 ×2 6 ) = 1 + 0 + 4 + 0 + 16 + 0 + 64 = (85) 10 Thus, Binary Number (1010101) 2 is equal to (85) 10 in decimal system.

Example 3: Divide (11110) 2 by (101) 2

Example 4: Add (11011) 2 and (10100) 2

Hence, (11011) 2 + (10100) 2 =  (101111) 2

Example 5: Subtract (11010) 2 and (10110) 2

Hence, (11010) 2 &#x2013 (10110) 2 = (00100) 2

Example 6: Multiply (1110) 2 and (1001) 2 .

Thus, (1110) 2 × (1001) 2 = (1111110) 2

FAQs on Binary Number System

1. what is a binary number system.

Binary Number System is one of the four number system that is used to represent the numbers using only two digits, “0” and “1”. In binary number system the digits are called ‘bits’. Binary Number System is used by computers to perform various calculations.

2. What is a B it?

A bit in Binary Number System is defined as a individual digits that holds the value ‘0’ or ‘1’.

3. What is a Nibble?

A group of four digits is called the Niblle.

4. What is the Binary Value of 10?

The binary value of 10 is (1010) 2

5. What are Types of Number Systems?

There are various types of number systems and some of them are, Binary Number System Octal Number System Decimal Number System Hexadecimal Number System

6. How to Calculate Binary Numbers?

Binary numbers are calculated from dicmal numbers by dividing the decimal number with 2 and writing the remainder. Then we arrange all the remainders from newest to oldest to get the binary number.

7. How to Add Binary Numbers?

Binary numbers are added by using the formulas written below, 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 (carry 1)

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Last modified on February 7th, 2024

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Binary number system.

The word binary comes from ‘Bi’ means 2. Thus, a binary number system consists of 2 numbers, 0 and 1. It starts with 0 and ends in 1 and, therefore, has a base 2. The base-2 system in the positional notation is represented as (11101) 2 .

It is widely used in making the latest computers and computer-based devices in their electronic circuits using logic gates. Each digit in the machine is referred to as a bit.

The numbers 0 to 20 in decimal are written in binary as follows:

Binary to Decimal Conversion

The binary number is converted to the decimal number by expressing each digit as the product of each number (1 or 2) to the power of 2 based on its place value.

If a binary number has n digits a n-1 ….a 3 a 2 a 1 a 0 , its corresponding decimal number is obtained as:

(a 0 ×2 0 ) + (a 1 ×2 1 ) + (a 2 ×2 2 ) +….

Let us convert the binary number 11101 to its corresponding decimal number.

The binary number 11101 is expressed as:

(11101) 2 = (1 × 2 4 ) + (1 × 2 3 ) + (1 × 2 2 ) + (0 × 2 1 ) + (1 × 2 0 )

Decimal to Binary Conversion

A decimal number is converted to its corresponding binary number by dividing the number by 2 until we get 1 as the quotient. The quotients are written from down to up.

Let us convert the decimal number 20 to its corresponding binary number.

Dividing the number by 2 in each step, we get:

Thus, the decimal number (20) 10 is expressed as (10100) 2

Operations on Binary Numbers

Like decimal numbers, binary numbers are used to perform mathematical operations: addition, subtraction, multiplication, and division.

Binary Addition

Binary numbers are added digit by digit to obtain the result of addition.

We will use the following 4 rules for addition:

Binary Subtraction

Like addition, binary subtraction is done digit by digit to obtain the result.

• 0 – 0 = 0
• 1 – 0 = 1
• 0 – 1 = 0
• 1 – 1 = 0

Binary Multiplication

The rules for multiplying 2 binary numbers are given below:

Binary Division

The rules for dividing 2 binary numbers are given below:

• 1 ÷ 0 = Meaningless
• 0 ÷ 0 = Meaningless

Click here to learn more about addition, subtraction, multiplication, and division in the binary number system.

Complement of a Binary Number 

• 1’s complement is obtained by inverting the digits of the binary number. For example, the complement of (110) 2 is (001) 2 .
• 2’s complement is given by inverting the digits of the number and adding 1 to the least significant bit. For example, 2’s complement of (111) 2 is (001) 2 .

Solved Examples

Convert the binary number (111001) 2 to its decimal number.

The given binary number is (111001) 2 To find the corresponding decimal number: = (1 × 2 0 ) + (1 × 2 1 ) + (1 × 2 2 ) + (0 × 2 3 ) + (0 × 2 4 ) + (1 × 2 5 ) = 1 + 2 + 4 + 8 + 16 + 32  = 57 Thus, the corresponding decimal number is (57) 10.

Convert the decimal number (134) 10 to its binary number.

The given decimal number is (134) 10 To find the corresponding binary number: 134 ÷ 2 = 81, R = 0 81 ÷ 2 = 40, R = 1 40 ÷ 2 = 20, R = 0 20 ÷ 2 = 10, R = 0 10 ÷ 2 = 5, R = 1 5 ÷ 2 = 2, R = 1 2 ÷ 2 = 1, R = 0 1 ÷ 2 = 0, R = 1 Thus, the corresponding binary number is (10110010) 2

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Binary Number System

A Binary Number is made up of only 0 s and 1 s.

There is no 2, 3, 4, 5, 6, 7, 8 or 9 in Binary!

Binary numbers have many uses in mathematics and beyond.

How do we Count using Binary?

It is just like counting in decimal except we reach 10 much sooner.

Well how do we count in Decimal?

The same thing is done in binary ...

What happens in Decimal?

And that is what we do in binary ...

See how it is done in this little demonstration (press play button):

Decimal vs Binary

Here are some equivalent values:

Binary numbers also have a beautiful and elegant pattern:

Here are some larger values:

"Binary is as easy as 1, 10, 11."

Now see how to use Binary to count past 1,000 on your fingers:

In the Decimal System there are Ones, Tens, Hundreds, etc

In Binary there are Ones, Twos, Fours, etc, like this:

This is 1×8 + 1×4 + 0×2 + 1 + 1×(1/2) + 0×(1/4) + 1×(1/8) = 13.625 in Decimal

Numbers can be placed to the left or right of the point, to show values greater than one and less than one.

Example: 10.1

• The "10" means 2 in decimal,
• The ".1" means half,
• So "10.1" in binary is 2.5 in decimal

You can do conversions at Binary to Decimal to Hexadecimal Converter .

The word binary comes from "Bi-" meaning two. We see "bi-" in words such as "bicycle" (two wheels) or "binocular" (two eyes).

A single binary digit (like "0" or "1") is called a "bit".

For example 11010 is five bits long.

The word bit is made up from the words " b inary dig it "

When you say a binary number, pronounce each digit.

Example: the binary number "101" is spoken as "one zero one" , or sometimes "one-oh-one" .

This way people won't confuse it with a decimal number.

How to Show that a Number is Binary

To show that a number is a binary number, follow it with a little 2 like this: 101 2

This way people won't think it is the decimal number "101" (one hundred and one).

Example: What is 1111 2 in Decimal?

• The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8)
• The next "1" is in the "2×2" position, so that means 1×2×2 (=4)
• The next "1" is in the "2" position, so that means 1×2 (=2)
• The last "1" is in the ones position, so that means 1
• Answer: 1111 = 8+4+2+1 = 15 in Decimal

Example: What is 1001 2 in Decimal?

• The "0" is in the "2×2" position, so that means 0×2×2 (=0)
• The next "0" is in the "2" position, so that means 0×2 (=0)
• Answer: 1001 = 8+0+0+1 = 9 in Decimal

Example: What is 1.1 2 in Decimal?

• The "1" on the left side is in the ones position, so that means 1.
• The 1 on the right side is in the "halves" position, so that means 1×(1/2)
• So, 1.1 is "1 and 1 half" = 1.5 in Decimal

Example: What is 10.11 2 in Decimal?

• The "1" is in the "2" position, so that means 1×2 (=2)
• The "0" is in the ones position, so that means 0
• The "1" on the right of the point is in the "halves" position, so that means 1×(1/2)
• The last "1" on the right side is in the "quarters" position, so that means 1×(1/4)
• So, 10.11 is 2+0+1/2+1/4 = 2.75 in Decimal

"There are 10 kinds of people in the world, those who understand binary numbers, and those who don't."

Binary Number System

Binary number system is used to define a number in binary system. Binary system is used to represent a number in terms of two numbers only, 0 and 1. The binary number system is used commonly by computer languages like Java, C++. As the computer only understands binary language that is 0 or 1, all inputs given to a computer are decoded by it into series of 0's or 1's to process it further. In this lesson we will learn how to convert a decimal number to its binary number and the conversion of binary number to decimal number.

What is Binary Number System?

"Bi" in Binary means "two". Hence, this draws back the line to the representation of a number in terms of 0 and 1 only. It is possible to express decimal numbers in terms of a binary number system easily. Decimal numbers and binary numbers have different notations. A decimal number is represented with a base of 10 while a binary number is represented with a base of 2. For example, 2 in decimal notation is represented as \((2)_{10}\). The binary number for 2 is represented as \((10)_{2}\). Hence, 10 is the binary number representation for the number 2.

Binary Number System Chart

The numbers from 1 to 10 can be expressed in a binary number system in the following way:

Binary to Decimal Conversion

The binary number can be converted to a decimal number by expressing each digit as a product of the given number 1 or 0 to the respective power of 2. If a binary number has n digits, B = \(a_{n-1}...a_{3}a_{2}a_{1}a_{0}\), the decimal number for it is given as, D = (a 0 ×2 0 ) + (a 1 ×2 1 ) + (a 2 ×2 2 ) + ... Let us understand this using an example. We can convert 10101 to the decimal number form in the following way: The binary number 10101 is expressed as \((10101)_{2}\) = (1 × 2 4 ) + (0 × 2 3 ) + (1 × 2 2 ) + (0 × 2 1 ) + (1 × 2 0 ) = \((21)_{10}\). Thus, the binary number 10101 is expressed as \((21)_{10}\).

Decimal to Binary Conversion

The decimal number can be converted to a binary number by dividing the given number by 2 until we obtain the quotient as 1. The numbers are written from downwards to upwards. Let us understand this using an example. We can convert 30 to a binary number form in the following way:

Operations on Binary Numbers

Binary addition.

We add the binary numbers digit by digit and obtain the answer to the addition . It is important to remember the table below while adding the two binary numbers.

Binary Subtraction

The binary numbers are subtracted digit by digit and the answer is obtained. The table below is kept in mind while subtracting two binary numbers.

Binary Multiplication

The rules to multiply any two binary numbers is given as:

1's and 2's complement of a Binary Number

• 1's complement of a binary number is given by inverting the digits of the binary number. For example, 1's complement of \((101)_{2}\) is \((010)_{2}\).
• 2's complement of a binary number is given by inverting the digits of the binary number and adding 1 to the least significant bit. For example, 2's complement of \((111)_{2}\) is \((001)_{2}\) which is obtained by taking 1's complement of \((111)_{2}\) and adding 1 to the least significant bit.

Tips to Remember

Here are some important points to remember on binary number system:

• A binary number consists of two numbers 0s and 1s.
• Binary numbers are represented with 2 at their base. For example, \((101)_{2}\).
• Each digit in a binary number is referred to as a bit. For example, \((111)_{2}\) is a three-bit binary system.
• Binary addition is also referred to as the "AND" operation.
• Binary multiplication is also called an "OR" operation.
• The binary subtraction can be done by taking 1's and 2's complement of the binary number.
• The most significant digit in a binary number represents the sign of a binary number which is used to perform signed binary operations. 1 represents a negative sign while 0 represents a positive sign.

Topics Related to Binary Number System

• 32 in Binary
• 128 in Binary
• 255 in Binary
• Binary to Decimal Calculator
• Binary Calculator
• Binary to Decimal Formula

Examples on Binary Number System

Example 1: Convert the decimal number \((162)_{10}\) in binary.

Solution: In order to obtain the binary number for 162, we can divide it continuously by 2.

\(\therefore\) The binary number for \((162)_{10}\) is \((10100010)_{2}\).

Example 2: Convert the binary number \((100101)_{2}\) to decimal number.

Solution: The binary number \((100101)_{2}\) = (1 × 2 0 ) + (0 × 2 1 ) + (1 × 2 2 ) + (0 × 2 3 ) + (0 × 2 4 ) + (1 × 2 5 ) = \((37)_{10}\) \(\therefore\) The binary number \((100101)_{2}\) is \((37)_{10}\).

Example 3: Prove that binary number \((1000100)_{2}\) can be converted to the decimal number \((68)_{10}\).

Solution: The binary number \((1000100)_{2}\) = (0 × 2 0 ) + (0 × 2 1 ) + (1 × 2 2 ) + (0 × 2 3 ) + (0 × 2 4 ) + (0 × 2 5 ) + (1 × 2 6 ) = 64 + 4 = \((68)_{10}\) \(\therefore\) The binary number \((1000100)_{2}\) can be converted to the binary number \((68)_{10}\).

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Practice Questions on Binary Number System

Faqs on binary number system.

The system of representation in which a number can be expressed in terms of only two digits (0 and 1) with base 2 is known binary number system.

Why Binary Number System is Used in Computers?

Computer systems always process the given instructions using 0 or 1 as they either exist in on state or off state. This makes it easier for them to process the information faster.

What Does 10101 mean in Binary Number System?

10101 means 21 in a binary number system.

How Do You Convert a Decimal Number to Binary Number System?

A decimal number can be converted to a binary number system by dividing the given number by 2 until we obtain the quotient as 1. The numbers are written from downwards to upwards.

How to Convert a Binary Number to a Decimal Number?

We can convert a binary number to a decimal number expressing each digit as a product of the given number 1 or 0 to the respective power of 2. If a binary number has n digits, B = \((a)_{n-1}\).. \((a)_{3}\) \((a)_{2}\) \((a)_{1}\) \((a)_{0}\), the decimal number for it is given as, D = ( \((a)_{0}\)×2 0 ) + ( \((a)_{1}\)×2 1 ) + ( \((a)_{2}\)×2 2 ) + ...

What Does 1011 mean in Binary Number System?

1011 means 11 in binary number system as 1 × 2 0 + 1 × 2 1 + 0 × 2 2 + 1 × 2 3 = 1+ 2+ 0 + 8 = 11.

How Do You Write 13 in Binary Number System?

13 means 1101 in the binary number system. We continuously divide 13 by 2 until the quotient is 1. In this case, the steps that would be followed are:

• 13/2 give 6 as a quotient and 1 as a remainder .
• 6/2 give 3 as a quotient and 0 as a remainder.
• 3/2 give 1 as a quotient and 1 as a remainder.
• Now, the number is written from the bottom to the top as \((13)_{10}\) = \((1101)_{2}\).

Introduction to the Binary System

Get introduced to the base symbols and representation rules of the binary system.

Base digits

Place value, msb and lsb, test yourself, what is the binary system.

The decimal system leverages the fact that humans have ten fingers to count with. The binary system does something similar in the case of computers.

An essential component of computers is the transistor , which has two states, on and off. These are sometimes also referred to as high and low states, and correspond to the presence or absence of voltage. Each state can be represented using one of two symbols— 0 0 0 and 1 1 1 —resulting in a numbering system with base 2.

The binary system is a two-base system.

The base digits for the binary system are simple: 0 to represent a low state and 1 to represent a high state.

Like in the decimal system, the rules for representing a number in binary come from the place values of the digits. We know that the rule was that place values correspond to ascending powers of 10 10 10 from right to left.

Note : For the binary system, the rule is that—you guessed it—place values correspond to ascending powers of 2 2 2 from right to left.

Note: Recall that binary digits are referred to as bits.

MSB is the most significant bit . It is the leftmost bit of a binary number and the one with the largest place values. Similarly, the LSB , or least significant bit , is the rightmost bit and the one with a place value of 1 1 1 . The same symbol 1 in different places can have different values. For example, in 001 001 001 , the MSB is 0 0 0 , and in 100 100 100 , the MSB is 1 1 1 and vice versa for LSB. These terms make referring to binary numbers easier and can also be used to talk about groups of bits. For example, we can say “four least significant bits” to refer to the four right most bits in our number.

This is all well and good, but you might be thinking that it was lucky for us that we had five places to start with for the example above. So, how would we know how many places an arbitrary number would take up to be represented in binary?

For that, we need to think about the range of a set of bits.

Let’s think about how many different numbers we can represent with two bits.

We can represent four numbers from 0 — 3 0—3 0—3 . Let’s now see how many numbers we can represent with three bits.

Note: We already know that we can have four different numbers with two bits, which could each either be 0 0 0 or 1 1 1 . If we add a third bit, it could also either be 0 0 0 or 1 1 1 . So, we may extend each of the four two-bit binary numbers with a 0 0 0 on the left and then with a 1 1 1 on the left, giving 4 × 2 = 8 4\times2 = 8 4 × 2 = 8 different numbers. So, each bit contributes to doubling the number. In the following illustration, we are showing only the ones where the leftmost bit is 1 1 1 .

Let’s try to come up with an expression for the range of bits. This means we want an expression to calculate how many (and what) numbers we can represent using n n n bits.

Can we use the patterns we observed in our two examples to do this? Let’s see.

A single digit ( n = 1 ) (n = 1) ( n = 1 ) binary number can have two different values ( 0 0 0 or 1 1 1 ). When we take the number of bits to n = 2 n = 2 n = 2 , the number of possible values increases to 2 × 2 = 2 2 2 \times 2 = 2^2 2 × 2 = 2 2 . When we take the number of bits to three digits, the number of possible values increases to 2 × 2 × 2 = 2 3 2 \times 2 \times 2 = 2^3 2 × 2 × 2 = 2 3 . Notice that for n n n binary digits, we have all the k k k numbers available that we did for ( n − 1 ) (n-1) ( n − 1 ) digits. This happens when we turn the n n n th MSB digit off. Additionally, we have the option of turning the n n n th digit on, which gives k k k more possible numbers. This brings our total to 2 k 2k 2 k . Thus, with every increase in n n n , we are multiplying the possible combinations by 2 2 2 . We then have:

Note: Remember that one of these values is always 0 0 0 , so the range of numbers that we can represent is 0 0 0 to 2 n − 1 2^n-1 2 n − 1 .

How many numbers can you represent in binary using 6 6 6 digits?

Number Systems

The number system is a way to represent or express numbers.  You have heard of various types of number systems such as the whole numbers and the real numbers. But in the context of computers, we define other types of number systems. They are:

• The decimal number system
• The binary number system
• The octal number system and
• The hexadecimal number system

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Binary Number System (Base 2)

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Practice Questions

Q 1: How would you represent 10111 in the decimal number system?

Q 2: The LSB and MSB in the following number are: 1220

A) 1 & 0

B) 0 & 1

Ans: A) 0 & 1

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• History of the Binary Number System

Table of Contents

In this article, we will learn about the history of the binary number system, following different scholars and mathematicians and how they discovered and used the binary system over the years.

The  binary number system  has been the most impactful numerical system in the history of technological development. Over hundreds of years, discoveries within binary theory led to the invention of electric circuitry and, consequently, the first computer ENIAC, seventy years ago.

Who discovered binary code? The discovery of the binary system is mainly attributed to  Gottfried Wilhelm Leibniz in the 17th century.  In his paper “Explanation of Binary Arithmetic”, Leibniz proposed a binary digit system. However, earlier civilizations in Egypt and China used the binary system for rudimentary counting and arithmetic.

Numeric Systems

Before we get into the history of the binary numeral system, let’s take a more in-depth look at the different types of numeral systems used today.

The binary numeral system is a positional number system represented by only two symbols – 0 and 1. The radix of the binary number system is 2, and this means that only two numeric values can appear in any representation of a number. The binary system is used by computers and electronic devices to store data and perform computations.

The decimal system , which we use every day, works on a system where every value has its own unique symbol from 0 – 9. People have always used the decimal number system because it is straight to the point and easy to understand. There are different ways to represent this base-10 numeral system: the Hindu-Arabic numerals we use, Roman numerals, and Brahmi numerals.

Then, there is also the hexadecimal system. It is a combination of decimal digits and English letters (known as ASCII in computer science). It has 16 possible values – numbers from 0 – 9 and letters from A – F (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). Hexadecimal numbers are used in computer programming to represent URI’s and color codes. For example, the color white has a hex code of #FFFFFF and the color black has a hex code of #000000.

How was the binary system invented?

In this section, we shall go through the various stages in the invention of the binary numeral system.

As much as Leibniz invented binary code, it is necessary to understand that there were obscure uses of this numeral system before Leibniz published his papers.

The Egyptians

The Egyptians appeared to use a form of binary code in their “Horus-Eye” fractions. The Horus Eye fractions were used to measure and count grains and liquids in 2400 BCE.

The I Ching , or Book of Changes , from China is believed to have been supernaturally revealed to a mythical Chinese Emperor Fu Xi in the third century BCE. I Ching symbols like hexagrams and trigrams were incremented using binary progression. The I Ching concepts had many similarities with the binary system, so much so that Leibniz himself was interested in this system of numeration. He talked to a French Jesuit working in China at the time, and he was fascinated by the fact that this religious text used a binary concept to communicate messages.

Pingala was an Indian scholar who invented some semblance of the binary system in 200 BC. Pingala’s binary system had a large number of similarities with the modern binary system. His version of the binary system used short and long syllables to represent 0’s and 1’s. However, his positional notation began at 1, yet the modern one starts at 0. Also, Pingala’s binary representation incremented towards the left, while the modern binary system increments towards the right.

Thomas Harriot

In the 1600s, English mathematician and scientist Thomas Harriot experimented with binary digits. He seemed to know a lot about the binary system. His notes show that he used 0 and 1 in counting, addition, subtraction, multiplication, and division. He also converted the binary system into the decimal system and vice versa. However, there was no known application for the binary system at that time, so Thomas Harriot didn’t publish his work. 

Francis Bacon

In 1605, Francis Bacon invented Bacon’s Bilateral Cipher , which has similarities with the binary system. The cipher he created was used to encrypt messages so that only the sender and receiver could read the message. He used the letters ‘a’ and ‘b’ instead of 0 and 1 to form a binary code for every letter in a message. Here is a table showing Francis Bacon’s cipher. (Use our text to binary converter to see the similarities!)

Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz was a 17th-century German mathematician and philosopher who invented the binary number system. In his article Explication de l’Arithmétique Binaire , he proposed and developed a system that used only the binary digits 0 and 1 for arithmetic operations. He is known to have said of his binary system, “ These operations are so easy that we shall never have to guess or apply trial and error, as we must do in ordinary division. Nor do we need to learn anything by rote. “

Leibniz’s binary system used the binary notation of 0 and 1 to represent all the numbers in the decimal system. He avoided multiplication tables and created simple rules for calculations. He has been termed the father of binary code , and rightfully so because he created the current binary numeral system that is used in all modern computers and devices.

The binary system after Leibniz’s discovery

After Leibniz discovered the binary numeral system, it wasn’t specifically used in electronic computers and devices until other people experimented with it and developed systems that are still used today. Let’s take a look at some influential people who took Leibniz’s discoveries to the next level.

George Boole

George Boole was an English mathematician and philosopher who invented Boolean algebra in the 1950s. He realized that since his concept worked on a TRUE or FALSE basis, it could be combined with the on or off binary system to coordinate the switches in a digital circuit.

Claude Shannon

In 1937, Claude Shannon wrote his master’s thesis on digital circuit design theory using knowledge from Boolean algebra that could be used to power digital circuits. It was very influential in the field of computing, but it is not the only discovery he made. He invented information theory , which is the study of the quantification, storage, and communication of information. Claude Shannon has been termed the father of information theory.

George Stibitz

George Stibitz is considered one of the fathers of the modern digital computer. In the 1930s, he discovered how to use electromechanical relays as the switching element in Boolean logic digital circuits. He developed the Model K computer (K for kitchen, where he made it,) which was the first relay-based computer ever. It could perform only binary addition, but it was the first step in developing modern computers.

What was the binary system used for?

Today, the binary system is used to operate digital circuits in computers . However, ages before mathematicians like Leibniz and George Boole made discoveries to use binary code in computers, people used it for other purposes.

The binary system is not complex, and it only makes sense that early civilizations would use a simple numeral system to count and keep track of their property and finances. The ancient Egyptians used the Horus-Eye fraction binary system to measure their grains and liquids.

The Chinese used their I Ching binary system for religious purposes. The I Ching used binary code to arrange religious symbols into meaningful messages for its followers.

The binary system was used for rudimentary counting and arithmetic. However, Leibniz saw its potential for greatness. Consequently, George Boole discovered that by combining Boolean algebra with the binary system, he could control digital circuits. Claude Shannon used both Leibniz’s and Boole’s concepts to create the first modern computer, which he called Model K.   

The Future of the binary system

In this article, we have learned about the history of the binary system. The binary number system has served us well throughout the decades. However, one might ask the question: Where is the binary number system headed?

With quantum computing gaining traction in the computing world, the binary number system might become obsolete. However, right now, it is doing a great job helping us stay connected all over the world.  

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History of the Binary System

The Binary System of numeration is the simplest of all positional number systems . The base - or the radix - of the binary system is 2, which means that only two digits - 0 and 1 - may appear in a binary representation of any number. The binary system is of great help in the Nim -like games: Plainim , Nimble , Turning Turtles , Scoring , Northcott's game , etc. More importantly, the binary system underlies modern technology of electronic digital computers. Computer memory comprises small elements that may only be in two states - off/on - that are associated with digits 0 and 1. Such an element is said to represent one bit - bi nary dig it .

The first electronic computer - ENIAC which stood for Electronic Numerical Integrator And Calculator - was built in 1946 at the University of Pennsylvania, but the invention of the binary system dates almost 3 centuries back. Gottfried Wilhelm Leibniz (1646-1716), the co-inventor of Calculus , published his invention in 1701 in the paper Essay d'une nouvelle science des nombres that was submitted to the Paris Academy to mark his election to the Academy. However the actual discovery occurred more than 20 years earlier.

According to the Oxford Encyclopedic Dictionary (see Earliest Known Uses of Some of the Words of Mathematics ), an entry BINARY ARITHMETIC first appeared in English in 1796 in A Mathematical and Philosophical Dictionary .

Binary numbers are written with only two symbols - 0 and 1. For example, a = 1101. Since symbols 0 and 1 are also a part of the decimal system and in fact of a positional system with any base, there's an ambiguity as to what 1101 actually stands for. To avoid confusion, the base is often written explicitly, like in a = (1101) 2 or b = (1101) 10 . In the decimal system, 1101 is interpreted as 1 thousand 1 hundred 1 , which is just a sum of powers of 10 with coefficients that are the digits of the number. More accurately,

(1101) 10 = 1·10 3 + 1·10 2 + 0·10 + 1

To represent numbers, the decimal system uses the powers of 10, whereas the binary system uses in a similar manner the powers of 2.

(1101) 2 = 1·2 3 + 1·2 2 + 0·2 + 1

The numbers are different. In fact,

(1101) 2 = 8 + 4 + 1 = 13    ( = (13) 10 .)

There are several problems with using more than one number system at the same time. Should we read (1101) 2 as 1 thousand 1 hundred 1 in binary ? Or, after some mental calculations, just 13 without mentioning the base? The latter possibility is overtaxing and unreasonable: why to use a system other than the decimal in writing while depending on the decimal in speech? The former is inappropriate altogether for etymological reasons. We might say thousand to indicate a 1 in the fourth position from the right regardless of the base of the system in use, but this would conflict with the etymology of the word thousand , and the same is true of the word hundred . Both are related to the base 10 and no other.

In Words of Mathematics we find the following entries:

hundred (numeral): a native English compound. The first element, hund , actually means "ten." It comes from dekt-tom , an extension of the more basic Indo-European root dekm "ten." The second element is from the Old English rad "number", so that hundred means literally the "tens-number" in the sense that it is ten times ten.

thousand (numeral): actually an English compound, thus-hund . The first component is related to English thumb and thigh , and means "swollen, large." The Indo-European root is teu- "to swell." Related borrowings from Latin are tumor and tumulus . The second component is the root found in hundred (q.v.) , which is based on the Indo-European root dekm- "ten." The literal meaning of thousand is "a swollen or big hundred" because it is ten times a hundred.

So how does one read (1101) 2 ? In practice, the not-so-glamorous "one one zero one" does a reasonably good job. One adds the word "binary" if the meaning is not clear from the context. This is probably close to the Ancients' usage. Just think of how the Romans pronounced, say MCMLXXXII?

Now let me ask a couple of deceptively simple questions. Is it true that every number has a binary representation? And if so, is the binary representation of a number unique?

Here's one possible answer. For a given number, there exists an algorithm that outputs its binary representation. Therefore every number has a binary representation. Since the algorithm is reversible, the binary representation defines the number uniquely. (The algorithm works for integers. Another one works for fractions.)

There is a problem though. The algorithm assumes that the given number has been already somehow represented, so that it receives one representation of the number and outputs another. If the original number was decimal, the algorithm performs conversion between its decimal and binary representations. It appears that the answer we gave in the preceding paragraph is conditional: if a number has a decimal representation, it also has a binary representation. If the former is unique, so is the latter.

However, does every number have a decimal representation? To be more specific, does every counting number have a decimal representation? This question is either silly or plain artificial. For is it not how we count the numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and so on. Who would doubt that in this manner we count all numbers? This is in fact the definition of counting numbers ( The Penguin Dictionary of Mathematics ):

counting number a number used in counting objects; i.e. one of the set of positive integers: 1, 2, 3, 4, an so on.

The sample sequence is short, but of course the intention is to the sequence of decimal representations: 1, 2, 3, 4, ..., 10, 11, 12, ... We count the numbers sequentially and, as we go along, we give them names according to certain rules. Those rules are the basis of the positional (decimal) system representation:

With the relevant rule noted in parentheses, let's count and see how the rules apply: 1, 2 (#2), 3 (#2), ..., 8 (#2), 9 (#2), 10 (##3-4), 11 (#2), ..., 18 (#2), 19 (#2), 20 (#3), ..., 98 (#2), 99 (#2), 100 (#3-4, one recursion), ...

The question of what a (counting) number is is quite delicate. Numbers can be defined axiomatically , which guarantees their existence independent of any naming convention. Numbers may also be thought of as collections of drum beats we produce while counting: one drum beat per count. Naming them was a great human invention. Naming them according to a positional system of numeration was probably a single most important mathematical achievement over the space of some 1000 years .

Whether one may skip a number while tapping a drum may deserve a philosophical discussion. I assume this is not possible. Rules 1-4 guarantee that all possible (decimal) number names will eventually be assigned in proper order. Going one step further with this line of reasoning, I claim that any positional numeration is exhaustive in the sense that any (counting) number has a unique representation in every base and any such representation corresponds to a certain number. Rules 1-4 must of cause be adapted to a specific base of numeration. In particular, the naming rules for the binary system appear as

The binary counting then goes thus: 1, 10 (##3-4), 11 (#2), 100 (##3-4, one recursion), 101 (#2), ..., 111 (#2), 1000 (##3-4, 2 recursions), ...

The foregoing discussion presents a longwinded argument to the effect that there is not that much difference between the decimal and the binary systems. Decimal representations are shorter than their binary counterparts, but, as far as the counting process is concerned, the name assignment follows essentially the same rules.

Binary representation, just because it only uses two digits has an interesting interpretation. Binary representation of a number is a sum of powers of 2. A power of two is included into the sum if the corresponding digit in the representation is 1. For example,

(13) 10 = (1101) 2 = 2 3 + 2 2 + 2 0 .

The fact that every number has a unique binary representation tells us that every number can be represented in a unique way as a sum of powers of 2. I wish to give an independent proof due to L. Euler (1707-1783) [ Dunham , p 166] of the latter result.

Euler was a master of infinite series and products . Their theory have been developed in the 19th century, but Euler used them with great skill a century earlier to obtain many remarkable results. So, here's one example.

Let P(x) = (1 + x)(1 + x 2 )(1 + x 4 )(1 + x 8 )..., which is an infinite product. Multiplying out the terms of the product results in an infinite series:

(1 + x)(1 + x 2 )(1 + x 4 )(1 + x 8 )... = 1 + αx + βx 2 + γx 3 + δx 4 + ...

where the coefficients α, β, γ, δ, ... are yet to be determined. Note that

P(x)/(1 + x) = (1 + x 2 )(1 + x 4 )(1 + x 8 )...

which is just P(x²) . In other words,

P(x) = (1 + x)(1 + αx 2 + βx 4 + γx 6 + δx 8 + ...)

Cross-multiplication yields another identity

P(x) = 1 + x + αx 2 + αx 3 + βx 4 + βx 5 + γx 6 + γx 7 + ...

Compare this with the original expansion P(x) = 1 + αx + βx 2 + γx 3 + δx 4 + ... As with finite polynomials, if two series are equal, their coefficients must coincide termwise. Wherefrom we obtain, α = 1, β = α, γ = α δ = β ε = β, ... which means all the coefficients in the expansion of P(x) are equal to 1. Therefore,

(1 + x)(1 + x 2 )(1 + x 4 )(1 + x 8 )... = 1 + x + x 2 + x 3 + x 4 + x 5 + ...

But what is the meaning of coefficients α, β, γ δ, ε, ...? Each tells us in how many ways the corresponding term (a power of x ) can be obtained as the product of powers of x with exponents 1, 2, 4, 8, 16, ... Since, x a x b = x a + b and all the coefficients were found to equal 1, this is the same as saying that every (counting) number - exponents on the right - have a unique representation as a sum of powers of 2.

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• Math Article

Number System

The number system or the numeral system is the system of naming or representing numbers. We know that a number is a mathematical value that helps to count or measure objects and it helps in performing various mathematical calculations. There are different types of number systems in Maths like decimal number system, binary number system, octal number system, and hexadecimal number system. In this article, we are going to learn what is a number system in Maths, different types, and conversion procedures with many number system examples in detail .  Also, check mathematics for grade 12 here.

What is Number System in Maths?

A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, multiplication and division.

The value of any digit in a number can be determined by:

• Its position in the number
• The base of the number system

Before discussing the different types of number system examples, first, let us discuss what is a number?

What is a Number?

A number is a mathematical value used for counting or measuring or labelling objects. Numbers are used to performing arithmetic calculations.  Examples of numbers are natural numbers, whole numbers, rational and irrational numbers, etc. 0 is also a number that represents a null value. 

A number has many other variations such as even and odd numbers, prime and composite numbers. Even and odd terms are used when a number is divisible by 2 or not, whereas prime and composite differentiate between the numbers that have only two factors and more than two factors, respectively.

In a number system, these numbers are used as digits. 0 and 1 are the most common digits in the number system, that are used to represent binary numbers. On the other hand, 0 to 9 digits are also used for other number systems. Let us learn here the types of number systems.

Types of Number Systems

There are various types of number systems in mathematics. The four most common number system types are:

• Decimal number system (Base- 10)
• Binary number system (Base- 2)
• Octal number system (Base-8)
• Hexadecimal number system (Base- 16)

Now, let us discuss the different types of number systems with examples.

Decimal Number System (Base 10 Number System)

The decimal number system has a base of 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands and so on. This system is expressed in decimal numbers . Every position shows a particular power of the base (10).

Example of Decimal Number System:

The decimal number 1457 consists of the digit 7 in the units position, 5 in the tens place, 4 in the hundreds position, and 1 in the thousands place whose value can be written as:

(1×10 3 ) + (4×10 2 ) + (5×10 1 ) + (7×10 0 )

(1×1000) + (4×100) + (5×10) + (7×1)

1000 + 400 + 50 + 7

Binary Number System (Base 2 Number System)

The base 2 number system is also known as the Binary number system wherein, only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under this system are known as binary numbers which are the combination of 0 and 1. For example, 110101 is a binary number.

We can convert any system into binary and vice versa.

Write (14) 10 as a binary number.

Base 2 Number System Example

∴ (14) 10 = 1110 2

Octal Number System (Base 8 Number System)

In the octal number system , the base is 8 and it uses numbers from 0 to 7 to represent numbers. Octal numbers are commonly used in computer applications. Converting an octal number to decimal is the same as decimal conversion and is explained below using an example.

Example: Convert 215 8 into decimal.

215 8 = 2 × 8 2 + 1 × 8 1 + 5 × 8 0

= 2 × 64 + 1 × 8 + 5 × 1

= 128 + 8 + 5

Hexadecimal Number System (Base 16 Number System)

In the hexadecimal system, numbers are written or represented with base 16. In the hexadecimal system, the numbers are first represented just like in the decimal system, i.e. from 0 to 9. Then, the numbers are represented using the alphabet from A to F. The below-given table shows the representation of numbers in the hexadecimal number system .

Number System Chart

In the number system chart, the base values and the digits of different number systems can be found. Below is the chart of the numeral system.

Number System Conversion

Numbers can be represented in any of the number system categories like binary, decimal, hexadecimal, etc. Also, any number which is represented in any of the number system types can be easily converted to another. Check the detailed lesson on the conversions of number systems to learn how to convert numbers in decimal to binary and vice versa, hexadecimal to binary and vice versa, and octal to binary and vice versa using various examples.

With the help of the different conversion procedures explained above, now let us discuss in brief about the conversion of one number system to the other number system by taking a random number.

Assume the number 349. Thus, the number 349 in different number systems is as follows:

The number 349 in the binary number system is 101011101

The number 349 in the decimal number system is 349.

The number 349 in the octal number system is 535.

The number 349 in the hexadecimal number system is 15D

Number System Solved Examples

Convert (1056) 16 to an octal number.

Given, 1056 16 is a hex number.

First we need to convert the given hexadecimal number into decimal number

= 1 × 16 3 + 0 × 16 2 + 5 × 16 1 + 6 × 16 0

= 4096 + 0 + 80 + 6

= (4182) 10

Now we will convert this decimal number to the required octal number by repetitively dividing by 8.

Therefore, taking the value of the remainder from bottom to top, we get;

(4182) 10 = (10126) 8

Therefore, 

(1056) 16 = (10126) 8

Convert (1001001100) 2 to a decimal number.

(1001001100) 2

= 1 × 2 9 + 0 × 2 8 + 0 × 2 7 + 1 × 2 6 + 0 × 2 5 + 0 × 2 4 + 1 × 2 3 + 1 × 2 2 + 0 × 2 1 + 0 × 2 0

= 512 + 64 + 8 + 4

Convert 10101 2 into an octal number.

 10101 2 is the binary number

We can write the given binary number as,

Now as we know, in the octal number system,

Therefore, the required octal number is (25) 8

Convert hexadecimal 2C to decimal number.

We need to convert 2C 16  into binary numbers first.

2C → 00101100

Now convert 00101100 2 into a decimal number.

101100 = 1 × 2 5  + 0 × 2 4 + 1 × 2 3  + 1 × 2 2  + 0 × 2 1 + 0 × 2 0

= 32 + 8 + 4

Video Lesson on Numeral System

Number System Questions

• Convert (242) 10 into hexadecimal. [ Answer: (F2) 16 ]
• Convert 0.52 into an octal number. [ Answer: 4121]
• Subtract 1101 2 and 1010 2 . [ Answer: 0010]
• Represent 5C6 in decimal. [ Answer:  1478]
• Represent binary number 1.1 in decimal. [ Answer: 1.5]

Also Check: Binary Operations

Computer Numeral System (Number System in Computers)

When we type any letter or word, the computer translates them into numbers since computers can understand only numbers. A computer can understand only a few symbols called digits and these symbols describe different values depending on the position they hold in the number. In general, the binary number system is used in computers. However, the octal, decimal and hexadecimal systems are also used sometimes.

More Topics Related to Number Systems

Frequently asked questions on number system, what is number system and its types.

The number system is simply a system to represent or express numbers. There are various types of number systems and the most commonly used ones are decimal number system, binary number system, octal number system, and hexadecimal number system.

Why is the Number System Important?

The number system helps to represent numbers in a small symbol set. Computers, in general, use binary numbers 0 and 1 to keep the calculations simple and to keep the amount of necessary circuitry less, which results in the least amount of space, energy consumption and cost.

What is Base 1 Number System Called?

The base 1 number system is called the unary numeral system and is the simplest numeral system to represent natural numbers.

What is the equivalent binary number for the decimal number 43?

How to convert 30 8 into a decimal number.

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1.3: Examples of Binary Numbers

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• Page ID 96029

• Jeffrey R. Chasnov
• Hong Kong University of Science and Technology

\(\begin{array}{cc}\text { Decimal } & \text { Binary } \\ 1 & 1 \\ 2 & 10 \\ 3 & 11 \\ 4 & 100 \\ 0.5 & 0.1 \\ 1.5 & 1.1\end{array}\)

Essay @ the Number System | Data Representation | Information Technology

Are you looking for an essay on the ‘Number System’? Find paragraphs, long and short essays on the ‘Number System’ especially written for school and college students.

Essay on the Number System

Essay Contents:

• Essay on the Binary Coded Decimal

Essay # 1. The Binary Number System:

In mid-1940s John von Neumann along with H.H. Goldstein and A.W. Buifks proposed a new number system using two digits called Binary Number System and it readily became the basic number system for computer operations.

A number system using two digits, under the basic concept, will have a base (radix) of two and its highest digit will be [Number of Digits in the System minus 1] or [2-1] = 1; the other digit being 0. Thus, the Binary Number System has two digits, 0 and 1, 0 representing the off-position and 1 representing the on-position of a two state device—the 0 and 1 are called Binary digits or Bits in short.

But since, a single bit can represent only two different positions, so the switches are always used in combinations to give larger number of options. When 4 bits, called nibble, are used together we can get 2 4 = 16 different combinations; with 8 bits, called a byte, we get 2 8 = 256 combinations; and with 16 bits, called a word, we get 2 16 = 65,536 positions. Incidentally, a kilo in binary system is 2 10 = 1024, and not 1000.

Since writing 16 or 32 bits of 0s and Is is a bit cumbersome, difficult to remember, with chances of mistakes being high, a short-hand version called Hexadecimal Number System has been developed, where 4 binary digits are represented by one digit of the hexadecimal system—which uses 16 digits : 0 to 9, and A to F, with A being 10 and F being 15, the others coming in between.

As far as the binary and the decimal systems are concerned, the digits 0 and 1 are common in both the systems, but there after they differ, as in the binary system the highest digit is 1 and in the decimal system it is 9. To get the value of two in the decimal system, we add 1 with 1 to get 2.

Similarly, in binary system we will have to add 1 with 1—but we cannot write 2, as we do in the decimal system. What is the way out? In the decimal system, when we add 1 to the highest digit of 9, we write 0 in unit’s place and carry over 1 to the ten’s place, the value of the digit changing to 10.

Thus we have:

Essay # 4 . Binary Coded Decimal:

No matter how scientific and useful a number system is, we always feel a bit comfortable working with the decimal number system. Hence, a number of code systems have been developed to integrate the two number systems of decimal and binary—so that we operate in decimal system and the computer continues with the binary system, the codes facilitating conversion from one form to the other.

These codes are called Binary Coded Decimals [BCD], A large number of such codes exist. Under these systems, individual decimal digits are converted to equivalent binary form, instead of the total decimal value being converted to its binary equivalent. The most popular one is called the 8-4-2-1 code.

In this system, each decimal number is expressed in binary form using its equivalent binary digit using 4 bits, like 0001 for 1, 0100 for 4, 1001 for 9. The weights given to the bit positions are 8, 4, 2, and 1, respectively and so it is called 8-4-2-1 code.

The decimal 386 in binary form using 8-4-2-1 code would be as: