Binary Representation in Digital Electronics

Digital electronics systems use binary numbers to understand and process data and instructions. The representation of data and information in binary number system is referred to as binary representation. It is a crucial concept in digital electronics that allows for the implementation and operation of digital devices and systems like computers, smartphones, data communication systems, and more.

In this article, we will explore different types of binary representations along with their basic description. But before that let’s have look into the basics of binary number system.

What is Binary Number System?

Binary number system is a base-2 number system which has two digits, 0 and 1, to represent all the possible values. Each digit of the binary number system is termed as a bit or binary digit.

In digital electronic systems, the binary number system acts as the backbone, and is used to represent two possible states, i.e. ON and OFF, of the system. In digital electronics, 0 represents a low voltage state, and 1 represents a high voltage state.

In any given binary number, there are two important terms namely, MSB (Most Significant Bit) and LSB (Least Significant Bit). For example, consider a binary number (1001101)2, here, the rightmost 1 is the MSB and the leftmost 1 is the LSB.

What is Binary Representation in Digital Electronics?

In digital electronics, the way of expressing information in the form of binary numbers, i.e. as a string of binary 0s and 1s is referred to as binary representation. Binary representation is an integral part of any digital system like computers, microprocessors, logic circuits, microcontrollers, and more. It provides an easier method to implement and operate digital systems.

Binary representation involves a positional notation, i.e. each bit in a binary information specifies a power of 2. For example, consider a piece of binary information (10011101)₂, then in this binary number, each bit is associated with a power of 2 as follows −

1	0	0	1	1	1	0	1
27	26	25	24	23	22	21	20

We can obtain the decimal value of this given binary number simply by multiplying its each bit with the corresponding power of 2 and adding the products.

For example,

(1 × 2⁷) + (0 × 2⁶) + (0 × 2⁵) + (1 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)

= 128 + 0 + 0 + 16 + 8 + 4 + 0 + 1 = (157)₁₀

Therefore, the 157 in decimal has a binary representation “10011101” in digital electronics.

Overall, binary representation is a very important concept in digital electronics, as it plays a vital in various operations like data storage, data processing and manipulation, arithmetic and logical operations, data transmission and communication, etc. In simple words, we can state that binary representation of data and information forms the foundation of digital systems and operations.

Types of Binary Representations in Digital Electronics

In digital electronics, we generally use the following two types of binary representations −

• Unsigned Binary Representation

• Signed Binary Representation

Let’s discuss both of these binary representations in detail.

Unsigned Binary Representation

Unsigned binary representation is a type of binary representation in which all the bits specify only the magnitude of the number. This binary representation is mainly used to represent only positive numbers and zero. Thus, the unsigned binary representation can be used for non-negative numbers.

For example, consider a decimal number 115. Then, its unsigned binary representation will be,

(115)₁₀ = (1110011)₂

In this representation, there are 7 bits and all the bits are representing the magnitude of the decimal number 115.

Signed Binary Representation

Signed binary representation is another type of binary representation used in digital electronics. This representation provides a way to represent positive, zero, and negative numbers.

In signed binary representation, the leftmost bit, called most significant bit (MSB), specifies the sign of the number. The remaining bits represent the magnitude or value of the number.

If the MSB is 1, then the given number is negative, and if it is 0, the given number is positive.

Singed binary representation is further classified into the following three types namely,

• Sign-Magnitude Representation

• 1’s Complement Representation

• 2’s Complement Representation

Sign-Magnitude Representation

In the sign-magnitude representation, the leftmost bit (MSB) is used to specify the sign of the number and the remaining bits specify the magnitude. Hence, the sign-magnitude representation of an unsigned binary number can be obtained just by added a sign bit at the leftmost side of the number.

If this sign bit is 1, then the given number is negative, and if the sign bit is 0, the given number will be positive.

For example, consider a positive decimal number +115. The magnitude of this number is 115 in decimal and 1110011 in binary. It has 7 bits that represent the magnitude. Since the given number is positive, then we have to include a sign bit 0 on leftmost side of the magnitude bits, i.e.

(+115)₁₀ = (0 1110011)₂

Therefore, the sign-magnitude binary representation of +115 is 0 1110011.

Similarly, the sign-magnitude binary representation of -115 is obtained by including a sign-bit 1 on leftmost side of the magnitude, i.e.,

(-115)₁₀ = (1 1110011)₂

1’s Complement Representation

The 1’s complement binary representation is another way of representing the signed-binary numbers. In 1’s complement representation, the leftmost bit is used as the sign bit, where 0 represents a positive sign and 1 represents a negative sign.

The 1’s complement of a binary number is obtained by inverting all the bits of the singed binary number. In digital electronics, the 1’s complement representation simplifies the subtraction operation of binary numbers.

For example, consider a negative decimal number -115. The magnitude of this number is 115 in decimal. The signed binary representation of 115 is 01110011. It has 8 bits, where the MSB bit is a 0, representing a positive sign. The 1’s complement of 0 is 1 and vice-versa. Therefore, the 1’s complement of 01110011 will be,

(10001100)₂ = (-115)₁₀

Here, the MSB is 1, representing the negative sign. Therefore, 10001100 is the 1’s complement representation of the number -115.

2’s Complement Representation

The 2’s complement representation is the most widely used signed binary representation in digital electronics. This representation also simplifies the binary subtraction. Similar to any other signed binary representation, the 2’s complement representation also has a sign bit on the left most side of the number.

The 2’s complement of a binary number is determined by taking its 1’s complement and then adding 1.

For example, consider a negative number -115 in decimal. Then, its 1’s complement is,

(-115)₁₀ = (10001100)₂

2’s complement of (-115)₁₀ = 1’s complement of (-115)₁₀ + 1

= 10001100 + 1 = 10001101

Hence, the 2’s complement representation of (-115)₁₀ is (10001101)₂.

Conclusion

Binary representation is a crucial concept in digital electronics used to implement digital systems to perform various operations like data storage, arithmetic and logical operations, data transfer, etc. Binary representation of information reduces the circuit complexity and cost of a digital system. Overall, the binary representation forms the foundation of digital electronics.