In this post, we will do a quick comparison of decimal vs binary number systems, highlighting all the similarities and differences between the two number systems for better understanding of their use case.

We are familiar with the decimal number system or the decimal numeration system since we use it every day for paying bills, grocery shopping, bank transactions, etc.,. In the decimal number system, we represent or express a given decimal number in Units, tens, hundreds, thousands, etc.,

On the other hand, our computers and other digital devices which we use every day make use of the binary number system to communicate, perform arithmetic and logical operations.

The decimal number uses 10 digits to represent different values whereas the Binary number uses only 2 digits (0 & 1) to represent any binary number.

We will further advance the discussion, and see what are the similarities and differences between the binary and decimal systems.

## Similarities between Binary and Decimal systems

Firstly, the binary and decimal systems can represent or express numbers with both integer parts and fractional parts equally. So any given number, for example, the number 14, in decimal system it is (14)_{10} and in the binary system it is expressed as (1110)_{2}. Here, subscript 10 & 2 are the radix of respective number systems.

Another important similarity between the binary and decimal system is that the value of a given digit (in a decimal number) or a given bit (in binary number) is determined by the position of the decimal digit from the decimal point or position of binary bit from the binary point. The farthest digit or binary bit from the decimal/binary point has the largest weight.

Example for the decimal system: 623.54 – the left-most digit starting from the decimal point (1) has the largest weight and the rightmost digit from the decimal point has the lowest weight or value(4).

Example for the binary system: 101.010 – the leftmost bit starting from the binary point(1) has the largest weight and the rightmost bit starting from the binary point towards the right has the least weight or value(0).

In addition to this, a similarity between binary and decimal systems worth mentioning is that the binary system is also capable of performing all the arithmetic operations like addition, subtraction, multiplication, and division just like we do in the decimal number system.

Let us list out all the similarities between binary and decimal systems in a tabular form to remember them easily.

Binary System | Decimal System |
---|---|

The binary system can represent integer binary numbers as well as fractional binary numbers. | The decimal system can also represent decimal numbers containing both integer and fraction components. |

The value or weight of a bit depends on the position of the bit from the radix point. | The value/weight of a digit in a given decimal number also depends on the position of the digit from the decimal point. |

Example of binary number with integer and fractional part: 101.010 (Five and Twenty Five hundredths) | Example of Decimal number: 5.25 (Five and Twenty Five hundredths) |

Representation of Decimal number & decimal number fraction. | Representation of Binary number and binary fraction. |

All arithmetic operations like addition, subtraction, multiplication & division can be performed on binary numbers. | Addition, subtraction, multiplication & division arithmetic operations can be performed on decimal numbers. |

## Differences between Binary and Decimal Systems

We see that Binary system and decimal system are nothing but two different number systems or numeration systems based on the number of digits used by each system. Both binary and decimal systems differ at various levels.

The fundamental difference between binary and decimal systems is the number of digits used to represent any given number. The binary system uses only 2 digits (0s & 1s) to represent binary numbers while the decimal system uses 10 digits from 0 to 9 (0,1,2,3,4,5,6,7,8,9) to represent any decimal number integer or fraction.

The binary system which uses only 2 digits is also referred to as base-two system, while the decimal system utilizing 10 digits is known as base-ten system.

The units in the binary system (0s and 1s) are known as bits which is more familiar in the computer world. The units in the decimal number system (0 to 9) are called digits.

The radix used in the binary system is 2, the radix used in the decimal system is 10. In other words, we can also say that in the binary system the values or weights are expressed as the power of 2 and in the decimal system, the values are expressed as the power of 10.

The radix point in the binary system is called a Binary Point and the radix point in the decimal system is referred to as Decimal Point.

In a binary system, the left-most bit which has the largest weight is referred to as the Most Significant Bit(MSB), and the rightmost bit from the binary point which has the least weight is referred to as Lease Significant Bit (LSB). In the decimal system, the left-most digit with the highest weight is called a Most significant digit (MSD) and the rightmost digit with the least weight is called a Least Significant Digit (LSD).

It is easier to learn the various arithmetic and logical operations in the binary system when compared to the decimal system since the binary system has only 2 digits (0 and 1).

On the contrary, it becomes tedious to remember and work with large binary numbers as compared to large decimal numbers.

Example: 65536 in decimal system is easy to remember than 10000000000000000 in binary system.

Decimal Numbers | Binary Numbers |
---|---|

Decimal numbers consist of combination ten digits or values from 0 to 9 (0,1,2,3,4,5,6,7,8,9). | Binary numbers are expressed or represented by the combination of only 2 digits or values 0 (zero) and 1 (one). |

The Radix for decimal number is 10, i.e., all the digits in a decimal number are expressed in terms of powers of 10. | The Radix for binary number is 2, i.e., all the digits or values in a binary number are expressed in terms of powers of 2. |

Example of decimal number: 25 (Twenty Five) | Example of Binary number: 11001 (Twenty Five) |

Representation of Decimal number & decimal number fraction. | Representation of Binary number and binary fraction. |

It is easier to represent and work with large decimal numbers. | It is pretty difficult to remember and work with large binary numbers as the number of bits required to represent increases drastically as the numbers grow. |

The radix point is called a decimal point. | The radix point is referred to as a binary point. |

In Decimal number system, we have Units, Tenths, Hundredths, Thousandths commonly referred to as decimal units. | In Binary number system, we have bits. Group of 4 bits is called a nibble, Group of 8 bits is called a Byte Group of 16 Bits is called a word. |

Applications Most common and used in everyday life. Used in all Financial transactions, banking operations. | Applications Most widely used for performing arithmetic and logical operations in all digital circuits. |

Note : Even though decimal numbers and binary numbers are different, the set of rules for addition, subtraction, multiplication and division applicable to decimal numbers is also applicable to binary numbers. But the way the these operations are performed is slightly different which we will learn in detail later.