Mastering the Number System: A Step-by-Step Tutorial
Number system (Definition, Type, Conversion & Example)
Number system (Definition, Type, Conversion & Example)
A number system defines how a number can be represented In this blog, we will discuss the different types of number systems and how to convert between them. A number can be represented differently in Different systems. Each number system is associated with a base or radix The most commonly used number system is the decimal system, also known as the base-10 system. The decimal number system is said to be of base or radix 10.A number in base r contains r digits 0,1,2,...,r-1The number is represented by the following ways (a)r
In the decimal system, there are 10 digits (0-9) used to represent numbers. This system is widely used in our everyday lives, as it is easy for humans to understand and work with.
Another important number system is the binary system, also known as the base-2 system. In the binary system, there are only 2 digits (0 and 1) used to represent numbers. This system is widely used in computers and digital electronics because it is easy for machines to understand and work with.
We can also convert numbers from one system to another. For example, we can convert a decimal number to a binary number by dividing the decimal number by 2 and keeping track of the remainder. We can then repeat this process until we have no more quotient. The remainders will then be arranged in the reverse order to get the binary equivalent of the decimal number.
We can also convert a binary number to a decimal number by multiplying each digit by 2 raised to the power of its position and adding all the products. For example, if we want to convert the binary number 1011 to decimal, we would calculate 1 x 2^3 + 0 x 2^2 + 1 x 2^1 + 1 x 2^0 = 8 + 2 + 1 = 11.
Different Type of Number Systems
Binary number System
Decimal number System
Octal number System
Hexadecimal Number System
Decimal Number System
The word decimal is derived from the Latin root decem (ten). In this system the base b = 10 and we use ten symbols.
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
The decimal number system, also known as the base-10 system, is a positional numeral system with 10 as its base. It uses the digits 0 through 9 to represent numbers. In a decimal number, each digit's position represents a power of 10, with the rightmost digit representing 10^0 (1), the next digit to the left representing 10^1 (10), the next representing 10^2 (100), and so on.
For example, the decimal number 237 is interpreted as follows:
The digit 7 is in the "ones" place, representing 7 * 10^0 = 7.
The digit 3 is in the "tens" place, representing 3 * 10^1 = 30.
The digit 2 is in the "hundreds" place, representing 2 * 10^2 = 200.
The decimal system is widely used in everyday life for counting, arithmetic operations, and representing real-world quantities. It contrasts with other numeral systems, such as binary (base-2), octal (base-8), and hexadecimal (base-16), which have different bases and use different sets of digits.
Binary Number System
The word binary is derived from the Latin root bini(or two by two).In this system the base b = 2 and we use only two symbols, S = {0, 1} The symbols in this system are often referred to as binary digits or bits. Binary numbers are numbers expressed in the base-2 numeral system, which uses only two symbols: 0 and 1. It is a fundamental system in digital electronics and computing because it aligns well with the binary nature of electronic switches, which can be in one of two states (on or off).
In binary, each digit's position represents a power of 2, with the rightmost digit representing 2^0 (1), the next digit to the left representing 2^1 (2), the next representing 2^2 (4), and so on.
For example, the binary number 1011 is interpreted as follows:
The rightmost digit (1) is in the 2^0 place, representing 1 * 2^0 = 1.
The next digit to the left (1) is in the 2^1 place, representing 1 * 2^1 = 2.
The next digit (0) is in the 2^2 place, representing 0 * 2^2 = 0.
The leftmost digit (1) is in the 2^3 place, representing 1 * 2^3 = 8.
Therefore, the binary number 1011 is equivalent to the decimal number 1 + 2 + 8 = 11.
Binary is essential in digital systems because it allows for straightforward representation of information using on/off states, making it efficient for computer processing and storage.
Octal Number System
The word octal is derived from the Latin root octo (eight).In this system the base b = 8 and we use eight symbols to represent a number. The set of symbols is:
S = {0, 1, 2, 3, 4, 5, 6, 7}
The octal number system, also known as base-8, is a positional numeral system that uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each digit's position represents a power of 8, with the rightmost digit representing 8^0 (1), the next digit to the left representing 8^1 (8), the next representing 8^2 (64), and so on.
For example, the octal number 237 is interpreted as follows:
The rightmost digit (7) is in the 8^0 place, representing 7 * 8^0 = 7.
The next digit to the left (3) is in the 8^1 place, representing 3 * 8^1 = 24.
The leftmost digit (2) is in the 8^2 place, representing 2 * 8^2 = 128.
Therefore, the octal number 237 is equivalent to the decimal number 7 + 24 + 128 = 159.
While octal was historically used in computing, it has been largely replaced by hexadecimal (base-16) due to the convenience of hexadecimal in representing binary data in a more compact form. Octal is still occasionally encountered, but it is not as prevalent as binary or hexadecimal in modern computing.
Hexadecimal Number System
The word hexadecimal is derived from the Greek root hex (six) and the Latin root decem (ten).
In this system the base b = 16 and we use sixteen symbols to represent a number.
The set of symbols is
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
The symbols A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively.
The symbols in this system are often referred to as hexadecimal digits.
The hexadecimal number system, often simply referred to as hex, is a base-16 numeral system. It uses sixteen digits: 0-9 for values 0 to 9, and A-F (or a-f) for values 10 to 15. Each digit's position represents a power of 16, with the rightmost digit representing 16^0 (1), the next digit to the left representing 16^1 (16), the next representing 16^2 (256), and so on.
The hexadecimal system is commonly used in computing and digital electronics because it provides a more concise representation of binary-coded values. Each hex digit corresponds to four binary digits (bits), making it easier to work with binary data in a more human-readable format.
For example, the hexadecimal number 1A3 is interpreted as follows:
The rightmost digit (3) is in the 16^0 place, representing 3 * 16^0 = 3.
The next digit to the left (A, which is 10 in decimal) is in the 16^1 place, representing 10 * 16^1 = 160.
The leftmost digit (1) is in the 16^2 place, representing 1 * 16^2 = 256.
Therefore, the hexadecimal number 1A3 is equivalent to the decimal number 3 + 160 + 256 = 419.
Hexadecimal is commonly used in programming, memory addresses, and representing color codes in HTML and other digital graphics applications.
A number system defines how a number can be represented.
What is Binary number system?
Binary numbers are numbers expressed in the base-2 numeral system, which uses only two symbols: 0 and 1
What is decimal number system ?
The word decimal is derived from the Latin root decem (ten). In this system the base b = 10 and we use ten symbols.S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
What is octal number system ?
The octal number system, also known as base-8, is a positional numeral system that uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.
What is Hexadecimal number system ?
The hexadecimal number system, often simply referred to as hex, is a base-16 numeral system. It uses sixteen digits: 0-9 for values 0 to 9, and A-F (or a-f) for values 10 to 15.
Conclusion
In conclusion, number systems are an important aspect of technology and mathematics. Understanding the different types of number systems and how to convert between them can be useful in various fields such as computer science and electrical engineering. In This Blog we have discussed about
Binary number System
Decimal number System
Octal number System
Hexadecimal Number System
With the knowledge of number system conversion, you can easily understand the concept of how a computer or machine works with numbers and can help you to solve problems efficiently.
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