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Updated on 27th June, 2023 , 9 min read
Number systems play a crucial role in mathematics and are essential for computer understanding. They allow us to represent numbers in different forms and facilitate arithmetic calculations. Numbers, as mathematical values used for counting and measuring, can be categorized into various types, such as natural numbers, whole numbers, rational numbers, irrational numbers, and more. Similarly, there are different number systems, each with unique properties. These include the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.
In this article, we will delve into the exploration of various number systems: binary, octal, decimal, and hexadecimal. We will examine the conversions between these systems and provide illustrative examples to enhance our comprehension of this concept.
A number system is a defined method of writing and representing numbers. It serves as a mathematical notation that uses consistent symbols or digits to express numbers from a specific set. The number system ensures a distinct representation for each number and reflects the arithmetic and algebraic properties of the figures. Moreover, it enables us to perform fundamental arithmetic operations such as addition, subtraction, multiplication, and division.
The value of a digit within a number can be determined based on three factors:
Numbers serve mathematical purposes such as counting, measuring, and labelling. They are essential for performing arithmetic calculations. Various types of numbers exist, including natural, whole, rational, and irrational numbers. Additionally, 0 represents a null value. Numbers also exhibit variations like even and odd numbers, as well as prime and composite numbers. The terms "even" and "odd" describe divisibility by 2, while "prime" and "composite" differentiate between numbers with two factors and those with more than two factors, respectively.
In number systems, these numbers function as digits. Binary numbers primarily use the digits 0 and 1. Other number systems utilize the digits 0 to 9. Let's explore the different types of number systems.
A number system refers to the consistent representation of numbers using digits or symbols. The value of a digit within a number depends on the digit itself, its position within the number, and the base of the number system being used. Each number in the system has a distinct representation and enables arithmetic operations such as addition, subtraction, and division.
There are various types of number system. The four main types of number system that we will discuss in brief are:
In the decimal number system, the base number is 10, and it employs a set of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This system is commonly utilized to express numbers in our everyday lives. When a number is presented without specifying its base, it is assumed to be in base 10. For instance, 72310, 3210, and 425710 are a few illustrations of numbers within the decimal number system.
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:
(1×103) + (4×102) + (5×101) + (7×100)
(1×1000) + (4×100) + (5×10) + (7×1)
1000 + 400 + 50 + 7
1457
The Binary number system, also referred to as the base 2 number system, utilizes only two digits: 0 and 1. In this system, the base-2 represents a radix of 2. The numbers represented within this system are called binary numbers, consisting exclusively of combinations of 0 and 1. For instance, 110101 is an example of a binary number.
It is possible to convert any number system into binary and vice versa.
For Example:
In the binary system, there are only two digits: 0 and 1. Each digit in a binary number is called a "bit." The position of each bit determines its value, based on powers of 2.
Let's take the binary number 10110. To understand its value, we'll assign powers of 2 to each bit, starting from the rightmost bit:
|
1 0 1 1 0 2^4 2^3 2^2 2^1 2^0 |
Calculating the value of each bit:
|
16 + 0 + 4 + 2 + 0 = 22 |
So, the binary number 10110 is equivalent to the decimal number 22.
The octal numeral system utilizes a base of 8 and consists of eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. One notable advantage of this system is its reduced digit count compared to various other systems, which helps minimize computational errors. The octal system does not include digits such as 8 and 9. Similar to binary, the octal numeral system finds application in minicomputers, employing digits ranging from 0 to 7.
Example:Convert 2158 into a decimal.
Solution:
2158 = 2 × 82 + 1 × 81 + 5 × 80
= 2 × 64 + 1 × 8 + 5 × 1
= 128 + 8 + 5
= 14110
The hexadecimal number system uses a base of 16 to represent numbers. Initially, numbers in the hexadecimal system are represented similarly to those in the decimal system, ranging from 0 to 9. However, beyond 9, additional symbols from the alphabet, specifically A to F, are used to represent the values. The table below illustrates the representation of numbers in the hexadecimal number system.
|
Hexadecimal |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
A |
B |
C |
D |
E |
F |
|
Decimal |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
Numbers can be expressed in various number systems, such as binary, decimal, hexadecimal, and more. It is also possible to convert a number from one system to another with relative ease. To learn the conversion process, refer to the comprehensive lesson on number system conversions. This lesson provides instructions on converting decimal to binary and vice versa, hexadecimal to binary and vice versa, as well as octal to binary and vice versa, accompanied by several examples.
Having covered the different conversion methods described above, let's briefly discuss the conversion from one number system to another using a random number as an example.
Let's consider the number 349. Here is how the number 349 appears in different number systems:
To convert a number from binary to the decimal system, the following steps are employed:
Step 1:Begin with the rightmost digit of the given binary number and multiply each digit by the corresponding exponent of the base.
Step 2:The exponents are assigned starting from 0 and incrementing by 1 as we move from right to left.
Step 3:Simplify the products obtained in the previous step and sum them up.
To illustrate these steps, let's use the following example, where we aim to convert a binary number to the decimal number system.
Solution:
Step 1:Identify the base of the given number.
Here, the base of 1001112 is 2.
Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time we move from right to left. Since the base is 2, we multiply the digits of the given number by 20, 21, 22, and so on from right to left.
Step 3:We just simplify each of the above products and add them.
Here, the sum is the equivalent number in the decimal number system of the given number. or,
We can use the following steps to make this process simplified.
100111 = (1 × 25) + (0 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (1 × 20)
= (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)
= 32 + 0 + 0 + 4 + 2 + 1
= 39
Thus, 1001112= 3910.
To convert a number from the decimal system to binary, octal, or hexadecimal, the following steps are utilized. In this instance, we will demonstrate the steps for converting a decimal number to the octal system.
Example:Let's convert the number 432010 to the octal system.
Solution:
Step 1:Determine the base of the target number system. In this case, we are converting to the octal system, so the base is 8.
Step 2: Divide the given number by 8, the base of the octal system. Record the quotient and remainder in the quotient-remainder form. Repeat this process by dividing each subsequent quotient by 8 until we obtain a quotient less than 8.
Step 3:The number in the octal system is obtained by reading the remainders and the last quotient from bottom to top.
Therefore, 432010 = 103408
To convert a number from binary, octal, or hexadecimal systems to another system, the conversion typically involves two steps. First, the number is converted from the original system to the decimal system. Then, the decimal number is converted to the desired system using the respective conversion process.
Example:Let's convert the binary number 10101111002 to the hexadecimal system.
Solution:
Step 1:Convert the given number to the decimal system using the conversion process explained earlier. Thus, 10101111002 = 70010 → (1)
Step 2: Convert the decimal number obtained in step 1 to the hexadecimal system.
In this case, we need to convert 70010 to the hexadecimal system using the provided process. Note that in the hexadecimal system, the numbers 11 and 12 are represented as B and C, respectively. Therefore, 70010 = 2BC16 → (2)
From equations (1) and (2), we conclude that 10101111002 is equal to 2BC16 in the hexadecimal system.
There are different types of numbers used in the number system. The table below shows the other types of numbers:
|
Number System |
Numbers |
|
Natural |
1, 2, 3, 4, ... |
|
Whole |
0, 1, 2, 3, 4, ... |
|
Integers |
..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... |
|
Rational |
1/3, 1/5, 1/8, ... |
|
Irrational |
√3, π, 1.732, √2, ... |
Note: The table lists different number systems and provides examples of numbers within each system. Natural numbers include positive integers, while whole numbers include natural numbers along with zero. Integers consist of both positive and negative whole numbers, including zero. Rational numbers are represented as fractions, and irrational numbers cannot be expressed as fractions and have non-terminating and non-repeating decimal representations.
In computer systems, the numeral system used is the binary system, which is a base-2 system. The binary system consists of two digits, 0 and 1, and is the fundamental building block for all computer operations. However, computer systems also utilize other numeral systems for convenience and ease of representation. Here are the commonly used numeral systems in computer systems:
These different numeral systems allow computers to represent and manipulate data efficiently, depending on the specific requirements of the application or task at hand.
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By - Nikita Parmar 2023-08-25 09:48:44 , 11 min readA number system is a way of representing and expressing numbers. It consists of a set of digits and rules for combining those digits to form numbers.
The common types of number systems include the decimal system (base-10), binary system (base-2), octal system (base-8), and hexadecimal system (base-16).
In the decimal system, there are ten digits (0-9). Each digit’s value is determined by its position in a number, starting from the rightmost digit, which has a position value of 1. The leftmost digit represents the highest place value.
The binary system uses only two digits, 0 and 1. It is widely used in digital electronics and computer systems. In binary, each digit’s value is a power of 2, based on its position.
The octal system uses eight digits (0-7). It is commonly used in computer programming. Each digit in octal represents a power of 8, based on its position.
The hexadecimal system uses sixteen digits (0-9 and A-F), where A-F represent values from 10 to 15. Hexadecimal numbers are often used in computer science and programming to represent large binary values concisely.
Yes, numbers can be converted between different number systems using conversion methods specific to each system. For example, binary to decimal conversion involves multiplying each digit by the corresponding power of 2 and summing the results.
Computers use the binary system because electronic circuits can easily represent two states (on/off or 0/1). Binary is fundamental to digital data storage, processing, and communication in computers.
Yes, there are other number systems such as the Roman numeral system, which uses letters to represent numbers, and the base-60 system used for measuring time (hours, minutes, seconds).
Understanding different number systems helps in various fields such as computer science, mathematics, and engineering. It enables efficient data representation, problem-solving, and communication within these disciplines.
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