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• Number System

What is Number System?

The Number System includes any of the numerous sets of symbols and the rules for using them to denote numbers, which are used to state how many objects are there in a given set. Thus, the idea of “oneness” can be denoted by the Roman numeral I, by the Greek letter alpha α which was the first letter used as a numeral, by the Hebrew letter aleph which is the first letter used as a numeral, or by the modern number 1, which is nothing but Hindu-Arabic in origin.

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Number System Definition

Number system is a mathematical presentation of numbers of a given set. For further discussion, let us understand number systems.

Most likely, the beginning system of the inscribed symbol in ancient Mesopotamia was a system of symbols for numbers. The present number systems are place-value systems. That is, the value of these numbers depends upon the position or place of the numbers in the representation; for example, the 3 in 30 and 300 represents 3 ten’s and 3 hundred, respectively. In the ancient systems, such as the Egyptian civilization, Roman, Hebrew, and Greek Number systems, did not have a positional characteristic, and it was a very complicated arithmetical calculation. Other systems though, including the Babylonian, one version each of the Chinese and Indian, as well as the Mayan system, did use the principle of place value. 

Number Systems Through the Ages

The development of numbers grew in different directions and has dissimilar versions. Some of the different versions of number systems were developed in Egypt, Babylon, Rome and some other versions are called Hindu-Arabic, Mayans, and the modern American number systems. It is believed that mathematical evolution had begun before the counting systems of numbers started.

Let us discuss the different divisions under the number system.

Unary Numeral System

The simplest number system of numerals is the unary numeral system, in which each and every natural number is represented by an equivalent number of symbols. They are denoted by /, for instance, then the number seven would be represented by /////// such that we must write seven /.

Tally marks represent one such system which is still in regular use. But, the urinary system can only be used for numbers which are of a smaller size with a small value, although this plays a related role in theoretical computer science. Do you know Elias gamma coding, which is generally used in data compression, denotes arbitrary-sized numbers by using unary to indicate the length of binary numbers?

For example, if / stands for one, * for ten and - for 100, then the number 304 can be compactly represented as --- //// and the number 123 as - ** /// without any need for zero and this is called sign-value notation.

Binary Numeral System

A binary number is used in the system of mathematics, and the world of science. In this system, the numbers are expressed in the form of a base-2 system or binary numerical system, which uses only two numbers that are 0's (zeros) and 1's (ones). The positional representation is done with a radix of 2 in the base-2 system.

In computers, the main system of numbers is based on the positional system in base 2 which is a binary numeral system, with two digits of binary, 0's and 1's. Positional systems received by grouping binary digits by three, which is the octal numeral system or four which is the hexadecimal numeral system, are commonly used. For very large numbers, bases 232 or 264, you can group the binary digits by 32 or 64, which is the portion of the machine word.

Number system represents a valuable set of numbers that consists of natural numbers, integers, real numbers, irrational numbers, rational numbers and so on.

Natural Numbers

The natural (or counting) numbers are from 1, 2, 3, 4, 5, 6, 7, 8, 9, etc. Natural numbers are infinite numbers. The set of natural numbers, {1, 2, 3, 4, 5, 6, 7, 8...}, is denoted by N in short form.

whole numbers start from 0 and include the natural numbers.

The addition of any 2 natural numbers is also a natural number (for instance, 4 + 5000 = 5004), and the product of any two natural numbers is also a natural number (4 × 3500 = 14000). The subtraction and division of any two or more natural numbers is not equal to natural numbers, though.

Integers are natural numbers consisting of the set of real numbers, their additive inverses and zero.

{..., -9, -8, -7, -6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}

The set of integers is denoted by J or Z in short form. These integer values can be represented on the number line. These numbers can be a whole number or decimal numbers. You can represent -6.5 on a number, 6 on a number line.

The addition, product, and difference between any two integers is also an integer. But this is not true for division.

Rational Numbers

Rational numbers are figures which can be formulated as a ratio between two integers. For instance, the fractions 13 and −11118 are both rational numbers. Rational numbers include all the integers since any integer is denoted by z.

All decimals which terminate are rational numbers (since 8.27 can be written as 827/100.) Decimals which are a recurrent pattern in nature after some point are also called a rational number.

Irrational Numbers

An irrational number is a number that cannot be written as a proportion (or fraction). The irrational number does not end or repeats in a decimal form. The ancient Greek mathematician discovered that not all integers are rational numbers; there are equations that cannot be solved using ratios of integers.

The square root of 2 is about 1.414, because 1.4142=1.999396, which is almost equal to 2. You will never get the exact number by squaring the fraction (or terminating decimal numbers). The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repetitive pattern

The golden ratio is another famous irrational number, a number which has great importance in biology:

1 + 5√2 = 1.61803398874989…

π(pi), the proportion of the perimeter of a circle to its diameter:

π = 3.14159265358979…

and e, the most significant number in calculus:

e = 2.71828182845904…

Real Numbers

Real numbers are the set of numbers containing all the irrational numbers and all the rational numbers. They consist of “all the numbers” on the number scale. Real numbers are infinite numbers just as there are infinitely many numbers in each of the other sets of numbers. But, the infinity of the real numbers is a bigger infinity.

Complex Numbers

Complex numbers are the set {a+bi | real numbers are a and b}, where the imaginary unit is i, -1

Complex numbers include the set of real numbers, which includes the set of both rational and irrational numbers. The real numbers, in the complex system, are denoted in the form a + 0i = a. a real number.

This set is always denoted by C in the short form. The set of complex numbers is significant because for any polynomial p(x) with real number coefficients, all the solutions of p(x) = 0 will be in C.

Important Number Systems

At present, there are four types of number system; these are:

Binary system

Decimal system

Octal System

Hexadecimal system

Number System Conversions

Binary system conversions.

Binary to Octal

Binary to Decimal

Binary to Hexadecimal

Example: 1100011

Octal means 8 

Starting from the least significant bit, we will make a group of three bits First pair = 011

Second group = 100

Third group = 001

Now, we know that 011 = 3, 100 = 4, and 001 = 1, so the equivalent octal number system will be (143) 8 .

Example: 1101001

Starting from the least significant bit from 2 0 , then 2 1 , till the nth most significant bit, i.e., 2ⁿ.

1 = 2³.1 = 8

0 = 2⁴ .0= 0

1 = 2⁵ = 32

1 = 2⁶ = 64

Now, adding all these:

= 1 + 0 + 0 + 8 + 0 + 32 + 64 = (105) 10

Example: 101010001

We know that hexadecimal means 16, so we will start from the least significant bit and make groups of 4 bits:

First pair = 0001 = 1

Second pair = 0101 = 5

Third pair = 0001 = 1

So, the hexadecimal equivalent will be (151) 16 .

Decimal System Conversions

Decimal to Binary

Decimal to Octal

Decimal to Hexadecimal

Example: (131) 10

Divide 131 by 2 and continue till you get the quotient as 0.

Divide 131 by 2, we get the quotient as 65 and remainder as 1

Divide 65 by 2, we get 32, remainder as 1

Divide 32 by 2, we get 16, and remainder as 0

Divide 16 by 2,we get 8 and remainder as 0

Divide 8 by 2, we get 4, and remainder as 0

Divide 4 by 2, then 2 by 2, and finally divide 1 by 2,with the quotient as 0, we get the remainder as 1

Starting from LSB to MSB, write the binary system so formed:

(10000011) 2

Decimal to Octal Conversion

Example: (425) 10

Divide 425 by 8 till you get the quotient as 0. 

Dividing 425 by 8, we get the quotient as 53, and the remainder as 1

Divide 53 by 8, we get the quotient as 6, and the remainder as 5

Divide 6 by 8 by taking the quotient as 0, we get the remainder as 6 

So, the octal equivalent of (425) 10 is (651) 8 .

Example: (650) 10

Divide 650 by 16, we get the quotient as 40 and the remainder as 10 or A

Divide 40 by 16, we get the quotient as 2 and the remainder as 8

Divide 2 by taking the quotient as 0, we get the remainder as 2

So, the hexadecimal equivalent of (650) 10 is (28A) 16 .

Octal System Conversions

Example: (79) 8 to () 10

= 7 Χ 8¹ + 9 Χ 8⁰= 56 + 9 = 65

So, the decimal equivalent of (79)) 8 is (65) 10 .

First 20 Binary number system conversions to Decimal, Octal, and Hexadecimal systems:

Number System Chart

Number system in computer.

In computers, the technique of presenting the work is the number s system. Computers use the decimal number system the most, while the information converted from higher-level to the lower one is in the form of bits, i.e., 0’s and 1’s, which is actually the binary system.

The most commonly used Number system is the decimal positional number system, the decimal referring to the use of 10 numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to construct all the required numbers. This discovery was made by the Indians. There are other two common number systems which are used in computers and computing science. They are the binary system, and these are denoted by 0's and 1's, and the hexadecimal system, which has 16 symbols (We can understand by the term Hex which is 16) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Fill in the Blanks:

1. The similar ___________of symbols may represent different symbols in various numeral systems (Ans: sequence)

2. ___________________________ developed by the Hindus in India, slowly radiated to other surrounding countries due to their commercial and military activities with India.

(Ans: The numerical system and the concept of zero)

3. The main system of numerals is based on ____________________in base 2 with two digits of binary, 0 and 1. (Ans: the positional system)

State True or False and Correct the Statements:

1. The natural numbers start from 0 to infinite numbers( ) (Ans: false)

2. The complex numbers include the set of real numbers, that is, which includes the set of both rational and irrational numbers. () (Ans: true )

3. The real numbers are the set of numbers containing all of the irrational numbers ( ) (Ans: false)

4. The integers are the natural numbers consisting of the set of real numbers, their additive inverses and zero. ( ) (Ans: true)

5. The quaternions were discovered by William. Rowan Hamilton( ) (Ans: true)

Tick the Correct Answers

1. Unary Numeral System is Denoted by_________________

(Ans: d. /)

2. Real Number Includes _________________________

a. Only rational numbers

b. Only irrational numbers

c. Both the rational and irrational numbers

d. None of the above

(Ans: c. both the rational and irrational numbers)

3. The Natural (or counting) Numbers are from_________________

b. 0, 1, 2, 3 ,4……………..

c. 1, 2, 3, 4………..

d. ….-2, -1, 0 , 1, 2 ,3……..

(Ans: c.1, 2, 3, 4………..)

4. The Complex Numbers are the Set {a + bi | __________________are a and b},

a. The rational numbers

b. The irrational numbers

c. The real numbers

d. The integer numbers

(Ans: c. the real numbers). 

5. Which Number System Cannot be Written as a Proportion?

a. Rational numbers

b. Irrational number

c. Real numbers

d. Integers numbers

(Ans: irrational numbers).

FAQs on Number System

1. Who Invented Number System?

Around the year 1679, the number was invented by Gottfried Leibniz, and he published about his development in his article named Explication de l'Arithmétique Binaire in the year 1703.

2. What is the Number System in Maths?

A number system is basically a way of presenting a given set of numbers. In mathematics, we can represent numbers in the form of symbols and digits in a consistent manner. The low-level language that is understandable by computers is actually the number system. There are four types of number systems, binary, decimal, octal, hexadecimal, etc.

3.   Who invented zero?

Indian mathematician Brahmagupta is said to have invented zero.

4.   Where did negative numbers come from?

Negative numbers are said to have originated in China.

5. What are the applications of the number system?

The most common application of the number system can be found in computer technology. It uses the binary number system. The base 2 number system is used in the process of digital encoding

6. What is the difference between rational and irrational numbers?

Rational number is defined as the ratio of two numbers, and it is expressed in the form p/q, where q is not equal to 0. Whereas, the irrational number cannot be expressed as the ratio of two numbers. 

NCERT Study Material

Number Systems

Number systems are systems in mathematics that are used to express numbers in various forms and are understood by computers. A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. Numbers have various categories like natural numbers, whole numbers, rational and irrational numbers, and so on. Similarly, there are various types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

In this article, we will explore different types of number systems that we use such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. We will learn the conversions between these number systems and solve examples for a better understanding of the concept.

What are Number Systems?

A number system is a system representing numbers. It is also called the system of numeration and it defines a set of values to represent a quantity. These numbers are used as digits and the most common ones are 0 and 1, that are used to represent binary numbers. Digits from 0 to 9 are used to represent other types of number systems.

Number Systems Definition

A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The numbers are represented in a unique manner and allow us to operate arithmetic operations like addition, subtraction, and division.

Types of Number Systems

There are different types of number systems in which the four main types are as follows.

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

We will study each of these systems one by one in detail after going through the following number system chart.

Number System Chart

Given below is a chart of the main four types of number system that we use to represent numbers.

Binary Number System

The binary number system uses only two digits: 0 and 1. The numbers in this system have a base of 2. Digits 0 and 1 are called bits and 8 bits together make a byte. The data in computers is stored in terms of bits and bytes. The binary number system does not deal with other numbers such as 2,3,4,5 and so on. For example: 10001 2 , 111101 2 , 1010101 2 are some examples of numbers in the binary number system.

Octal Number System

The octal number system uses eight digits: 0,1,2,3,4,5,6 and 7 with the base of 8. The advantage of this system is that it has lesser digits when compared to several other systems, hence, there would be fewer computational errors. Digits like 8 and 9 are not included in the octal number system. Just like the binary, the octal number system is used in minicomputers but with digits from 0 to 7. For example, 35 8 , 23 8 , and 141 8 are some examples of numbers in the octal number system.

Decimal Number System

The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10. The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example, 723 10 , 32 10 , and 4257 10 are some examples of numbers in the decimal number system.

Hexadecimal Number System

The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means the numbers 10-15 of the decimal number system respectively. This system is used in computers to reduce the large-sized strings of the binary system. For example, 7B3 16 , 6F 16 , and 4B2A 16 are some examples of numbers in the hexadecimal number system.

Conversion of Number Systems

A number can be converted from one number system to another number system using number system formulas. Like binary numbers can be converted to octal numbers and vice versa, octal numbers can be converted to decimal numbers and vice versa, and so on. Let us see the steps required in converting number systems.

Steps for Conversion of Binary to Decimal Number System

To convert a number from the binary to the decimal system, we use the following steps.

• Step 1: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
• Step 2: The exponents should start with 0 and increase by 1 every time we move from right to left.
• Step 3: Simplify each of the above products and add them.

Let us understand the steps with the help of the following example in which we need to convert a number from binary to decimal number system.

Example: Convert 100111 2 into the decimal system.

Step 1: Identify the base of the given number. Here, the base of 100111 2 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 as we move from right to left. Since the base is 2 here, we multiply the digits of the given number by 2 0 , 2 1 , 2 2 , 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 × 2 5 ) + (0 × 2 4 ) + (0 × 2 3 ) + (1 × 2 2 ) + (1 × 2 1 ) + (1 × 2 0 )

= (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)

= 32 + 0 + 0 + 4 + 2 + 1

Thus, 100111 2 = 39 10 .

Conversion of Decimal Number System to Binary / Octal / Hexadecimal Number System

To convert a number from the decimal number system to a binary/octal/hexadecimal number system, we use the following steps. The steps are shown on how to convert a number from the decimal system to the octal system.

Example: Convert 4320 10 into the octal system.

Step 1: Identify the base of the required number. Since we have to convert the given number into the octal system, the base of the required number is 8.

Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the quotient-remainder form. Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.

Step 3: The given number in the octal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

Therefore, 4320 10 = 10340 8

Conversion from One Number System to Another Number System

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems, we first convert it into the decimal system, and then we convert it to the required systems by using the above-mentioned processes.

Example: Convert 1010111100 2 to the hexadecimal system.

Step 1: Convert this number to the decimal number system as explained in the above process.

Thus, 1010111100 2 = 700 10 → (1)

Step 2: Convert the above number (which is in the decimal system), into the required number system (hexadecimal).

Here, we have to convert 700 10 into the hexadecimal system using the above-mentioned process. It should be noted that in the hexadecimal system, the numbers 11 and 12 are written as B and C respectively.

Thus, 700 10 = 2BC 16 → (2)

From the equations (1) and (2), 1010111100 2 = 2BC 16

• Indian Numeral System
• International Number System
• Binary Calculator
• Binary to Octal Conversion
• Octal to Binary
• Decimal to Binary
• Binary to Decimal
• Decimal to Hexadecimal
• Hexadecimal to Decimal

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Number Systems Examples

Example 1: Convert 300 10 into the binary number system with base 2.

Solution: 300 10 is in the decimal system. We divide 300 by 2 and note down the quotient and the remainder. We will repeat this process for every quotient until we get a quotient that is less than 2.

The equivalent number in the binary system is obtained by reading all the remainders and just the last quotient from bottom to top as shown above.

Thus, 300 10 = 100101100 2

Example 2: Convert 5BC 16 into the decimal system.

Solution: 5BC 16 is in the hexadecimal system. We know that B = 11 and C = 12 in the hexadecimal system. So we get the equivalent number in the decimal system using the following process:

Thus, 5BC 16 = 1468 10

Example 3: Convert 144 8 into the hexadecimal system.

Solution: The base of 144 8 is 8. First, we will convert this number into the decimal system as follows:

Thus, 144 8 = 100 10 → (1). Now we will convert this into the hexadecimal system as follows:

Thus, 100 10 = 64 16 → (2)

From the equations (1) and (2), we can conclude that: 144 8 = 64 16

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Practice Questions on Number Systems

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FAQs on Number Systems

What are number systems with examples.

A number system is a system of writing or expressing numbers. In mathematics, numbers are represented in a given set by using digits or symbols in a certain manner. Every number has a unique representation of its own and numbers can be represented in the arithmetic and algebraic structure as well. There are different types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. Some examples of numbers in different number systems are 10010 2 , 234 8 , 428 10 , and 4BA 16 .

What are the Different Types of Number Systems?

There are four main types of number systems:

What are the Conversion Rules of Number Systems?

To convert a number from binary/octal/hexadecimal system to a decimal number system, we use the following steps:

• 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.
• Simplify each of the above products and add them.

To convert a number from decimal system to binary/octal/hexadecimal system, we use the following steps:

• Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form.
• Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.
• The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems:

• We first convert it into the decimal system.
• Then we convert it to the required system.

What are the Uses of Each Number System?

There are different purposes of each number system, such as:

• The binary number system is used to store the data in computers.
• The advantage of the octal number system is that it has fewer digits when compared to several other systems, hence, there would be fewer computational errors.
• The decimal number system is the system that we use in daily life.
• The hexadecimal number system is used in computers to reduce the large-sized strings of the binary system.

What is the Importance of Number Systems?

Number systems help in representing the numbers in a small symbol set. Binary numbers are mostly used in computers that use digits like 0 and 1 for calculating simple problems. The number systems also help in converting one number system to another.

How are Number Systems Classified?

The number systems can be classified mainly into two categories: Positional and Non-positional number systems. For positional number systems, each digit is associated with a weight and its examples are binary, octal, decimal, etc. In non-positional number systems, the digit values are independent of their positions and its examples are gray code, cyclic code, aroma code, etc.

Why are Different Number Systems Used in Computers?

Computers cannot understand human languages, so to understand the commands and instructions given to the computers by programmers, different number systems are used such as the binary system, the octal system, the decimal system, and so on.

What is Number System?

A number system , or numeral system, is a method for representing numbers using symbols, crucial in mathematics and programming for data representation. It involves using digits to construct numbers, where each digit's value is determined by its position and the base value of the system. While computers primarily use the binary system of 0s and 1s, number systems are broadly applied in various fields for calculations, measurements, and labeling, among other uses. There are several types of number systems, each suited to different applications.

A number is nothing but symbols like 1, 2, 3, .. that can be used to perform calculations on them. A number can be written in a numeric form as well as in word form.

A number is a mathematical object used to count, measure, and label. It is a fundamental concept in mathematics, serving as a building block for various operations and theories. Numbers can be represented in different forms, such as integers, fractions, and decimals, and they can be positive, negative, or zero. They are essential for performing arithmetic operations like addition, subtraction, multiplication, and division, and they play a critical role in shaping mathematical principles and applications in various fields.

Types of Numbers

When we are talking about numbers, we mainly refer to the decimal number systems (having numbers between 0 and 9`). So, there are various types of numbers in the decimal number system. Let us discuss them briefly.

Positive Numbers : The numbers that are greater than zero are known as positive numbers. In other words, the numbers that are present on the right-hand side of the zero on the number line are termed positive numbers. Example: 1, 2, 3, 4, etc.

Negative Numbers : The numbers that are lesser than zero are known as negative numbers. In other words, the numbers that are present on the left-hand side of the zero on the number line are termed negative numbers. Example: -1, -2, -3, -4, etc.

Natural Numbers : The most basic types of numbers that range from 1 to positive infinity are called natural numbers. Natural numbers are represented by N . Natural numbers are positive numbers without zero. Example: 1, 2, 3, 4, etc.

Whole Numbers : The numbers that range between 0 to positive infinity are called whole numbers. Whole numbers are represented by W . Whole numbers are positive numbers including zero or natural numbers added with zero. Example: 0, 1, 5, 8 , etc.

Fractional Numbers : Fractional numbers are a part of whole numbers that are represented in the form of p/q, where p is known as the numerator and q is known as the denominator. Example:`` 1/2, 5/3, 3/4, /7/8`, etc.

Integer Numbers : The integer numbers are a collection of positive values, negative values, and zero. Integer numbers include decimal point numbers but fractional numbers (numbers of the form numerator/denominator ) is not a part of integers numbers. Integer numbers are represented by Z . The range of integer numbers is from negative infinity to positive infinity including the number zero. Example: 1.5, -3.6, 7, -8, 0 , etc.

Prime Numbers : The number that can be completely divided by one and the number itself only is called a prime number. In other words, we can say that prime numbers have only two factors i.e. one and the number itself. Example: 3, 17, 23, 67 etc.

Composite Numbers : The number that can be completely divided by one or more numbers apart from 1 and the number itself is called a composite number. In other words, we can say that composite numbers have more than two factors. Example: 32, 8, 25, 6 etc.

Even Numbers : The positive or negative numbers that can be completely divided by the number 2 are known as even numbers. Example: 2, 4, 18, 26 , etc.

Odd Numbers : The positive or negative numbers that cannot be completely divided by the number 2 are known as even numbers. Example: 3, 13, 11, 17 , etc.

Rational Numbers : The number that can be represented in the form of p/q where p and q both are integers and q is not equal to 0, is called a rational number. Example: 10/2, 0.5 , etc.

Irrational Numbers : The number that cannot be represented in the form of p/q where p and q both are integers, is called an irrational number. Example: the value of pi, the value of root 2 , etc.

Real Numbers : Real numbers are the combination of rational numbers and irrational numbers. The real numbers are represented by R . Hence, real numbers can be both positive and negative.

Refer to the complete chart of numbers given below for a better understanding.

Note: All the fractional numbers are rational numbers but vice versa is not true.

Types of Number System

Now, there are various types of number systems that have their use cases. Let us discuss them briefly.

Refer to the complete chart of number systems given below for a better understanding.

The decimal number system or the base 10 number system uses 10 digits ranging from 0 to 9 to represent any number. The first position from the rightmost part of a decimal number is known as one's place or unit's place, then the second position is known as ten's place, and so on. Every decimal number is represented by the power of base 10 .

For example , the number 1457 is represented as:

In the above example:

• 7 is present at the unit's place.
• 5 is present at ten's place.
• 4 is present at hundred's place.
• 1 is present at thousand's place.

The binary number system or the base 2 number system uses only two digits i.e. 0 and 1 to represent any number. For example, the decimal number 14 is represented as 1110 in the binary number system. We need to keep dividing the decimal number by 2 until the quotient becomes 0 and along with that, we need to remember the remainder. When the remainders obtained in the division are represented in reverse order, a binary number is generated. Let us see how?

The octal number system or the base 8 number system uses 8 digits ranging from 0 to 7 to represent any number. The octal number is very commonly used the computer systems.

Every octal number is represented by the power of base 8 .

For example , the number 215 is represented as:

4. Hexadecimal

The hexadecimal number system or the base 16 number system uses 16 digits ranging from 0 to 9 (i.e. 10 digits) and A to F (i.e. 6 digits ) to represent any number. Refer to the chart shown below to know the representation of hexadecimal numbers.

We can convert a decimal number into a hexadecimal number. We need to keep dividing the decimal number by 16 until the remainder becomes 0 and along with that, we need to remember the remainder. When the remainders obtained in the division are represented in reverse order, the hexadecimal number is generated. Let us see how?

Refer to the image provided below to have a quick look at the base value of various number systems.

Refer to the next section to know how to convert decimal number systems into other number systems along with the implementation in C++, Java, and Python.

Conversion of One Another

We can convert one number system to another number system. in this section, we will be learning about how to convert a decimal number system into the other number system with that we will be learning how to convert other number systems into the decimal number system.

1. Definition

As we have discussed earlier:

• The binary number system uses only two digits i.e. 0 and 1 to represent any number.
• The decimal number system uses ten digits from 0 to 9 to represent any number.
• The octal number system uses 8 digits ranging from 0 to 7 to represent any number.
• The hexadecimal number system uses 16 digits ranging from 0 to 9 (i.e. 10 digits) and A to F (i.e. 6 digits) to represent any number.

Let's Now Learn about Conversion Between Number Systems

Converting decimal to binary :.

• Step 1 : Divide the decimal number by 2, the base for binary numbers.
• Step 2 : Record the remainder; this becomes the least significant digit (LSD) of the binary number.
• Step 3 : Use the quotient as the new dividend and divide again by 2.
• Step 4 : The new remainder becomes the next digit of the binary number, positioned to the left of the previous digit.
• Repeat steps 3 and 4 until the quotient is 0. The last remainder obtained is the most significant digit (MSD).

Example : Convert decimal 13 to binary.

• 13 ÷ 2 = 6, remainder 1 (LSD)
• 6 ÷ 2 = 3, remainder 0
• 3 ÷ 2 = 1, remainder 1
• 1 ÷ 2 = 0, remainder 1 (MSD)
• Binary equivalent: 1101

Converting Decimal to Octal :

• Step 1 : Divide the decimal number by 8, the base for octal numbers.
• Step 2 : The remainder becomes the least significant digit of the octal number.
• Step 3 : The quotient is used as the new dividend, divided again by 8.
• Step 4 : The new remainder becomes the next octal digit, positioned to the left of the previous digit.
• Continue until the quotient is 0. The final remainder is the most significant digit.

Example : Convert decimal 95 to octal.

• 95 ÷ 8 = 11, remainder 7 (LSD)
• 11 ÷ 8 = 1, remainder 3
• 1 ÷ 8 = 0, remainder 1 (MSD)
• Octal equivalent: 137

Converting Decimal to Hexadecimal :

• Step 1 : Divide the decimal number by 16, the base for hexadecimal numbers.
• Step 2 : The remainder becomes the least significant digit of the hexadecimal number. Use 0-9 for remainders 0-9 and A-F for remainders 10-15.
• Step 3 : Use the quotient as the new dividend and divide by 16.
• Step 4 : The new remainder becomes the next hexadecimal digit, to the left of the previous one.
• Repeat steps until the quotient is 0, with the last remainder being the most significant digit.

Example : Convert decimal 254 to hexadecimal.

• 254 ÷ 16 = 15, remainder 14 (E, LSD)
• 15 ÷ 16 = 0, remainder 15 (F, MSD)
• Hexadecimal equivalent: FE

Converting Binary to Decimal :

• Step 1 : Starting from the right (least significant bit, LSB), multiply each binary digit by 2 raised to its position number, counting from 0.
• Step 2 : Sum up the results of these multiplications to get the decimal equivalent.

Example : Convert binary 11101011 to decimal.

• Calculation: (1 * 2^7 + 1 * 2^6 + 1 * 2^5 + 0 * 2^4 + 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0)
• Result: 128 + 64 + 32 + 0 + 8 + 0 + 2 + 1 = 235
• Decimal equivalent: 235

Converting Binary to Octal :

• Step 1 : Group the binary digits into sets of three, starting from the right. If necessary, pad the leftmost group with zeros to make it a trio.
• Step 2 : Convert each trio into its octal equivalent, which will range from 0 to 7.

Example : Convert binary 11101011 to octal.

• Grouping: 1 110 101 011
• Converting each group: 1 is 1, 110 is 6, 101 is 5, 011 is 3
• Octal equivalent: 1653

Converting Binary to Hexadecimal :

• Step 1 : Divide the binary number into groups of four, starting from the right. Pad the leftmost group with zeros if it's not already a quartet.
• Step 2 : Convert each group into its hexadecimal equivalent, which will be in the range 0-9 and A-F for 10-15.

Example : Convert binary 1110101101101 to hexadecimal.

• Grouping: 1110 1011 0110 1
• Converting each group: 1110 is E, 1011 is B, 0110 is 6, 0001 is 1 (after padding)
• Hexadecimal equivalent: 1E6B

Converting Octal to Decimal :

• Step 1 : Starting from the right (least significant digit, LSD), multiply each octal digit by 8 raised to its position number, counting from 0.
• Step 2 : Sum up these products to get the decimal equivalent.

Example : Convert octal 247 to decimal.

• Calculation: (2 * 8^2 + 4 * 8^1 + 7 * 8^0)
• Result: 128 + 32 + 7 = 167
• Decimal equivalent: 167

Converting Octal to Binary :

• Step 1 : Write down each octal digit separately.
• Step 2 : Convert each octal digit into a corresponding group of three binary digits (bits).
• Step 3 : Concatenate these binary groups to form the complete binary number.

Example : Convert octal 247 to binary.

• Octal digits: 2, 4, 7
• Binary equivalents: 010 (for 2), 100 (for 4), 111 (for 7)
• Binary equivalent: 010100111

Converting Octal to Hexadecimal :

• Step 1 : Convert the octal number to binary by following the steps for octal to binary conversion.
• Step 2 : Group the binary digits into sets of four, starting from the right. If necessary, pad the leftmost group with zeros to make it a quartet.
• Step 3 : Convert each binary group into its hexadecimal equivalent, using 0-9 for values 0-9 and A-F for values 10-15.

Example : Convert octal 5456 to hexadecimal.

• Convert to binary: 5456 octal = 101 100 101 110 binary
• Group into fours: 0101 1001 0111
• Convert each group: 0101 is 5, 1001 is 9, 0111 is 7
• Hexadecimal equivalent: 597

Converting Hexadecimal to Decimal :

• Step 1 : Understand the decimal values of hexadecimal symbols. For digits 0-9, the values are the same, and for letters A-F, they represent 10-15 respectively.
• Step 2 : Multiply each digit of the hexadecimal number by 16 raised to its position number, starting from the right (LSB to MSB).
• Step 3 : Sum up all the results from step 2 to get the decimal equivalent.

Example : Convert hexadecimal 8EB4 to decimal.

• Calculation: (8 * 16^3 + 14 * 16^2 + 11 * 16^1 + 4 * 16^0)
• Result: 36532
• Decimal equivalent: 36532

Converting Hexadecimal to Binary :

• Step 1 : Convert each hexadecimal digit to its decimal equivalent.
• Step 2 : Convert each decimal digit to a 4-bit binary group.
• Step 3 : Concatenate all the 4-bit groups to get the full binary number.

Example : Convert hexadecimal B2E to binary.

• Decimal equivalents: B=11, 2=2, E=14
• Binary conversions: B= 1011 , 2= 0010 , E= 1110
• Binary equivalent: 101100101110

Converting Hexadecimal to Octal :

• Step 1 : Convert the hexadecimal number to binary by following the steps in the Hexadecimal to Binary conversion.
• Step 2 : Group the binary digits into sets of three, starting from the right. If necessary, pad the leftmost group with zeros.
• Step 3 : Convert each binary group to its octal equivalent.
• Step 4 : Concatenate these octal digits to get the octal equivalent.

Example : Convert hexadecimal B2E to octal.

• First, convert B2E to binary: 101100101110
• Group into threes: 010 110 010 111
• Convert to octal: 2=010, 6=110, 2=010, 7=111
• Octal equivalent: 2627
• Number systems represent numbers through symbols, pivotal in math and computing, with variations like binary (0s and 1s) crucial for digital systems.
• Positive and negative numbers denote values above and below zero, respectively, on the number line.
• Natural numbers start from 1, whole numbers include zero, and integers encompass both along with negative numbers.
• Prime numbers have only two factors (1 and themselves), while composite numbers have additional divisors.
• Rational numbers can be expressed as fractions, whereas irrational numbers cannot be represented as simple fractions.
• Decimal, binary, octal, and hexadecimal systems are used based on their applicability, with decimal for general arithmetic and others in specialized computing contexts.

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Computer Number Systems

Jul 26, 2014

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Computer Number Systems. ACSL. All computers – from large mainframes to hand-held micros – ultimately can do one thing: detect whether an electrical signal is “on” or “off”.

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Computer Number Systems ACSL

All computers – from large mainframes to hand-held micros – ultimately can do one thing: detect whether an electrical signal is “on” or “off”. • Computer programs in any language are converted by various pieces of systems software into sequences of bits (Binary digITs) which correspond to sequences of on/off (equivalently TRUE/FALSE or 1/0) signals. • Proficiency in the binary number system is essential to understanding how a computer works.

Since binary numbers representing moderate values quickly become rather lengthy, bases eight (octal) and sixteen (hexadecimal) are frequently used as short-hand. • Octal numbers group binary numbers in bunches of 3 digits and convert the triplet to a single digit between 0 and 7, inclusive. For example, 10010101102 = 001 001 010 1102 = 11268.

Hexadecimal numbers group binary numbers by fours, and convert the quadruplet to a single digit in the range 0, 1, 2 …, 9, A, B, C, D, E, F. For example, 101101101001012 = 0010 1101 1010 01012 = 2DA516.

Sample Problem 1 • Solve for X. • X 16 = 36768 One method of solution is to convert 36768 into base 10, and then convert that number into base 16 to yield the value of X. An easier solution, less prone to arithmetic mistakes, is to convert from octal (base 8) to hexadecimal (base 16) through the binary (base 2) representation of the number: 36768 = 011 110 111 1108 = 0111 1011 11108 = 7BE16

Sample Problem 2 • Solve for X. • X 16 = FEED16 – 6ACE16 The rightmost digit becomes F, because 1D-E=F. Next, D-C=1 (the E becomes a D because we had to borrow from it to do the units’ subtraction), E-A=4, and then F-6=9. Combining these results of each column, we get a final answer of 941F16.

Sample Problem 3 • In the ACSL computer, each “word” of memory contains 20 bits representing 3 pieces of information. The most significant 6 bits represent Field A; the next 11 bits, Field B; and the last 3 bits represent Field C. For example, the 20 bits comprising the “word” 1814916 has fields with values of 616 , 2916 and 116 . What is Field B in E1B7D16 ? (Express your answer as a base 16 number.)

Sample Problem 3 Solution • E 1 B 7 D = 1110 0001 1011 0111 1101 • = 1110 00 01 1011 0111 1 101 • Field B = 01 1011 0111 1 • = 011 0110 1111 • = 3 6 F16

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