The number system is important from the point of view of understanding how data is represented before it can be processed by any digital system, including a digital computer. There are two basic ways to represent the numerical values of the various physical quantities that we are constantly dealing with in our daily lives. The arithmetic value that is used to represent the quantity and is used to do the calculations is defined as NUMBERS. A symbol like “4, 5, 6” that represents a number is known as a**numbers**. Without numbers, it is not possible to count things, date, time, money, etc. these numbers are also used for measuring and labeling. The properties of numbers make them useful for performing arithmetic operations on them. These numbers can be written in numerical forms as well as words.

For example, 3 is written as three in words, 35 is written as thirty-five in words, etc. Students can write the numbers from 1 to 100 in words to learn more. There are different types of numbers, which we can learn. They are whole and natural numbers, even and odd numbers, rational and irrational numbers, etc.

### Number and its types.

The numbers used in mathematics are mostly decimal number systems. In the decimal number system, the digits used are 0 to 9 and base 10 is used. There are many types of numbers in the decimal number system, below are some of the types of numbers mentioned,

- The numbers that are represented to the right of zero are called
**Positive Numbers**. The value of these numbers increases as you move to the right. Positive numbers are used for addition between numbers. Example: 1, 2, 3, 4. - The numbers that are represented on the left side of zero are called
**negative numbers**. The value of these numbers decreases as you move to the left. Negative numbers are used for subtraction between numbers. Example: -1, -2, -3, -4. **Natural numbers**they are the most basic type of numbers that go from 1 to infinity. These numbers are also called Positive Numbers or Counting Numbers. Natural numbers are represented by the symbol N.**integer numbers**are basically the natural numbers, but also include 'zero'. Whole numbers are represented by the symbol W.**whole**are the set of integers plus the negative values of the natural numbers. Whole numbers do not include fractional numbers, that is, they cannot be written in**a/b**mold. The range of integers is from infinity at the negative end to infinity at the positive end, including zero. Integers are represented by the symbol**z****rational numbers**are the numbers that can be represented as a fraction, that is, a/b. Here a and b are integers and b≠0. All fractions are rational numbers, but not all rational numbers are fractions.**Irrational numbers**are the numbers that cannot be represented as fractions, that is, they cannot be written as a/b.- Numbers that have no factors other than 1 and the number itself are called
**Prime numbers.**All numbers that are not prime numbers are called as**composed numbers**except 0. Zero is neither a prime nor a composite number.

### What is a number system?

A number system is a method of displaying numbers in writing, which is a mathematical way of representing the numbers in a given set, using the numbers or symbols mathematically. The writing system for denoting numbers using digits or symbols logically is defined as**Numerical system.**The number system represents a useful set of numbers, reflects the arithmetic and algebraic structure of a number, and provides a standard representation. The digits from 0 to 9 can be used to make all the numbers. With these digits, anyone can create infinite numbers. For example, 156,3907, 3456, 1298, 784859, etc.

### Types of number systems

Depending on the base value and the number of digits allowed, number systems are of various types. The four common types of number systems are:

**Decimal number system****Binary Number System****Octal Numerical System****hexadecimal number system**

**Decimal number system**

The number system with a base value of 10 is called the decimal number system. It uses 10 digits i.e. 0-9 to create numbers. Here, each digit in the number is in a specific place value place with product of different powers of 10. Here, the place value is labeled from right to left as the first place value called units, the second from the left as tens, and so on Hundreds, Thousands, etc. Here, ones have a place value of 100, tens have a place value of 101, hundreds have a place value of 102, thousands have a place value of 103, and so on.

For example, 10264 has place values like,

(1 × 10

^{4}) + (0 × 10^{3}) + (2 × 10^{2}) + (6 × 10^{1}) + (4 × 10^{0})= 1 × 10000 + 0 × 1000 + 2 × 100 + 6 × 10 + 4 × 1

= 10000 + 0 + 200 + 60 + 4

= 10264

**Binary Number System**

The number system with base value 2 is called the binary number system. It uses 2 digits i.e. 0 and 1 to create numbers. The numbers formed with these two digits are called binary numbers. Binary number system is very useful in electronic devices and computer systems because it can be easily realized using only two states ON and OFF i.e. 0 and 1.

Decimal numbers 0-9 are represented in binary as: 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001

For example, 14 can be written as 1110, 19 can be written as 10011, 50 can be written as 110010.

**Example of 19 in** **Binary system**

Here 19 can be written as 10011

**BENEFITS**

Logical operations are the backbone of any digital computer, although solving a computer problem can also involve an arithmetic operation. George Boole's introduction of the mathematics of logic laid the foundation for the modern digital computer. He reduced the mathematics of logic to a binary notation of '0' and '1'.

Another advantage of this number system was that all types of data could be conveniently represented in terms of 0 and 1.

In addition, the basic electronic devices used for the implementation of the hardware can be conveniently and efficiently operated in two different modes.

circuits needed to perform arithmetic operations.

**Octal Numerical System**

Octal numbering system is one where the base value is 8. It uses 8 digits i.e. 0-7 to create octal numbers. Octal numbers can be converted to decimal values by multiplying each digit by the place value and adding the result. Here the place values are 80, 81, and 82. Octal numbers are useful for representing UTF8 numbers. Example,

(135)

_{10}can be written as (207)_{8}(215)

_{10}can be written as (327)_{8}

**hexadecimal number system**

The number system with a base value of 16 is called the hexadecimal number system. It uses 16 digits to create its numbers. Digits 0-9 are considered digits in the decimal number system, but digits 10-15 are represented as A-F, i.e. 10 is represented as A, 11 as B, 12 as C, 13 as D, 14 as E and 15 as F. Hexadecimal numbers are useful for handling memory address locations. The hexadecimal number system provides a condensed way of representing large stored and processed binary numbers. Examples,

(255)

_{10}can be written as (FF)_{sixteen}(1096)

_{10}can be written as (448)_{sixteen}(4090)

_{10}can be written as (FFA)_{sixteen}

HEXADECIMAL0123456789ABCDmiFDECIMAL0123456789101112131415

### examples of problems

**Question 1: Convert (18) _{10}as a binary number?**

**Solution:**

then (18)

_{10}= (1001)_{2}

**Question 2: Convert 325 _{8}in a decimal?**

**Solution:**

325

_{8}= 3 × 8^{2}+ 2 × 8^{1}+ 5 × 8^{0}= 3 × 64 + 2 × 8 + 5 × 1

= 192 + 16 + 5

= 213

_{10}

**Question 3: Convert (2056) _{sixteen}in an octal number?**

**Solution:**

Here (2056)

_{sixteen}is in hexadecimal formLet's first convert it to decimal form from hexadecimal.

(2056)

_{sixteen}= 2 × 16^{3}+ 0 × 16^{2}+ 5 × 16^{1}+ 6 × 16^{0}= 2 × 4096 + 0 + 80 + 6

= 8192 + 0 + 80 + 6

= (8278)

_{10}Now convert this decimal number to octal number by dividing it by 8

Therefore, it will take the value of the remainder of 20126

(8278)

_{10}= (20126)_{8}Therefore, (2056)

_{sixteen}= (20126)_{8}

**Question 4: Convert (101110) _{2}in octal number.**

**Solution:**

Given (101110)

_{2}a binary number, to convert it to an octal number

OCTAL NUMBER BINARY NUMBER 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111 Using the table above, we can write the given number as,

101 110 i.e.

101 = 5

110 = 6

So (101110)

_{2}in octal number is (56)_{8}

my personal notes*arrow_fall_up*