All About Number System

A number system is a structure for representing numbers. It defines a system of values to denote a quantity and is also regarded as the numeration system. The most common integers are 0 and 1, which denote binary numbers, and are employed as digits. Digits spanning from 0 to 9 are used to represent other types of number classification.

The number system is characterised as the consistent representation of numbers using numerals or other symbols. A digit, its order in the numeral, and the base of the number system formula can all be used to determine the value of any digit in a number. 

Number System and its kinds

The numbers have a distinct representation that allows us to perform mathematical operations like addition, division, and subtraction.

Similarly, other varieties of number systems, such as the decimal number system, the octal number system, the binary number system, and the hexadecimal number system, have different characteristics.

These four varied types of categorizations have different bases, such as:

• Binary Number System has a base of 2

• Octal comes with base 8

• 10 is the base of a decimal number system

• hexadecimal has 16 as the base number

Binary Number System

The binary number system uses only two digits: 0 and 1. The numerals in this system have a base of two. Bits are the digits 0 and 1, and a byte is made up of 8 bits. Bits and bytes are used to store data in computers. The numbers like 2,3,4,5, etc., are neglected by the binary numeral system. For example, under the binary number system, 100102, 1011012, and 10100012 are some instances of numbers.

Octal Number System

The octal number system employs the digits 0,1,2,3,4,5,6, and 7 on an eight-digit basis. Because this system has fewer digits than others, it has less computational errors. The octal numeral system does not include the numbers 8 and 9. In minicomputers, the octal number system is used in the same way as binary, but with digits spanning from 0 to 7. 378, 338, and 1218 are examples of octal numerals.

Decimal Numeral System

The decimal number system has ten digits: 0,1,2,3,4,5,6,7,8, and 9, with 10 as the basis. In everyday life, the decimal numeral system is the most prevalent manner of expressing numbers. The base of any number that is expressed without the need for a base is 10. 72310, 3210, and 425710 are just a few examples of numbers in the decimal number system.

Hexadecimal Number System

There is a total combination of sixteen alphabets and digits, i.e., ranging from 0-9 and A-F. They have 16 regarded as the base number. The alphabets A,B,C,D,E,F are declared as the digits 10,11,12,13,14,15 of the decimal numeral classification. For minimising the occurrence of the large string, this approach is widely used in computers.

For instance, 8B216, 7F16, 5B6A16 can be the types of the hexadecimal number system.

Rules for Conversion

A number can be converted from one numerical system to another. Binary numbers can be converted to octal numbers, which can then be converted to decimal numbers, and so on. 

Converting Binary or Hexadecimal or Octal into Decimal Number System

For the transformation of any digit of any number system, such as binary or octal or hexadecimal, into decimal, we need to follow the given set of procedures. So as to efficiently convert the numeral into the decimal system.

Illustration 1: Conversion 1011112 into decimal number system

Step 1: Identification of the number at the base, so here it is 2 in the question

Step 2: Now multiply every digit of the number given with the exponent of the base and start with the rightmost digit. The value of the exponents must begin with 0 and should show an increment of 1 each time we move from R to L. The base is 2 here, therefore multiplying the digits by the numeral given by 20, 21, 22, etc., from right to left.

Step 3: Now simplify the above and add them together

The sum here in this illustration is the equivalent numeral represented in the decimal system of the number given. 

101111 = (1 x 25) + (0 x 24) + (1 x 23) + (1 x 22) + (1x 21) + (1 x 20)

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

= 32 + 0 + 8 + 4 + 2 + 1

= 47

Therefore, 1011112 = 4710

So by this technique of the number system formula, we can easily convert any number system into the desired one without much difficulty.

Conclusion

The following approach can be utilised to transform any given numeral in any of the four number systems like hexadecimal, octal, decimal, or binary. The problems based on the number system are common in all types of examinations in which quantitative aptitude is a part.

So, while converting any digit into a different number system, the doers should keep in mind that the base should be properly verified before putting that into the equation as insertion of the wrong base number will convert the digit into the wrong number system.

Therefore, it is advised to proficiently observe the base and then frame the equation and solve accordingly. This will decrease the chances of committing silly mistakes in the question.