Test: Types of Number System - GATE MCQ

Concept:
A number system includes the number of independent digits used in the number system (the base), the place values of the different digits constituting the number, and the maximum numbers that can be written with the given number of digits.
Octal numbers: These numbers use digits from 0 to 7, total of 8 digits, and hence they are called octal number system. Octal numbers have base 8.
Hexadecimal numbers: The numbers which have base 16. It uses 16 different digits to represent the numbers. It is denoted as h16, where h is a hexadecimal number. It may be a combination of alphabets and numbers. Thus, it includes numbers from 0 to 9 and alphabets A to F.

Calculation:
Hexadecimal to Octal Conversion:
Given, (FB2)₁₆ is a hexadecimal number.
F → 1111, B → 1011, 2 → 0010
1111 1011 0010
Now group them from right to left, each having 3 digits.
111, 110, 110, 010
111 → 7, 110 → 6, 110 → 6, 010 → 2
Hence, FB2₁₆ = 7662₈

Convert the given b-radix into a decimal system.
(154)_b / (14)_b = (8)₁₀
(1 x b² + 5 x b¹ + 4 x b⁰)₁₀ / (1 x b¹ + 4 x b⁰)₁₀ = (8)₁₀ (b² + 5 b + 4) / (b + 4) = 8
(b² + 5 b + 4) = 8 (b + 4) 
b² + 5b + 4 = 8b + 32
b² - 3b - 28 = 0
b² - 7b + 4b - 28 = 0
b(b - 7) + 4(b - 7) = 0
b = -4 or b = 7
b = 7 Here b value should be greater than 5 because it maximum digit in b radix.
Hence the correct answer is 7.

Concept:
Another number system to Decimal
In each and every representation of numbers with different bases, the maximum value in a number system with the base ‘r’ is r – 1. Since numbers vary from 0 to r – 1.
To convert any number which is in a different base to a decimal number system we use binary-weighted representation.
Eg: let the number be ( abc⋯ ⋯ yz)_r
Now to convert the above number into the decimal number system
a × r^n-1 + b × r^n-2 + ⋯ ⋯ + y × r¹ + z × r⁰
If we convert all numbers into decimals then we can perform normal addition and subtraction etc.
Calculation:
>Octal to decimal conversion →
(10)₈ = 1 × 8¹ + 0 × 8⁰ = 8 + 0 = (8)₁₀

Concept:
Another number system to Decimal
In each and every representation of numbers with different bases, the maximum value in a number system with the base ‘r’ is r – 1. Since numbers vary from 0 to r – 1.
To convert any number which is in the different base to decimal number system we use binary-weighted representation.
Eg: let the number be ( abc⋯ ⋯ yz)_r
Now to convert the above number into the decimal number system
a × r^n-1 + b × r^n-2 + ⋯ ⋯ + y × r¹ + z × r⁰
If we convert all numbers into decimal then we can perform normal addition and subtraction etc.

Application:
Given:
(110)_x = (132)₄,
The decimal equivalent of this number will be:
1 x x² + 1 x x¹ + 0 x x⁰ = 1 x 4² + 3 x 4 + 2 x 4⁰
x² + x + 0 = 16 + 12 + 2
x² + x - 30 = 0 
On solving this quadratic equation we'll get:
x = 5, -6
Base can't be a negative so;
x = 5

Concept:
Hexadecimal number: In this, value of the base is 16. Each digit is represented by 4-bit binary no.
Octal number: For octal number, value of base is 8. Each digit of an octal number is represented by 3-bit binary no.
Octal number = 657
Binary representation for this number (each digit of a octal number is converted into 3 binary bits) 
So, 657 in binary is equivalent to 110 101 111
Now group this binary number into 4 bits starting from right to left. 
i.e. 0001 1010 1111
Hexadecimal representation for this number is : 1AF

Concept:
Decimal to binary:

• Take decimal number as dividend.
• Divide the number by 2.
• Get the integer quotient for the next iteration.
• Get the remainder (it will be either 0 or 1 because of divisor 2).
• Repeat the steps until the quotient is equal to 0
• Write the remainders in reverse order (which will be equivalent binary number of given decimal number).

Decimal to binary: (fractional part)

• Take decimal number as multiplicand.
• Multiple this number by 2 (2 is base of binary so multiplier here).
• Store the value of integer part of result in an array (it will be either 0 or 1 because of multiplier 2).
• Repeat the above two steps until the number became zero.
• Write these resultant integer part

Calculation:
Binary of  57:

Now, write remainder from bottom to up (in reverse order), this will be 111001 which is equivalent binary number of decimal integer 57.
Convert decimal fractional number 0.375 into binary number.
Here, decimal fraction: 0.375

Now, write these resultant integer part, this will be 0.0110 which is equivalent binary fractional number of decimal fractional 0.375.
∴ 57.375 can be written as 111001.011 in binary
Hence, option (1) is correct.

Formula:
If we covert x decimal to binary, divide x successively by 2 until the quotient is 0.

Calculation:
Divide 23 successively by 2 until the quotient is 0:
23/2 = 11, remainder is 1 (LSB)
11/2 = 5, remainder is 1
5/2 = 2, remainder is 1
2/2 = 1, remainder is 0
1/2 = 0, remainder is 1 (MSB)
Read from the bottom (MSB) to top (LSB) as 10111
∴ 10111 is the binary equivalent of decimal number 23

A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression that uses only two symbols: typically "0" and "1". The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit or binary digit.
Decimal system, also called Hindu-Arabic number system or Arabic number system, in mathematics, a positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It also requires a dot to represent decimal fractions.

Decimal system to binary system:
(0)₁₀ = (0)₂
(1)₁₀ = (1)₂
(2)₁₀ = (10)₂
(3)₁₀ = (11)₂
(4)₁₀ = (100)₂
(5)₁₀ = (101)₂
(6)₁₀ = (110)₂
(7)₁₀ = (111)₂
(8)₁₀ = (1000)₂
(9)₁₀ = (1001)₂
(10)₁₀ = (1010)₂....
Hence the correct answer is two.

Concept:
The binary number system with only two independent digits, 0 and 1, is a base-2 number system. All larger binary numbers are represented in terms of ‘0’ and ‘1’.
The decimal number is equal to the sum of binary digits (D_n) times their power of 2 (2ⁿ).
Decimal = - - - + D₂ × 2² + D₁ × 2¹ + D₀ × 2⁰ - - -

Calculation:
Given binary number = 11101₂
The equivalent decimal representation will be:
Decimal = 1 × 2⁴ + 1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰
Decimal = 16 + 8 + 4 + 0 + 1
Decimal = (29)₁₀

Octal Number: 

• The number to the Base 8 is called octal numbers.
• It uses the numbers from 0 to 7.
• The numbers 8 and 9 are not included in the octal number system.
• It is denoted by a₈ where a is a number with digits 0 to 7.

Conversion from Binary to Octal:
An octal Equivalent of a binary number is obtained by grouping 3 bits from right to left.

So, Octal Equivalent: 321.
Binary to Octal code: