Introduction to Different Number Systems (Decimal, Binary, Octal, and Hexadecimal) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making PDF Download

Some important number systems are as follows.

• Decimal number system
• Binary number system
• Octal number system
• Hexadecimal number system

Decimal number System

The Decimal Number System consists of ten digits from 0 to 9. These digits can be used to represent any numeric value. The base of decimal number system is 10. It is the most widely used number system. The value represented by individual digit depends on weight and position of the digit.

Let's take a closer look at something most of us take for granted.  Counting in decimal numbers.  For example, when you see the decimal number:  

4275

it means that you sum the following to get a concept of the total quantity of things represented by that number: 

which is the same as: 

(Remember that in mathematics, any number  raised to the zero power equals 1)

The number that is the on the far right is in the "ones" place holder (called the least significant digit).  To the left of that is the "tens" place holder.  To the left of that is the "hundreds" place holder, and so on.  They are all powers of 10.  Whatever digit is sitting in a particular place holder, you multiply that place holder amount ( power of 10) by the digit, and then add them all up. 

= 4 thousand, 2 hundred, and seventy-five 
= 4275

Binary Number System

Digital computer represents all kinds of data and information in the binary system. Binary Number System consists of two digits 0 and 1. Its base is 2. Each digit or bit in binary number system can be 0 or 1. A combination of binary numbers may be used to represent different quantities like 1001. The positional value of each digit in binary number is twice the place value or face value of the digit of its right side. The weight of each position is a power of 2.

In binary (Base-2), all the columns are powers of 2.  Think of the following table as the four least significant digit place holders for the binary number system just like the tables above using the decimal number system: 

The far right colum is the "ones" place holder.  To the left of that is the "twos" place holder.  To the left of that is the "fours" place holder (2² = 4).  To the left of that is the "eights" place holder (2³ = 8).  They are all powers of 2.  Whatever digit is sitting in a particular place holder, you multiply that place holder amount ( power of 2) by the digit, and then add them all up.  However, remember that a binary digit has only two options, 0 or 1, so it's very easy.

For example, let's look at the binary number: 

1011

You would this  interpret in decimal as: 

The total is:  11 (in decimal)  which equals   8 + 0 + 2 + 1

In other words,

1011₂  =  11₁₀

(Note: the subscripted number indicates the number base) 

Octal Number System

Octal Number System consists of eight digits from 0 to 7. The base of octal system is 8. Each digit position in this system represents a power of 8. Any digit in this system is always less than 8. Octal number system is used as a shorthand representation of long binary numbers. The number 6418 is not valid in this number system as 8 is not a valid digit.

Let's dig right in:

• Convert 357₁₀ to the corresponding base-eight number.

I will do the usual sequential division, this time dividing by 8 at each step:

Once I got to the "5" on top, I had to stop, because 8 doesn't divide into 5.

Then the corresponding octal number is 545₈.

• Convert 545₈ to the corresponding decimal number.

I will follow the usual procedure, listing the digits in one row and then counting off the digits from the RIGHT in the next row, starting at zero:

Then I'll do the usual addition and multiplication:

5×8² + 4×8¹ + 5×8⁰

= 5×64 + 4×8 + 5×1

= 320 + 32 + 5

= 357

Then the corresponding decimal number is 357₁₀.

Hexadecimal Number System

The Hexadecimal Number System consists of 16 digits from 0 to 9 and A to F. The alphabets A to F represent decimal numbers from 10 to 15. The base of this number system is 16. Each digit position in hexadecimal system represents a power of 16. The number 76416 is valid hexadecimal number. It is different from 76410 which is seven hundred and sixty four. This number system provides shortcut method to represent long binary numbers.

• Convert 357₁₀ to the corresponding hexadecimal number.

Here, I will divide repeatedly by 16, keeping track of the remainders as I go. (You might want to use some scratch paper for this.)

Reading off the digits, starting from the top and wrapping around the right-hand side, I see that:

357₁₀ = 165₁₆.

• Convert 165₁₆ to the corresponding decimal number.

List the digits, and count them off from the RIGHT, starting with zero:

Remember that each digit in the hexadecimal number represents how many copies you need of that power of sixteen, and convert the number to decimal:

1×16² + 6×16¹ + 5×16⁰

= 1×256 + 6×16 + 5×1

= 256 + 96 + 5

= 357

Then 165₁₆ = 357₁₀.