DECIMAL NUMBER SYSTEM – MINDSPARK

DECIMAL NUMBER SYSTEM

Number system is a system of representing numbers. In a number system, we can represent a number by its base. A base of 2 means it is a binary number, base of 8 means it is an octal number and if the base is 10 it is called the decimal number system. A base of 16 means it is a hexadecimal.

Decimal number system is used in various computer applications and is also called the base 10 number system. It consists of 10 digits from 0 to 9. 

In this number system every value has the place value of power 10 and can be written as: 

(82)_{10}=8 \times 10^{1}+2 \times 10^{0}

The digits to the right side of the decimal point are written with their decreasing power of 10.

For example, if we have 125.6 then we can write it as: 

(125.6)_{10}=1 \times 10^{2}+2 \times 10^{1}+5 \times 10^{0}+6 \times 10^{-1}

HOW TO READ A DECIMAL NUMBER?

When we write a number, say 145.68, the number on the left side of the decimal point is read just like we would read in case of no decimal. Here, 145 = 100 + 40 + 5 which is one hundred forty five. The number to the right of the decimal is actually a number with decreasing power so 6 is actually 6-tenth and 8 is 8-hundredth. We read these numbers individually. We read the number 145.68 as one hundred forty five point six eight.  

CONVERTING OTHER BASES INTO DECIMAL NUMBER SYSTEM

1. Converting Binary to Decimal

When we have binary numbers, which have the base 2, we can easily convert them to decimals with base 10.

Consider the number 1101_2 which is a binary number. We have to convert it to a decimal number with base 10. 

We simply multiply the digits with decreasing power of 2 as follows-

1 \times 2^{3}+1 \times 2^{2}+0 \times 2^{1}+1 \times 2^{0}

=8+4+0+1

=13

Therefore,1101_{2}=13_{10}.

2. Converting Octal to Decimal

Octal number has a base of 8. Every digit of the number is multiplied by the decreasing power of 8. 

Consider the number 112_{8}which is an octal number. We will convert it to decimal number as follows – 

1 \times 8^{2}+1 \times 8^{1}+2 \times 8^{0}

=64+8+2

=74 

Therefore,112_{8}=74_{10}.

3. Converting Hexadecimal to Decimal

A hexadecimal number has the base of 16. So, if we have to convert the number into decimals, we will multiply by the decreasing power of 16. 

If we have the number 14_{16}. We will convert it to decimals like this –

1 \times 16^{1}+4 \times 16^{0}

=16+4

Therefore,14_{16}=20_{10}.

CONVERTING DECIMALS INTO OTHER BASES 

1. Decimals to Binary

When we converted binary to decimals, we multiplied by the decreasing power of 2. Here, we divide by 2 till the quotient is 0. 

Suppose we have a number 22_{10}.

Divide 22 by 2, we get:

\frac{22}{2}= 11 Remainder – 0 (most significant bit)

\frac{11}{2}= 5 remainder – 1

\frac{5}{2}= 2 remainder – 1

\frac{2}{2}= 1 remainder – 0

\frac{1}{2} = 0 remainder – 1 (least significant bit)

We will write this number from the most significant bit to the least significant bit from left to right. The binary equivalent of 22_{10}=10110_{2}.

2. Decimals to Octal numbers

In the case of converting decimals to octal numbers, we divide by 8 until we reach 0 as the quotient.

Consider the number 65_{10}.

We will start by dividing with 8.

\frac{65}{8}= 8 Remainder – 1 (most significant bit)

\frac{8}{8}= 1 Remainder – 0

\frac{1}{8}= 0 remainder – 1 (least significant bit)

Therefore, the octal number for 65_{10} \text { is } 101_{8}.

3. Decimals to Hexadecimals

When we convert decimals to hexadecimals we divide by 16.

We consider the number 110_{10}. We divide the number by 16.

\frac{110}{6}= 6 remainder – 14

\frac{6}{16}= 0 remainder – 6

We replace the numbers 10, 11, 12, 13, 14 with alphabets A, B, C, D, E respectively. 

So, we get 110_{10}=6 E_{16}.