# Various Conversion Methods of Binary, Decimal and Hexadecimal

We mention here various conversion methods like binary to decimal, hexadecimal to decimal, hexadecimal to binary conversion are provided below in detail:

## Binary to decimal conversion

A binary number can be expressed as the sum of the product of each of this digit and the digit’s place value.

For example,

110101 = 1 x 2⁵ + 1 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2⁰

110.101 = 1 x 2² + 0 x 2¹ + 1x 2⁰ + 1 x 2⁻¹ + 1 x 2⁻² + 0 x 2⁻³

Since each power of two it weighted by either 0 or 1, and the binary number is the sum of the place values in which the bit 1 appears. This sum gives the decimal equivalent of the binary number.

The table lists the binary representations of the integers from 0 to 20, with the place value of the bits shown at the top.

Binary representation of decimal number

Example: Covert a) 10101 and b) 101.110 to decimal.

Ans: a) To convert 101012 to its decimal equivalent, write the appropriate place value over each bit and then and up those powers of two which are weighted by 1:

b) To convert 101.110₂ to its decimal equivalent, use the decimal values of the powers of two

## Decimal to binary conversion

It is possible to find binary representation of a decimal number N by converting its integral part (NF ) separately.

Example: Covert 109.78125 to binary number

Ans: a) to convert the integral part N1, that is 109, to binary, divide N1  and each successive quotient by 2, nothing the remainders, as follows:

The zero quotients indicate the end of the divisions. The sequence of remainders from the bottom to up, as indicated by the arrow, yields the required binary equivalent. That is, N1 = 109= 11011012.

In practice, the above divisions may be shown as:

Here the division processes are stopped when the quotient, 1, is less than the divisor 2. Again the arrow indicates the sequence of bits that give the binary equivalent of the number.

b) To convert, the fractional part NF, that is 0.78125, to its binary equivalent, multiply NF and each successive fractional part by 2, noting an integral part of the product, as follows:

The zero fractional part indicates the end of the multiplications. It is observed that the integral part of any product can only be 0 or 1. The sequence of integral parts as

Indicated by the arrow, yields the required binary number.

That is, NF,=0.78125=0.110012

In practice, the above multiplications may be shown as:

It is observed that the integral part of each product is underlined and is not used in the next multiplication. Again the arrow indicates the sequence of digits that give the required binary representation.

The binary equivalent of N is simply the sum of the binary equivalents of integral and fractional parts.

N= N1 + NF = 1101101.11001

Example:  Convert 13.6875 to decimal

Remark: the binary equivalent of a decimal fraction does not always terminate. For example, for the decimal number 0.6.

This shows that the above four steps will be repeated again and again. That is,

Decimal 0.6= 0.1001 1001 1001… …

Conversion between the hexadecimal and decimal systems is accomplished using the algorithms of section 2.3 with b=16. Additionally one has to know how to handle the hexadecimal digits A, B, C, D, E, and F.

One can also convert from hexadecimal to decimal by decimal by decimal evaluation of the expanded hexadecimal form.

Example: covert 73D516  to its decimal equivalent.

Example: covert 39.B816  to its decimal equivalent.

Example: covert the decimal number 9719 to its hexadecimal equivalent.

The base of a hexadecimal number is 16. Divide the number and each successive quotient by 16, noting the remainders, as follows:

Here the sequence of remainders, with F for reminder 15, in reverse order gives the hexadecimal form the decimal number.

Example: Covert the decimal fraction 0.78125 to its hexadecimal equivalent.

For this conversion apply the integral-part algorithm, with base 16, as follows:

In this case, a zero fractional part is reached. The sequence of integral parts, with C for 12 gives the required hexadecimal form.

Example: covert the decimal number 9719.78125 to its hexadecimal form.

From the prevision example we know, 971910 = 251710  and. 7812510 = C816

Combining the two parts, 9717.7812510=25F7.C816

This is accomplished exactly as octal-binary conversion, except that 4-bit equivalents are now involved.

Example: convert to binary form a) 3D5916, b) 27.A3C16.

Replacing each hexadecimal digit by its 4-bit representation

a) 3D 5916 = 0011 1101 0101 1001

That is, 3D 5916 = 111101010110012

b) Similarly, 27.A3C16 = 0010 0111.1010 0011 1100

That is, 27. A3C16 =100111.10100011112

Example: convert the binary numbers

a) 101101001011102,

b) 11100.10110110112

Partitioning each binary number into 4-bit blocks to the left and right of the fractional points and then replacing each 4-bit block by its equivalent hexadecimal digit,

a) Binary 0010 1101 0010 1110 Hexadecimal 2D2E

That is, 2d2E16 is the required hexadecimal form.

b) Similarly, 0001 1100.1011 0110 11002 = 1C.B6C16

Hence, 1C.B6C16  is the required hexadecimal form.

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## 2 comments for “Various Conversion Methods of Binary, Decimal and Hexadecimal”

1. Deangelo
June 12, 2014 at 7:18 PM

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• rasel
July 3, 2015 at 8:43 AM

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