Number System Base and Number System Conversion by Eliza Joanne

Number System Base

Note: 
We do not need to memorize each binary digit for its equivalent decimal conversion as there is a clear pattern of repetition as shown above.

Decimal number

• Base of 10
• Composed of 10 digits (0 to 9)

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Binary number

• Base of 2
• Consist of only digits 0 and 1

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Hexadecimal number

• Base of 16
• Composed of alphanumeric values 0 to F
• Suitable to represent binary numbers with a multiple of 4 bits 

Number System Conversion

Decimal to binary

The conversion of decimal (base 10) to binary (base 2) is a simple division of the decimal number by two. The remainder from the division by 2 makes up a bit of the binary number.
Here are some examples (using the most common method) :

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For the decimal part .3125, we take the decimal and multiply by 2:

                                0.3125 x 2 = 0.625

Next we take .625 and do the same multiplication step:

                                0.625 x 2 = 1.25

The multiplication step is repeated by taking .025:

*Note: we only take the decimal part.

                                0.25 x 2 = 0.50

The step is again repeated until we obtain the value 1.00. We stop here because we have zero as the decimal part value.

Hence,                      0.50 x 2 = 1.00

The whole number part of the result is the binary digit.

Therefore, the binary value for 57.3125 is 110011.0101₂

Binary to decimal and hexadecimal to decimal

To convert binary number to decimal number, we add the product of the values and the digits of each bit. The values of each bit are 2 to the power of that bit.

Example:

Likewise for hexadecimal to decimal.
Example:

Binary to hexadecimal

To convert a binary number to hexadecimal, we divide it into groups of four bits starting from the right. If the number bits is not a multiple of four we add extra 0 bits to the left. This action is called padding.

Examples:

As there are 8 bits, we simply divide it to groups of four bits and find its equivalent hexadecimal value.








The number of bits above is not a multiple of 4, therefore we add two extra 0 bit to the left and then divide it to groups of 4 bits.

Hexadecimal to binary

This conversion is probably the easiest of all. We just take the hexadecimal value and substitute it with its corresponding binary digits.

For example:

4A₁₆ = 01001010₂

D64₁₆ = 110101100100₂

These topics may not be the heart of Computer Organization & Architecture but without this knowledge, understanding upcoming topics will be difficult. These are the basics in Computer Organization & Architecture.
Thank You.