Number Systems

A writing system for expressing numbers. Each number system defines a set of symbols that each represent a specific value.

Base (or radix)

Number of symbols defined by a number system.

Commonly used number systems

• Base 10 - $0\text{ - }9$
• Base 2 - $0,1$
• Base 8 - $0\text{ - }7$
• Base 16 - $0\text{ - }9, \text{A - F}$

Conversion between number systems

10 —> n

Integer part

• Repeatedly divide the number (and the quotients) by $n$ until reaching 1
• Write the remainders in reverse order

Fractional part

• Repeatedly multiply by $n$ until fractional part reaches 0
• Write the integer parts in normal order

n —> 10

Multiply each digit by its positional value, and sum those values. Positional value is $n^k$ where $k$ is the position.

2 —> 8

• Split the given binary number into length 3 parts (prepend 0s if required)
• Convert each part to octal
• Join those together

2 —> 16

• Split the given binary number into length 4 parts (prepend 0s if required)
• Convert each part to hexagonal
• Join those together

16 —> 2

Convert each digit to 4-bit binary and join them together.

8 —> 2

Convert each digit to 3-bit binary and join them together.

8 <—> 16

Convert the number to base 2 or 10 and then conver to the target base.

Caution

These are required in s1:

• Addition, subtraction in base 2, 8, 16
• Multiplication, division in base 2

But I don’t know how to include them in a easy-to-understand way. 🌝

Confusion about unit prefixes

In computing, the prefix kilo —just like other prefixes— has been used to refer either $2^{10}$ or $10^3$ depending on the context.

• $10^3$ - Marketing of disk capacities (by disk manufacturers)
• $2^{10}$ - Memory capacities, and file sizes, disk capacities by operating systems

To avoid this confusion, 2 unit prefixes are used while measuring amounts of data.

• SI prefixes Defined by ISO. Based on powers of $10^3$. Examples: kilo, mega, giga.
• Binary prefixes Defined by IEC. Based on powers of $2^{10}$. Examples: kibi, mebi, gibi.