Other number systems used in computer programming

In system level programming (= low level programming), programmers often needs to use the binary representation
Writing out binary numbers is very tedious (long):
Instead of the binary repr., system level programmers often use:
to shorten the print out of binary numbers

Example:

Binary representation Octal representation 1101110111010101 156725

The Octal Number System

The Octal Number System:

Has 8 digits: 0, 1, 2, 3, 4, 5, 6, 7
The value of a digit at the k^th position is weighted by 8^k

Example:

153₍₈₎ = 107₍₁₀₎ ^^^ ||| ||+----- 3 * 8⁰ = 3 * 1 = 3 |+------ 5 * 8¹ = 5 * 8 = 40 +------- 1 * 8² = 1 * 64 = 64 + ---------- 107₍₁₀₎

Displaying/printing binary numbers

Displaying binary numbers is very verbose:

Binary numbers has many digits...

Displaying/printing binary numbers

Binary numbers are often displayed as octal/hexadecimal numbers for brevity:

because conversion between binary <--> octal and binary <--> hexadecimal are very easy

How to convert between binary representation <--> octal representation

Converting between binary representation <--> octal representation:

How to convert a binary number (= representation) → an octal number (= representation):
How to convert an octal number (= representation) → a binary number (= representation):

Example on how to convert: binary numbers ⇒ octal numbers

Conversion table between 1 octal digit and 3 binary digits:

Octal digit Binary digits ----------- ------------- 0 <--> 000 1 <--> 001 2 <--> 010 3 <--> 011 4 <--> 100 5 <--> 101 6 <--> 110 7 <--> 111 (Leading 0's can optionally be truncated)

Example: 11111101₍₂₎ ===> 375₍₈₎

Note: you must start grouping the binary digits from right → left

Note: you may need to add leading 0 bits to the binary number to convert the last octal digit

Example on how to convert: octal numbers ⇒ binary numbers

Conversion table between 1 octal digit and 3 binary digits:

Octal digit Binary digits ----------- ------------- 0 <--> 000 1 <--> 001 2 <--> 010 3 <--> 011 4 <--> 100 5 <--> 101 6 <--> 110 7 <--> 111 (Leading 0's can optionally be truncated)

Example: 174253₍₈₎ = 1111100010101011₍₂₎ (in 16 bits)

Note: you may need to drop some leading 0 bits from the binary number to get the desired length in the binary number

The octal dump application in UNIX

From Wikipedia:

I will demo the UNIX od command in class using these commands:

cd /home/cs255001/demo/dump od -t o1 bin-file // -t o1 means: display 1 byte data in octal // Adding z will print corresponding characters if possible

Each octal number in the output represents 8 bits (1 byte)