CS 101
Computer Science 1 in Java
Prof. Alvarez

Example of the Binary Representation of Numbers

Here is an example of binary codes for integers and floating-point numbers in a portion of a computer program. The language used for the program below is not Java, but rather assembly language, which is much closer to the level understood by the hardware inside the computer.

Example Program

The example program appears in the rightmost column below. The memory of a computer like the one used for this example is a sequence of consecutively numbered cells, each of which can store a 16-bit "word". A given program occupies a certain collection of words in memory. The address of a statement in the program can be described by giving the number of words from the start of the program, or offset, of the first word of the statement. The leftmost column below shows the address offsets of the corresponding program statements in the rightmost column. The second column from the left shows the hexadecimal encodings of the program statements themselves. 
 0000				.data
 0000 0003				piAsAnInt dw 3
 0002 FFFD				negPiAsAnInt dw -3
 0004 4048F5C3				piToTwoDecimals dd 3.14

 0000				.code
 0000				main proc
 0000  B8 ---- R			mov ax, @data
 0003  8E D8				mov ds, ax
 0005  B4 09				mov ah, 9
 0007  BA 0004 R			mov dx, offset piToTwoDecimals
 000A  CD 21				int 21h
 000C				main endp

Numerical Representation in the Example Program

The above program defines three variables called piAsAnInt, negPiAsAnInt, and piToTwoDecimals respectively. The decimal values of these variables appear in the right column near the top of the file. The binary encodings of these values appear in the second column from the left. Thus, the decimal value -3 is represented by the hexadecimal value FFFD. In binary, this is 1111111111111101, which is obtained from the positive value 0000000000000011 by complementing each bit (ones complement), and then adding 1 to obtain the twos complement. The binary encoding of the floating point value 3.14 is a bit more involved. It can be obtained by using the IEEE 754 standard for floating point representation.