Converting between integer data types

Recall:

There are many different 2s complement codes

Example: in Java

The 8 bits 2s complement codes is used in the byte data type
The 16 bits 2s complement codes is used in the short data type
The 32 bits 2s complement codes is used in the int data type
The 64 bits 2s complement codes is used in the long data type

Conversion between different 2s complement codes

When a variable is assigned to another variable of a different data type, the 2s complement code must be converted

Example:

Value 8 bits 2s compl code 16 bits 2s compl code ------------------------------------------------------------- -3 11111101 1111111111111101

Converting between integer data types

Example of a Java program when conversion of 2s complement codes takes place:

public static void main(String[] arg) { byte b; short s; int i; b = -3; // Stores 11111101 in b s = b; // Converts 11111101 to 1111111111111101 i = b; // Converts 11111101 to 11111111111111111111111111111101 }

DEMO: /home/cs255001/demo/java/ShowIntConv.java

The need to convert between different data representations

Background information on the CPU:

The CPU performs arithmetic (and logic) operations
Arithmethic (and logic) operations uses two operate on binary numbers in the CPU
Circuitries in the CPU can only operate on binary numbers of the
same length
Example:
00000011 + 00000100 0000000000001000 - 0000000000000111

Consequently:

When we operate on values from 2 different representations:
We must convert one representation into the other data representation before we can perform the operation.

The data representation conversion problem explained in diagrams

The computer can represent signed values using the 8 bits 2s complement code:

The computer can perform arithmetic (e.g.: +) operation on two 8 bits 2s complement codes

The data representation conversion problem explained in diagrams

The computer can represent signed values using the 16 bits 2s complement code:

The computer can perform arithmetic (e.g.: +) operation on two 16 bits 2s complement codes

The data representation conversion problem explained in diagrams

However:

The computer can not perform arithmetic (e.g.: +) operation on mixed 2s complement codes

The data representation conversion problem explained in diagrams

Solution:

The computer first converts one 2s complement represention to the other 2s complement represention

The goal of a conversion operation

The goal of conversion is:

Find the representation in another encoding method that represent the same value (or approximately the same value) as the original encoding method

Examples:

8 bits 16 bits Value 2s complement code 2s complement code ------------------------------------------------------ 3 00000011 0000000000000011 -3 11111101 1111111111111101

Question:
How do you convert between two different 2s complement codes ???

How to convert a byte representation to an int representation

List of values and their 2s compl representation in byte and int data type:

Value byte repr int repr ------- ---------- --------------------------------- 127 01111111 00000000000000000000000001111111 ... 3 00000011 00000000000000000000000000000011 2 00000010 00000000000000000000000000000010 1 00000001 00000000000000000000000000000001 0 00000000 00000000000000000000000000000000 -1 11111111 11111111111111111111111111111111 -2 11111110 11111111111111111111111111111110 -3 11111101 11111111111111111111111111111101 ... -128 10000000 11111111111111111111111110000000

Can you see how to convert an byte repr of some value into an int repr ?

How to convert a byte representation to an int representation

Conversion: Extend the sign bit of the byte representation to fill 32 bits

The sign bit is the left-most bit in the byte representation

Value byte repr int repr ------- ---------- --------------------------------- 127 01111111 00000000000000000000000001111111 ... 3 00000011 00000000000000000000000000000011 2 00000010 00000000000000000000000000000010 1 00000001 00000000000000000000000000000001 0 00000000 00000000000000000000000000000000 -1 11111111 11111111111111111111111111111111 -2 11111110 11111111111111111111111111111110 -3 11111101 11111111111111111111111111111101 ... -128 10000000 11111111111111111111111110000000

We call this operation: sign extension

How to convert an int representation to a byte representation

Notice that:
for the value in the range [−128, 127], the last 8 bits in the int repr and byte repr are equal:

Value byte repr int repr ------- ---------- --------------------------------- ... 128 Not possible 00000000000000000000000100000000 127 01111111 00000000000000000000000001111111 ... 3 00000011 00000000000000000000000000000011 2 00000010 00000000000000000000000000000010 1 00000001 00000000000000000000000000000001 0 00000000 00000000000000000000000000000000 -1 11111111 11111111111111111111111111111111 -2 11111110 11111111111111111111111111111110 -3 11111101 11111111111111111111111111111101 ... -128 10000000 11111111111111111111111110000000 -129 Not possible 11111111111111111111111101111111 ...

How to convert an int representation to a byte representation

Conversion: truncate the int repr to a byte by keeping the right-most 8 bits

However: this conversion is only safe when the value of the int repr is in the range [−128, 127]

Value byte repr int repr ------- ---------- --------------------------------- ... 128 Not possible 00000000000000000000000100000000 127 01111111 00000000000000000000000001111111 ... 3 00000011 00000000000000000000000000000011 2 00000010 00000000000000000000000000000010 1 00000001 00000000000000000000000000000001 0 00000000 00000000000000000000000000000000 -1 11111111 11111111111111111111111111111111 -2 11111110 11111111111111111111111111111110 -3 11111101 11111111111111111111111111111101 ... -128 10000000 11111111111111111111111110000000 -129 Not possible 11111111111111111111111101111111 ...

Where do you find int to byte conversion operations used ?

One place where int to byte conversion (and vice versa) is performed is in assignment statements:

int i; byte b; i = b; // Converts byte repr in b tnto int repr // before storing the representation in i b = (byte) i; // Converts int repr in i tnto byte repr // before storing the representation in b

Quiz on int --> byte casting

Quiz:

Given that: 4100 (decimal) is represented by the 2s complement code 00000000000000000001000000000100.
What is the value of the byte variable b in this statement:

Quiz on int --> byte casting

Quiz:

Given that: 4100 (decimal) is represented by the 2s complement code 00000000000000000001000000000100.
What is the value of the byte variable b in this statement:

Answer: 4 (because b contains 00000100)
Demo: /home/cs255001/demo/java/JavaByteIntCast.java

How to convert a short representation to an int representation

List of values and their 2s compl representation in short and int data type:

Value short repr int repr ------- ----------------- --------------------------------- 32767 0111111111111111 00000000000000000111111111111111 ... 3 0000000000000011 00000000000000000000000000000011 2 0000000000000010 00000000000000000000000000000010 1 0000000000000001 00000000000000000000000000000001 0 0000000000000000 00000000000000000000000000000000 -1 1111111111111111 11111111111111111111111111111111 -2 1111111111111110 11111111111111111111111111111110 -3 1111111111111101 11111111111111111111111111111101 ... -32768 1000000000000000 11111111111111111000000000000000

Can you see how to convert an short repr of some value into an int repr ?

How to convert a short representation to an int representation

Conversion: Extend the sign bit of the short representation to fill 32 bits

The sign bit is the left-most bit in the short representation

Value short repr int repr ------- ----------------- --------------------------------- 32767 0111111111111111 00000000000000000111111111111111 ... 3 0000000000000011 00000000000000000000000000000011 2 0000000000000010 00000000000000000000000000000010 1 0000000000000001 00000000000000000000000000000001 0 0000000000000000 00000000000000000000000000000000 -1 1111111111111111 11111111111111111111111111111111 -2 1111111111111110 11111111111111111111111111111110 -3 1111111111111101 11111111111111111111111111111101 ... -32768 1000000000000000 11111111111111111000000000000000

This operation is: sign extension

How to convert an int representation to a short representation

Notice that:
for the value in the range [−32768, 32767], the last 16 bits in the int repr and short repr are equal:

Value short repr int repr ------- ----------------- --------------------------------- 32767 0111111111111111 00000000000000000111111111111111 ... 3 0000000000000011 00000000000000000000000000000011 2 0000000000000010 00000000000000000000000000000010 1 0000000000000001 00000000000000000000000000000001 0 0000000000000000 00000000000000000000000000000000 -1 1111111111111111 11111111111111111111111111111111 -2 1111111111111110 11111111111111111111111111111110 -3 1111111111111101 11111111111111111111111111111101 ... -32768 1000000000000000 11111111111111111000000000000000

How to convert an int representation to a short representation

Conversion: truncate the int repr down to 16 bits keeping the right-most bits

However: this conversion is only safe when the value of the int repr is in the range [−32768, 32767]

Value short repr int repr ------- ----------------- --------------------------------- 32767 0111111111111111 00000000000000000111111111111111 ... 3 0000000000000011 00000000000000000000000000000011 2 0000000000000010 00000000000000000000000000000010 1 0000000000000001 00000000000000000000000000000001 0 0000000000000000 00000000000000000000000000000000 -1 1111111111111111 11111111111111111111111111111111 -2 1111111111111110 11111111111111111111111111111110 -3 1111111111111101 11111111111111111111111111111101 ... -32768 1000000000000000 11111111111111111000000000000000

Quiz on int --> short casting

Quiz:

Given that: 262150 (decimal) is represented by the 2s complement code 00000000000001000000000000000110.
What is the value of the short variable s in this statement:

Quiz on int --> short casting

Quiz:

Given that: 262150 (decimal) is represented by the 2s complement code 00000000000001000000000000000110.
What is the value of the short variable s in this statement:

Answer: 6 (because s contains 0000000000000110)
Demo: /home/cs255001/demo/java/JavaShortIntCast.java