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The meaning of a given sequence of bits within a computer depends on the context. In this section we describe how to represent integers in binary, decimal, and hexadecimal and how to convert between different representations.

We also describe how to represent negative integers and floating-point numbers. Since Babylonian times, people have represented integers using positional notation with a fixed base. The most familiar of these systems is decimal , where the base is 10 and each positive integer is represented as a string of digits between 0 and 9. When the base is 2, we represent an integer as a sequence of 0s and 1s.

In this case, we refer to each binary base 2 digit—either 0 or 1—as a bit. You need to know how to convert from a number represented in one system to another. Converting between hex and binary.

Given the hex representation of a number, finding the binary representation is easy, and vice versa, because 16 is a power of 2. To convert from hex to binary, replace each hex digit by the four binary bits corresponding to its value. Conversely, to convert from binary to hex, prepend leading 0s to make the number of bits a multiple of 4, then group the bits 4 at a time and convert each group to a single hex digit.

Converting from decimal to base b. It is slightly more difficult to convert an integer represented in decimal to one in base b because we are accustomed to performing arithmetic in base The easiest way to convert from decimal to base b by hand is to repeatedly divide by the base b , and read the remainder upwards.

For example, the calculations below convert from the decimal integer to binary and to hexadecimal 16E. Parsing and string representation. Converting a string of characters to an internal representation is called parsing. First, we consider a method parseInt to parse integers written in any base.

Next, we consider a toString method to compute a string representation of an integer in any given base. These are simplified versions of Java's two-argument Integer. The following table contains the decimal, 8-bit binary, and 2-digit hex representations of the integers from 0 to The first operations that we consider on integers are basic arithmetic operations like addition and multiplication. In grade school you learned how to add two decimal integers: Repeat with the next digit, but this time include the carry bit in the addition.

The same procedure generalizes to any base by replacing 10 with the desired base. We can represent only 2 n integers in an n -bit word. For example, with bit words, we can represent the integers from 0 to 65, Wee need to pay attention to ensure that the value of the result of an arithmetic operation does not exceed the maximum possible value.

This condition is called overflow. For addition of unsigned integers, overflow is easy to detect: The grade-school algorithm for multiplication works perfectly well with any base.

Detecting overflow is a bit more complicated than for unsigned integers. To negate a two's complement integer, flip the bits and then add 1. This explains the bounds on values of these types and explains the behavior on overflow in Java that we first observed in Section 1. The IEEE standard defines the behavior of floating-point number of most computer systems. For simplicity, we illustrate with a bit version known as half-precision binary floating point or binary16 for short.

The same essential ideas apply to the bit and bit versions used in Java, which we refer to as binary32 and binary The real-number representation that is commonly used in computer systems is known as floating point. It is just like scientific notation, except that everything is represented in binary. Just as in scientific notation, a floating-point number consists of a sign , a coefficient , and an exponent.

The first bit of a floating-point number is its sign. The sign bit is 0 is the number is positive or zero and 1 if it is negative. The remaining 10 bits are devoted to the coefficient. The normalization condition implies that the digit before the decimal place in the coefficient is always 1, so we need not include that digit in the representation. The bits are interpreted as a binary fraction, so 1. Encoding and decoding floating-point numbers. Given these rules, the process of decoding a number encoded in IEEE format is straightforward.

The process of encoding a number is more complicated, due to the need to normalize and to extend binary conversion to include fractions. Java uses binary32 for type float 32 bits, with 8 bits for the exponent and 23 bits for the fraction and binary64 for type double 64 bits, with 11 bits for the exponent and 53 bits for the fraction.

This explains the bounds on values of these types and explains various anomolous behavior with roundoff error that we first observed in Section 1. Java code for manipulating bits.

Java defines the int data type to be a bit two's complement integer and support various operations to manipulate the bits. Binary and hex literals. You can specify integer literal values in binary by prepending 0b and in hex by prepending 0x. You can use literals like this anywhere that you can use a decimal literal. Shifting and bitwise operations.

Java supports a variety of operations to manipulate the bits of an integer: Shift left and right: For shift right, there are two versions: Here are a few examples: One of the primary uses of such operations is shifting and masking , where we isolate a contiguous group of bits from the others in the same word. Use a shift right instruction to put the bits in the rightmost position. If we want k bits, create a literal mask whose bits are all 0 except its k rightmost bits, which are 1.

Use a bitwise and to isolate the bits. The 0s in the mask lead to zeros in the result; the 1s in the mask specify the bits of interest. In the following example, we extract bits 9 through 12 from the bit int.

To process text, we need a binary encoding for characters. The basic method is quite simple: The following table is a definition of ASCII that provides the correspondence that you need to convert from 8-bit binary equivalently, 2-digit hex to a character and back. For example, 4A encodes the letter J. Unicode is a bit code that supports tens of thousands. The dominant implementation of Unicode is known as UTF UTF-8 is a variable-width character encoding that uses 8 bits for ASCII characters, 16 bits for most characters, and up to 32 bits for other characters.

The encoding rules are complicated, but are now implemented in most modern systems such as Java so programmers generally need not worry much about the details. Big Endian, little endian. Computers differ in the way in which they store multi-byte chunks of information, e. This consists of the two bytes 70 and F2, where each byte encodes 8 bits. The two are two primary formats, and they differ only in the order or " endianness " in which they store the bytes.

Big endian systems store the most significant bytes first, e. Little endian systems store the least significant bytes first, e. This format is more natural when manually performing arithmetic, e. Intel , Intel Pentium, Intel Xeon use this format. Exercises Convert the decimal number 92 to binary.

Convert the hexadecimal number BB23A to octal. Add the two hexadecimal numbers 23AC and 4B80 and give the result in hexadecimal. Assume that m and n are positive integers.

What is the only decimal integer whose hexadecimal representation has its digits reversed. Develop an implementation of the toInt method for Converter. Develop an implementation of the toChar method for Converter.

Creative Exercises IP address. Write a program IP. That is, take the bits 8 at a time, convert each group to decimal, and separate each group with a dot. For example, the binary IP address should be converted to Web Exercises Excel column numbering. Write a function elias that takes as input an integer N and returns the Elias Gamma code as a string. The Elias Gamma code is a scheme to encode the positive integers. To generate the code for an integer N, write the integer N in binary, subtract 1 from the number of bits in the binary encoding, and prepend that many zeros.

For example, the code for the first 10 positive integers is given below.