## Number Systems

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There are infinite ways to represent a number. The four commonly associated with modern computers and digital electronics are: Decimal base 10 is the way most human beings represent numbers. Decimal is sometimes abbreviated as dec. Binary base 2 is the natural way most digital circuits represent and manipulate numbers. Binary numbers are sometimes represented by preceding the value with '0b', as in 0b Binary is sometimes abbreviated as bin. Octal base 8 was previously a popular choice for representing digital circuit numbers in a form that is more compact than binary.

Octal is sometimes abbreviated as oct. Hexadecimal base 16 relationship among binary octal decimal and hexadecimal number systems currently the most popular choice for representing digital circuit numbers in a form that is more compact than binary.

Hexadecimal numbers are sometimes represented by preceding the value with '0x', as in 0x1B Hexadecimal is sometimes abbreviated as hex.

All four number systems are equally capable of representing any number. Furthermore, a number can be perfectly converted between the various number systems without any loss of numeric value. At first blush, it seems like using any number system other than human-centric decimal is complicated and unnecessary. However, since the job of electrical and software engineers is to work with digital circuits, engineers require number systems that can best transfer information between the human world and the digital circuit world.

It turns out that the way in which a number is represented can make it easier for the engineer to perceive the meaning of the number as it applies to a digital circuit.

In other words, the appropriate number system can actually make things less complicated. Almost all modern digital relationship among binary octal decimal and hexadecimal number systems are based on two-state switches. The switches are either on or off. Because the fundamental information element of digital circuits has two states, it is most naturally represented by a number system where each individual digit has two states: For example, switches that are 'on' are represented by '1' and switches that are 'off' are represented by '0'.

It is easy relationship among binary octal decimal and hexadecimal number systems instantly comprehend the values of 8 switches represented in binary as It is also easy to build a circuit to display each switch state in binary, by having an LED lit or unlit for each binary digit.

As digital circuits grew more complex, a more compact form of representing circuit information became necessary. That means three binary digits convert neatly into one octal digit. That means four binary digits convert neatly into one hexadecimal digit. Unfortunately, decimal base 10 is not a whole power of 2. So, it is not possible to simply chunk groups of binary digits to convert the raw state of a digital circuit into the human-centric format.

Number Systems There are infinite ways to represent a number. Fundamental Information Element of Digital Circuits Almost all modern digital circuits are based on two-state switches.

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We can use this fact to convert between binary, octal, and hexadecimal systems, as shown in Figure G. The procedure for converting an octal to a binary is shown by the arrow marked a.

We can prove that replacing each octal digit by its 3-bit equivalent binary value gives the right result:. Analogously, we can convert a hexadecimal number to its equivalent binary number by replacing each digit in the hexadecimal number by its 4-bit equivalent binary value, as shown by the arrow marked b. To convert a binary to it octal equivalent, we reverse the procedure outlined earlier see arrow marked c in Figure G.

The bits in the binary number are grouped into 3-bit groups from right to left. Each such group is replaced by its equivalent octal digit. This corresponds to reversing the computation shown above. Analogously, we can convert a binary to a hexadecimal number by replacing each 4-bit group by its equivalent hex digit see arrow marked d in Figure G.

Home Certification A programmer's guide to java certification. Converting between Binary, Octal, and Hexadecimal The procedure for converting an octal to a binary is shown by the arrow marked a.

We can prove that replacing each octal digit by its 3-bit equivalent binary value gives the right result: Writing the Second Edition. Basics of Java Programming. Declarations and Access Control. Control Flow, Exception Handling, and Assertions. Nested Classes And Interfaces. Taking the SCPJ2 1. Objectives for the SCPJ2 1. Declarations and Access Control Chapter 4. Garbage Collection Chapter 8. Language Fundamentals Chapter 2. Operators and Assignments Chapter 3.

Fundamental Classes in the java. The Collections Framework Chapter Objectives for the Java 2 Platform Upgrade Exam. Flow Control, Assertions, and Exception Handling. Annotated Answers to Review Questions. Solutions to Programming Exercises. Number Systems and Number Representation. Important Terms of Agreement. Tenure of the License Agreement. Limited Warranty Under the License. Remedies Provided Under the License. Liabilities Under the License.