Binary Representation
Binary representation is a method of representing numbers using only two digits, 0 and 1. In computing and digital systems, binary representation is used to store and process data. It is based on the binary number system, where each digit's place value is a power of 2. This system is fundamental to the operation of digital electronics and computer systems.
7 Key excerpts on "Binary Representation"
eBook - ePubElectronics Explained
The New Systems Approach to Learning Electronics
- Louis E. Frenzel(Author)
- 2010(Publication Date)
- Newnes(Publisher)
...The symbols 0 and 1 are called binary digits or bits. For example, the 6-bit number 101101 represents the decimal number 45. Your understanding of how digital circuits, microprocessors, and related equipment process data is tied directly to an understanding of the binary number system. The reason for using binary numbers in digital equipment is the ease with which they can be implemented. The electronic components and circuits used to represent and process binary data must be capable of assuming two discrete states to represent 0 and 1. Examples of two-state components are switches and transistors. When a switch is closed or on, it can represent a binary 1. When the switch is open or off, it can represent a binary 0. A conducting transistor may represent a 1, whereas a cut-off transistor may represent a 0. The representation may also be voltage levels. For example, a binary 1 may be represented by +3.3 V and a binary 0 by 0 V as previously shown in Figure 5.1. Binary-to-decimal Conversions The binary system is similar to the decimal system in that the position of a digit in a number determines its weight. Recall that in the decimal system the weights are powers of 10. The right-most digit is units or 1’s, and then 10’s, 100’s, 1000’s, and so on, moving from right to left from one digit to the next. The position weights of a binary number are powers of 2: 2 0 = 1 2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 16 2 5 = 32 2 6 = 64 2 7 = 128 2 8 = 256 The position weights of an 8-bit binary number follow: The most significant digit or bit (MSB) and the least significant bit (LSB) are identified. Now let’s evaluate the decimal quantity associated with a given binary number, 101101. You simply multiply each bit by its position weight and add the values to get the decimal equivalent, 45. You can see that positions with a 0 bit have no effect on the value. Therefore, they can be ignored...
eBook - ePub- Herbert L. Blitzer, Karen Stein-Ferguson, Jeffrey Huang(Authors)
- 2010(Publication Date)
- Academic Press(Publisher)
...Using digital signal for communication can prevent signal drifts. DATA REPRESENTATION IN DIGITAL SYSTEMS One important aspect of digital system design is how information ultimately will be converted into a bit pattern. Numeric values are the most prevalent and natural type of data representation from an analog source. When converting numeric information to binary codes, it is necessary to have a good mapping strategy, so that the data can be well represented relative to how it will be used. Numeric information can be categorized into three data types: Integer Negative value Floating point representation This section discusses the representation of these three data types. Integer Representation: Decimal versus Binary or Hexadecimal To introduce binary data representation, it is useful to dissect the more common decimal system we all know. The decimal number 123 is based on powers of 10 (hence the term decimal, from the Latin for ten). It is really 1 times 10 2 plus 2 times 10 1 plus 3 times 10 0. or 1 * 100 + 2 * 10 + 3 * 1 = 100 + 20 + 3 = 123. Each place holder is multiplied by 10 raised to a power consistent with its place in the string of digits, and the value in each place holder can be any whole number between zero and 9. The powers of 10 increase by one, going up from zero, as one progresses to the left. Similar rules apply in the binary system but the values are different. Each place holder is multiplied by 2 raised to some power, and the value in each place holder can be any whole number between zero and one. The powers of 2 increase by one, going up from zero, as one progresses to the left. Binary is represented by only zeros and ones. A binary pattern (number) can be converted to a decimal number and a decimal number can be converted to binary. The binary number system is sometimes called the base-2 system, and base might be indicated by 2, b, B, or Bin. So 1011 2 or 1011 B are both binary numbers...
eBook - ePubDigital Design
Basic Concepts and Principles
- Mohammad A. Karim, Xinghao Chen(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...Each binary digit is known as a bit. The large number of bits present in the Binary Representation presents no significant problem since in most digital systems most operations occur in parallel. Table 1.1 Correspondence Table The advantage of using the binary system is that there are many devices that manifest two stable states. Examples of such binary states include on–off, positive–negative, and high–low choices. One of the two stable states can be assigned the value logic 1 while the other can be treated as logic 0. Bit values are often represented by voltage values. We define a certain range of voltages as logic 1 and another range of voltages as logic 0. A forbidden voltage range typically separates the two permissible ranges. The approximate nominal values used for many technologies are 5 V for a binary value of 1 and 0 V for a binary value of 0. Since there can be numbers represented in different systems, it can be confusing at times. It is thus prudent whenever we are working with a particular number system to use the radix as its subscript. Examples of a few binary numbers and their decimal equivalents are as. follows: 1 0 1 1 1 2 = 1 × 1 6 + 1 × 4 + 1 × 2 + 1 × 1 = 2 3 1 0 1 0 1. 1 1 1 1 2 = 1 × 4 + 1 × 1 + 1 x 0. 5 + 1 × 0. 2 5 + 1 × 0. 1 2 5 + 1 × 0. 0 6 2 5 = 5. 9 3 7 5[--=PL. GO-SEPARATOR=--]1 0 1 1 0 0. 1 0 0 1 2 = 1 × 8 + 1 × 4 + 1 × 0. 5 + 1 × 0. 0 6 2 5 = 1 2. 5 6 2 5 1 0 Similar to what is known as a decimal point in decimal number system, the radix point in a binary number system is called a binary point. Octal (radix 8) and hexadecimal (radix 16) numbers are used primarily as convenient shorthand representations for binary numbers because their radices are powers of 2. Converting a multibit number to either octal or hexadecimal (hex for short) equivalent thus reduces the need for lengthy indecipherable strings. The octal number system uses digits 0 through 7 of the decimal system...
eBook - ePub- J. Crowe, Barrie Hayes-Gill(Authors)
- 1998(Publication Date)
- Butterworth-Heinemann(Publisher)
...2 Arithmetic and digital electronics 2.1 INTRODUCTION Many of the applications of digital electronic circuits involve representing and manipulating numbers as binary code (i.e. O’s and l’s). For instance in order to input any analogue value (e.g. a voltage or temperature) into a digital circuit it must be first encoded as a binary value, whilst subsequent arithmetic performed on such an input must be carried out by further digital circuits. The way in which some arithmetic operations are implemented as digital electronic circuits is considered in the next chapter. Here, as a prelude to this, some of the many ways in which numbers can be represented as binary code are introduced, followed by a description of how to perform binary arithmetic; that is addition, subtraction, multiplication and division on numbers represented only by O’s and l’s. 2.2 BASES-2, 10 AND 16 (BINARY, DECIMAL AND HEXADECIMAL) Numbers are most commonly represented using the 10 digits 0 to 9, that is in base-10 (or decimal). This widespread use is linked to our possession of 10 fingers and their value as a simple counting aid. However, from a purely mathematical viewpoint the base system used for counting is unimportant (indeed before metrification in Europe (use of base-10 in all weights and measures) many other bases were common). In digital electronics the only choice of base in which to perform arithmetic is base-2, that is binary arithmetic, using the only two digits available, 0 and 1. 1 Before continuing it is necessary to consider how to convert numbers from one base to another. 2.2.1 Conversion from base- n to decimal In order to do this it is essential to realise what a number expressed in any base actually represents. For example the number 152 10 in base-10 represents 2 the sum of 1 hundred, 5 tens and 2 units giving 152 10 units...
eBook - ePub- John Stonham(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...Numerical representation of information 1 Objectives □ To distinguish between analogue and digital quantities. □ To investigate the binary number system. □ To define and examine binary-coded decimal numbers. □ To specify the essential properties of a position sensing code. □ To introduce the concepts of error detection and correction in binary data. In almost all activities, we are constantly dealing with quantities or measurements. This information is expressed in the form of numbers and can be processed in a digital system, provided it can be represented in an electronic form. Information is almost always encoded in binary in a digital system and a wide range of codes exist. The choice of a particular code is influenced by the type of operation to be carried out on the data. In this chapter, methods of representing numerical data in binary will be introduced. It is essential that the designer is familiar with fundamental coding techniques, as the form in which information is represented has significant influence on the design, performance and reliability of a digital system. Analogue and digital data The first step in any data processing operation is to obtain information about the objects or phenomena of interest. The acquisition of information usually involves taking measurements on some property or characteristic of a system under investigation. In order to evaluate and assess the system, the measurements are monitored, scaled, compared, combined, or operated on in various ways. It is therefore essential that we have standard ways of representing our information. Measurements can be divided into two broad categories. An analogue measurement is continuous and is a function of the parameter being measured. Conversely, a digital quantity is discrete and it can only change by fixed units. All drips are assumed to be the same size Figure 1.1 shows two beakers being filled, one from a dripping tap and the other from a trickling tap...
eBook - ePub- Don Davis, Eugene Patronis, Pat Brown(Authors)
- 2013(Publication Date)
- Routledge(Publisher)
...W. Tukey. A device with two stable positions, such as a relay or a flip-flop circuit can store one bit of information. N such devices can store N bits, since the total number of possible states is 2 N and log 2 2 N = N. If the base 10 is used the units may be called decimal digits. Fig. 5-6 gives the definition of binary. Figure 5-6 Binary choices. Since M = log 10 N /log 10 2= 3.32 log 10 N, a decimal digit is about 3⅓bits. A digital wheel on a desk computing machine has ten stable positions and therefore has a storage capacity of one decimal digit. In analytical work where integration and differentiation are involved, the base e is sometimes useful. The resulting units of information will be called natural units (Nats). Change from the base a to base b merely requires multiplication by log b a. Mathematically the number of bits is defined by: 5-14 log 2 P = B i t s and 5-15 2 B i t s = P where, P is the number of possibilities. 5.6.1 The Physical Dimensions of One Bit One Nat of information has the area of a square exactly two Planck lengths on a side. One Nat equals 0.693 bits, One Planck length equals 5-16 L P = h ¯ × G C = 1.616252 (81) × 10 − 35 m where, G is the gravitational constant, h ¯ is the reduced Planck constant, where h ¯ = (h 2 π) C is the speed of light. Planck. area equals 5-17 L P 2 = h ¯ × G C = 2.61223 × 10 − 70 m 2 5-18 1.0 Nat = (2 × L P) 2 = 1.044909 (2.61223) × 10 − 69 m 2 5-19 1.0 bit = 1.0 Nat ln 2 = 1.507485 (2.61223) × 10 − 69 m 2 1 bit = 3.88263 × 10 − 35 m the length of one side of the bit square 5-20 1 Nat =. 3.232506(3.88263) × 10 − 35 m the length of one side of the Nat square 5-21 b i t P l N a t P l = 1.201122 × 2 P l = 2.402244 P l 5-22 (2.4 P l) 2 2 P l 2 = 0.693 bits per Nat. 5.6.2 Reading Binary Numbers Table 5-2, Binary to Decimal to Hexadecimal to Octal allows you to see the essential simplicity of binary coding. Pick any decimal number and see how the 1s add up on the exponential scales at the top of the chart...
eBook - ePub- S W Amos, Mike James(Authors)
- 2000(Publication Date)
- Newnes(Publisher)
...This is illustrated below but it is worth stressing now that the logic convention chosen should always be stated or implicit on diagrams of logic circuits. The tendency in the design of digital equipment is to favour positive logic and if no logic convention is indicated on a diagram it can normally be assumed that positive logic is used. Positive logic is used in all the diagrams in this chapter. Binary scale The advantage of labelling the two significant voltage levels 0 and 1 is that it simplifies the process by which logic circuits are able to carry out mathematical and other operations. Arithmetical operations, for example, can be performed by using the binary scale of numbers which has only two digits 0 and 1. Conventional counting uses the scale of 10 (the decimal scale) and in numbers the digits are arranged according to the power of 10 they represent. For example the number 4721 (four thousand, seven hundred and twenty-one) means, if written out in full: 4 × 10 3 + 7 × 10 2 + 2 × 10 1 + 1 × 10 0 i. e. 4000 + 700 + 20 + 1 = 4721 Similarly in the binary scale of counting, the digits (0 or 1) in a number are arranged according to the power of 2 they represent. For example the binary number 110101 means, if written out in full: 1 × 2 5 + 1 × 2 4 + 0 × 2 3 + 1 × 2 2 + 0 × 2 1 + 1 × 2 0 i. e. 32 + 16 + 0 + 4 + 0 + 1 = 53 The first nine numbers in the binary scale are as follows: binary number decimal equivalent 1 1 10 2 11 3 100 4 101 5 110 6 111 7 1000 8 1001 9 It is common practice, however, with very large numbers to translate each digit of the decimal number separately into the binary scale. This is known as the binary-coded decimal system and in it the number 4721 would be coded as follows: 100 : 111 : 10 : 1 i. e. 4 : 7 : 2 : 1 This system has the advantage that after a little experience binary-coded numbers can be translated into decimal form on inspection...
