Slide 1 Chapter 15:Digital Electronics ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 2 215-1a Introduction• Modern computers are composed mostly of digital circuits.• It is critical to understand the building blocks of digital circuits that comprise computer and computer-related equipment.• Digital circuits are composed of devices and circuits that are conventional in nature and repeat many of the structures we have looked at in this class. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 3 3Digital Concepts and Terminology• There are fundamental concepts and terms related to digital technology that are essential to master in order to understand their operation.•The binary numbering system forms the basis for all digital logic systems and is critical to the understanding of digital concepts. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________Slide 4 4Interpretation of Binary Numbers• To understand the binary numbering system, a comparison between binary and decimal is useful.Multiply each digit by the value of the position weightMultiply each digit by the value of the position weightDetermination of value23, 22, 21, 20, 2-1, ...8, 4, 2, 1, 1/2…103, 102, 101, 100, 10-1, ...1000, 100, 10, 1, .1, …Weights of digit positionsTwo: 0 and 1Ten: 0,1,2,3,4,5,6,7,8,9Number of digitsSubscript 2 (e.g. 110112)Subscript 10 (e.g. 25910)Indication of number systemBinary SystemDecimal SystemCharacteristic ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 5 5Weighted Values Decimal/Binary1 2 7101 0 0 0 1 1 1 12 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 6 6Binary Numbers• Binary number allows only two digits, 0 and 1.• A single binary digit is called a bit.• An eight-bit digit would be: 11010111.• Microprocessors work with a variety of bit widths, including 4, 8, 16, 32, 64, 128, and more.•The radix pointseparates the integer from the fractional portion of the number. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________Slide 7 7Numerical Sequences• Counting in binary is very simple and follows the same rules as decimal numbers. In decimal, we count by incrementing the least significant digit to the next highest digit value.• It works the same way with binary numbers, except the highest digit value is 1.• The table to the right counts to 10 using 4 bits of binary101010100191000801117011060101501004001130010200011Binary EquivalentDecimal NumberEnd 15-1a ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 8 815-1b Converting Binary to Decimal• It is important to be able to convert binary numbers to their decimal equivalent.• To convert a binary number to its equivalent decimal value:– Multiply the weight of each column by the digit value in that column.– Sum the products obtained. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 9 9Binary to Decimal1 0 0 1 1 0 0 1 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________Slide 10 10Converting Decimal to Binary• If we have a number expressed as decimal number, we should be able to convert it to a binary number.• One procedure for decimal-to-binary conversion is:– Divide the decimal number by 2.– Write the remainder down as a bit in the converted number beginning with the least significant bit (LSB).– Repeat the first two steps until a quotient of 0 is obtained. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 11 11Decimal to Binary (divide by two)• Convert 10010to binary ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 12 12Decimal to Binary•10010 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________Slide 13 13Converting between Binary and Hex• To increase the efficiency and simplicity of working with binary numbers, they are often converted to hexadecimal or base 16.• Conversion between binary and hexadecimal is simple:– Begin at the radix and mark off groups of 4 binary digits.– Replace each group of four bits with the hexadecimal digit.• See table on following slide for examples. ___________________________________ ___________________________________