# Digital Fundamentals

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2 Decimal Numbers The position of each digit in a weighted number system is assigned a weight based on the base or radix of the system. The radix of decimal numbers is ten, because only ten symbols (0 through 9) are used to represent any number. The column weights of decimal numbers are powers of ten that increase from right to left beginning with 10 0 =1: For fractional decimal numbers, the column weights are negative powers of ten that decrease from left to right:

3 Decimal Numbers Decimal numbers can be expressed as the sum of the products of each digit times the column value for that digit. Thus, the number 9240 can be expressed as or (9 x 10 3 ) + (2 x 10 2 ) + (4 x 10 1 ) + (0 x 10 0 ) 9 x 1, x x x 1 Express the number as the sum of values of each digit = (4 x 10 2 ) + (8 x 10 1 ) + (0 x 10 0 ) + (5 x 10-1 ) +(2 x 10-2 )

4 Binary Numbers For digital systems, the binary number system is used. Binary has a radix of two and uses the digits 0 and 1 to represent quantities. The column weights of binary numbers are powers of two that increase from right to left beginning with 2 0 =1: For fractional binary numbers, the column weights are negative powers of two that decrease from left to right:

5 Binary Numbers Summary A binary counting sequence for numbers from zero to fifteen is shown. Notice the pattern of zeros and ones in each column. Digital counters frequently have this same pattern of digits: Counter Decoder Decimal Number Binary Number

6 Binary Conversions The decimal equivalent of a binary number can be determined by adding the column values of all of the bits that are 1 and discarding all of the bits that are 0. Convert the binary number to decimal. Start by writing the column weights; then add the weights that correspond to each 1 in the number ½ ¼ ¼ = 37¼

7 Binary Conversions You can convert a decimal whole number to binary by reversing the procedure. Write the decimal weight of each column and place 1 s in the columns that sum to the decimal number. Convert the decimal number 49 to binary. The column weights double in each position to the right. Write down column weights until the last number is larger than the one you want to convert

8 Binary Conversions You can convert a decimal fraction to binary by repeatedly multiplying the fractional results of successive multiplications by 2. The carries form the binary number. Convert the decimal fraction to binary by repeatedly multiplying the fractional results by x 2 = carry = 0 MSB x 2 = carry = x 2 = carry = x 2 = x 2 = carry = 1 carry = 0 Answer = (for five significant digits)

9 Binary Conversions You can convert decimal to any other base by repeatedly dividing by the base. For binary, repeatedly divide by 2: Convert the decimal number 49 to binary by repeatedly dividing by 2. You can do this by reverse division and the answer will read from left to right. Put quotients to the left and remainders on top. Answer: Continue until the last quotient is Quotient Decimal number remainder base

10 Binary Addition The rules for binary addition are = 0 Sum = 0, carry = = = 0 Sum = 1, carry = 0 Sum = 1, carry = = 10 Sum = 0, carry = 1 When an input carry = 1 due to a previous result, the rules are = 01 Sum = 1, carry = = 10 Sum = 0, carry = = 10 Sum = 0, carry = = 10 Sum = 1, carry = 1

12 Binary Subtraction The rules for binary subtraction are 0-0 = = = = 1 with a borrow of 1 Subtract the binary number from and show the equivalent decimal subtraction / / / = 14

13 1 s Complement The 1 s complement of a binary number is just the inverse of the digits. To form the 1 s complement, change all 0 s to 1 s and all 1 s to 0 s. For example, the 1 s complement of is In digital circuits, the 1 s complement is formed by using inverters:

14 2 s Complement The 2 s complement of a binary number is found by adding 1 to the LSB of the 1 s complement. Recall that the 1 s complement of is (1 s complement) To form the 2 s complement, add 1: Input bits Adder Output bits (sum) Carry in (add 1) (2 s complement) 1

15 Signed Binary Numbers Summary There are several ways to represent signed binary numbers. In all cases, the MSB in a signed number is the sign bit, that tells you if the number is positive or negative. Computers use a modified 2 s complement for signed numbers. Positive numbers are stored in true form (with a 0 for the sign bit) and negative numbers are stored in complement form (with a 1 for the sign bit). For example, the positive number 58 is written using 8-bits as (true form). Sign bit Magnitude bits

16 Signed Binary Numbers Summary Negative numbers are written as the 2 s complement of the corresponding positive number. The negative number -58 is written as: -58 = (complement form) Sign bit Magnitude bits An easy way to read a signed number that uses this notation is to assign the sign bit a column weight of -128 (for an 8-bit number). Then add the column weights for the 1 s. Assuming that the sign bit = -128, show that = -58 as a 2 s complement signed number: Column weights: = -58

17 Floating Point Numbers Summary Floating point notation is capable of representing very large or small numbers by using a form of scientific notation. A 32-bit single precision number is illustrated. S E (8 bits) F (23 bits) Sign bit Biased exponent (+127) Magnitude with MSB dropped Express the speed of light, c, in single precision floating point notation. (c = x 10 9 ) In binary, c = In scientific notation, c = x S = 0 because the number is positive. E = = = F is the next 23 bits after the first 1 is dropped. In floating point notation, c =

18 Arithmetic Operations with Signed Numbers Using the signed number notation with negative numbers in 2 s complement form simplifies addition and subtraction of signed numbers. Rules for addition: Add the two signed numbers. Discard any final carries. The result is in signed form. Examples: = = = = = = -3 Discard carry = = = -9

19 Arithmetic Operations with Signed Numbers Note that if the number of bits required for the answer is exceeded, overflow will occur. This occurs only if both numbers have the same sign. The overflow will be indicated by an incorrect sign bit. Two examples are: = = = -126 Discard carry = = = +2 Wrong! The answer is incorrect and the sign bit has changed.

20 Arithmetic Operations with Signed Numbers Rules for subtraction: 2 s complement the subtrahend and add the numbers. Discard any final carries. The result is in signed form. Repeat the examples done previously, but subtract: (+30) (+15) 2 s complement subtrahend and add: = = = (+14) (-17) = = = +31 (-1) (-8) = = = +7 Discard carry Discard carry

21 Hexadecimal Numbers Hexadecimal uses sixteen characters to represent numbers: the numbers 0 through 9 and the alphabetic characters A through F. Large binary number can easily be converted to hexadecimal by grouping bits 4 at a time and writing the equivalent hexadecimal character. Express in hexadecimal: Group the binary number by 4-bits starting from the right. Thus, 960E Decimal Hexadecimal Binary A B C D E F

22 Hexadecimal Numbers Hexadecimal is a weighted number system. The column weights are powers of 16, which increase from right to left. Column weights{ Express 1A2F 16 in decimal. Start by writing the column weights: A 2 F 16 1(4096) + 10(256) +2(16) +15(1) = Decimal Hexadecimal Binary A B C D E F

23 Octal Numbers Octal uses eight characters the numbers 0 through 7 to represent numbers. There is no 8 or 9 character in octal. Binary number can easily be converted to octal by grouping bits 3 at a time and writing the equivalent octal character for each group. Express in octal: Group the binary number by 3-bits starting from the right. Thus, Decimal Octal Binary

24 Octal Numbers Octal is also a weighted number system. The column weights are powers of 8, which increase from right to left. Column weights{ Express in decimal. Start by writing the column weights: (512) + 7(64) +0(8) +2(1) = Decimal Octal Binary

25 BCD Binary coded decimal (BCD) is a weighted code that is commonly used in digital systems when it is necessary to show decimal numbers such as in clock displays. The table illustrates the difference between straight binary and BCD. BCD represents each decimal digit with a 4-bit code. Notice that the codes 1010 through 1111 are not used in BCD. Decimal Binary BCD

26 BCD You can think of BCD in terms of column weights in groups of four bits. For an 8-bit BCD number, the column weights are: What are the column weights for the BCD number ? Note that you could add the column weights where there is a 1 to obtain the decimal number. For this case: =

27 BCD A lab experiment in which BCD is converted to decimal is shown. Summary

28 Gray code Gray code is an unweighted code that has a single bit change between one code word and the next in a sequence. Gray code is used to avoid problems in systems where an error can occur if more than one bit changes at a time. Decimal Binary Gray code

29 Gray code Summary A shaft encoder is a typical application. Three IR emitter/detectors are used to encode the position of the shaft. The encoder on the left uses binary and can have three bits change together, creating a potential error. The encoder on the right uses gray code and only 1-bit changes, eliminating potential errors. Binary sequence Gray code sequence

30 ASCII ASCII is a code for alphanumeric characters and control characters. In its original form, ASCII encoded 128 characters and symbols using 7-bits. The first 32 characters are control characters, that are based on obsolete teletype requirements, so these characters are generally assigned to other functions in modern usage. In 1981, IBM introduced extended ASCII, which is an 8- bit code and increased the character set to 256. Other extended sets (such as Unicode) have been introduced to handle characters in languages other than English.

31 Parity Method Summary The parity method is a method of error detection for simple transmission errors involving one bit (or an odd number of bits). A parity bit is an extra bit attached to a group of bits to force the number of 1 s to be either even (even parity) or odd (odd parity). The ASCII character for a is and for A is What is the correct bit to append to make both of these have odd parity? The ASCII a has an odd number of bits that are equal to 1; therefore the parity bit is 0. The ASCII A has an even number of bits that are equal to 1; therefore the parity bit is 1.

32 Cyclic Redundancy Check Summary The cyclic redundancy check (CRC) is an error detection method that can detect multiple errors in larger blocks of data. At the sending end, a checksum is appended to a block of data. At the receiving end, the check sum is generated and compared to the sent checksum. If the check sums are the same, no error is detected.

33 Selected Key Terms Byte Floating-point number Hexadecimal Octal BCD A group of eight bits A number representation based on scientific notation in which the number consists of an exponent and a mantissa. A number system with a base of 16. A number system with a base of 8. Binary coded decimal; a digital code in which each of the decimal digits, 0 through 9, is represented by a group of four bits.

34 Selected Key Terms Alphanumeric ASCII Parity Cyclic redundancy check (CRC) Consisting of numerals, letters, and other characters American Standard Code for Information Interchange; the most widely used alphanumeric code. In relation to binary codes, the condition of evenness or oddness in the number of 1s in a code group. A type of error detection code.

35 1. For the binary number 1000, the weight of the column with the 1 is a. 4 b. 6 c. 8 d Pearson Education

36 2. The 2 s complement of 1000 is a b c d Pearson Education

37 3. The fractional binary number 0.11 has a decimal value of a. ¼ b. ½ c. ¾ d. none of the above 2008 Pearson Education

38 4. The hexadecimal number 2C has a decimal equivalent value of a. 14 b. 44 c. 64 d. none of the above 2008 Pearson Education

39 5. Assume that a floating point number is represented in binary. If the sign bit is 1, the a. number is negative b. number is positive c. exponent is negative d. exponent is positive 2008 Pearson Education

40 6. When two positive signed numbers are added, the result may be larger that the size of the original numbers, creating overflow. This condition is indicated by a. a change in the sign bit b. a carry out of the sign position c. a zero result d. smoke 2008 Pearson Education

41 7. The number 1010 in BCD is a. equal to decimal eight b. equal to decimal ten c. equal to decimal twelve d. invalid 2008 Pearson Education

42 8. An example of an unweighted code is a. binary b. decimal c. BCD d. Gray code 2008 Pearson Education

43 9. An example of an alphanumeric code is a. hexadecimal b. ASCII c. BCD d. CRC 2008 Pearson Education

44 10. An example of an error detection method for transmitted data is the a. parity check b. CRC c. both of the above d. none of the above 2008 Pearson Education

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