ENEL 353 F13 Section 02 Slides for Lecture 2 slide 2/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 3/21 Previous Lecture Today’s Lecture Slides for Lecture 2 ENEL 353: Digital Circuits — Fall 2013 Term Number systems: decimal, binary, octal, and hexadecimal. Steve Norman, PhD, PEng Related material in Harris & Harris (our course textbook): ◮ Introduction to ENEL 353 ◮ Sections 1.1, 1.2, 1.3: general background on digital ◮ Course policies Electrical & Computer Engineering Schulich School of Engineering design University of Calgary ◮ Sections 1.4.1, 1.4.2, 1.4.3: closely related to today’s lecture 11 September, 2013 slide 4/21 slide 5/21 slide 6/21 ENEL 353 F13 Section 02 Slides for Lecture 2 ENEL 353 F13 Section 02 Slides for Lecture 2 ENEL 353 F13 Section 02 Slides for Lecture 2 Number systems The decimal system The decimal system 702.53 stands for ... What is the general pattern used for a number in the decimal What does 702.53 mean? 7 × 10 2 system? Seven hundred and two point five three, of course! Why even + 0 × 10 1 Ten is called the base or the radix of the decimal system. ask the question? + 2 × 10 0 It’s probably not a coincidence that most of us have ten We’ve all used numbers in this form so often that it’s possible + 5 × 10 − 1 fingers and that ten is the radix of the number system we use we’ve forgotten the underlying rules for the number system + 3 × 10 − 2 in daily life! we use every day. The plural of radix is radices . This an example of the decimal system , by far the most common system used by humans for representing numbers. ENEL 353 F13 Section 02 Slides for Lecture 2 slide 7/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 8/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 9/21 About the word “decimal” Other radices? General format for a base r number Write it as two lists of digits separated by a dot: For any integer r ≥ 2, you can create a number system with In technical discussion of number systems, decimal means radix r . Such a system would be called a “base r ” number d n d n − 1 d n − 2 · · · d 1 d 0 . d − 1 d − 2 · · · d − p “base ten, with numbers written using digits 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9.” system. Each digit d k must be ≥ 0 and < r . You would need symbols for digits ranging from 0 up to r − 1. The word decimal does NOT mean “has a dot to separate the Mathematically, the number is . . . integer part of a number from the fraction part.” If r ≥ ten, you can just use the digits we already have. n So 28 + 79 = 107 is an example of decimal addition, even If r > ten, you have to make up new symbols for ten, eleven, � d k r k though you don’t see any “decimal points”. twelve, etc. k = − p
ENEL 353 F13 Section 02 Slides for Lecture 2 slide 10/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 11/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 12/21 Textbook notation for indicating a radix Base two numbers, also called binary numbers Conversion from base r to base ten Numbers in the course textbook, course notes, quizzes, tests, Base two is important because it allows design of fast and etc., will often have a base ten subscript to make it clear what Just use the formula: efficient electronic circuits for arithmetic: addition, n radix is in use. � d k r k subtraction, multiplication, etc. So 17 10 is just the number we all know as seventeen. Usually k = − p A binary digit is called a bit . So a bit is a thing that can have it would be written as 17, with no subscript. We’ve already used it for 1011 . 01 2 . one of two values: 0 or 1. What are 17 8 and 17 16 in base ten? Another example: What is 235 16 in base ten? Binary number example: What is 1011 . 01 2 in base ten? How about 18 8 and 19 8 ? slide 13/21 slide 14/21 slide 15/21 ENEL 353 F13 Section 02 Slides for Lecture 2 ENEL 353 F13 Section 02 Slides for Lecture 2 ENEL 353 F13 Section 02 Slides for Lecture 2 Conversion from base ten to base r : integers only Conversion from base ten to base r: numbers with Conversion from base ten to base r : more about fractions numbers with fractions This algorithm is important to know in ENEL 353; make sure you get lots of practice with it. The algorithm is not guaranteed to terminate in a finite The algorithm is: Using base ten, repeatedly divide by r until number of steps! For example, try to convert 0 . 6 10 to We won’t need this algorithm in ENEL 353, but it’s nice to you get a quotient of zero. The remainders you get along binary . . . know that it exists. the way are the digits of the base r number. The algorithm is: Using base ten, repeatedly multiply by r . multiplication product integer part remark Example: Convert 13 10 to binary. The integer parts of the numbers you get along the way are 0 . 6 × 2 1.2 1 0 . 1 2 so far Example: Convert 87 10 to base eight. the digits of the base r number. Fractional parts are used in 0 . 2 × 2 0.4 0 0 . 10 2 so far subsequent multiplications. 0 . 4 × 2 0.8 0 0 . 100 2 so far (Note that this method matches the paragraph that starts, 0 . 8 × 2 1.6 1 0 . 1001 2 so far Example: Convert 0 . 6875 10 to binary. “Working from the right . . . ,” in Example 1.5 on page 13 of 0 . 6 × 2 1.2 1 Uh-oh . . . a cycle! Harris & Harris.) ENEL 353 F13 Section 02 Slides for Lecture 2 slide 16/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 17/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 18/21 Conversion from base ten to base r : more about Octal and hexadecimal systems Conversion between binary, octal, and hex numbers with fractions The radix 8 system is called octal and so uses digits 0, 1, 2, 3, Conversion is easy! 3 bits make one octal digit; 4 bits make 4, 5, 6 and 7. one hex digit. The algorithm tells us that The radix 16 system is called hexadecimal or “hex” and uses Example: Express 11010 2 in octal and hex. 0 . 6 10 = 0 . 1001 1001 1001 1001 · · · 2 . digits 0, 1, 2, . . . , 8, 9, and, um, wait a minute, what comes The problem is NOT that the algorithm is somehow defective. More examples: Express 153 8 and 5 D 16 in binary. after 9? So what is the real problem here? One more example: Convert 487 10 to hex and octal. Example: What is 3 A 9 . C 16 ?
ENEL 353 F13 Section 02 Slides for Lecture 2 slide 19/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 20/21 ENEL 353 F13 Section 02 Slides for Lecture 2 slide 21/21 Learn these tables! Hex gets used a lot! Next Lecture octal bit hex bit hex bit It’s a convenient way for humans to describe binary data. digit pattern digit pattern digit pattern Example: HTML color encoding uses two hex digits for 0 000 0 0000 8 1000 ◮ Addition of binary unsigned integers. brightness of each of red, green, and blue. #FFA500 (FF 16 1 001 1 0001 9 1001 means maximum red, A5 16 means quite a lot of green, and ◮ Representation of signed integers. 2 010 2 0010 A 1010 00 16 means no blue at all) is a shade of orange . . . 3 011 3 0011 B 1011 ◮ Related reading in Harris & Harris: Sections 1.4.4 to 4 100 4 0100 C 1100 1.4.6. 5 101 5 0101 D 1101 + + = 6 110 6 0110 E 1110 7 111 7 0111 F 1111
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