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ITEC 352

ITEC 352. Lecture 6 Back to binary. Review. Decoders 7 segment display Complexity of wiring Questions?. Outline. Homework Binary May be short depending on what you remember from Discrete Math. Questions. How can you tell if a number is negative in binary?

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ITEC 352

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  1. ITEC 352 Lecture 6 Back to binary

  2. Review • Decoders • 7 segment display • Complexity of wiring • Questions?

  3. Outline • Homework • Binary • May be short depending on what you remember from Discrete Math

  4. Questions • How can you tell if a number is negative in binary? • How can you tell if a number is 3.5743 in binary? • How can you tell if a set of binary digits is a String? • What do these questions lead you to believe about binary numbers?

  5. Numeral systems • Numeral Vs. Number • same as difference between a word vs. the things it refers to. • numeral is a group of symbols that represents a number. • E.g., 15 can be represented as: • 15, Fifteen, XV (roman) • What numeral system do we use everyday ?

  6. Numeral system (2) • Any numeral system is characterized by the number of digits used to represent numbers. • E.g., • Unary system: ? • Binary system: ? • Octal: ? • Decimal: ? • The numeral system is called the base.

  7. Numeral system (3) • If we had lots of numeral systems in use, things will get confusing: • E.g., What is: 20 + 10 = ? • Is it: • 30 ? • 24? • 12?

  8. Number Systems • To make things easier for us: we use decimal number system as our base. • Every number in any other base is converted to decimal for us to be able to understand. • How do we do this conversion?

  9. Radix • Determines the value of a number, by assigning a weight to the position of each digit. • E.g., Number 481 • start all positions from 0. • Position of “1” : 0; weight of position: 1 • Position of “8” : 1; weight of position: 10 • Position of “4” : 2; weight of position: 100 • Hence number: 4*100 + 8 * 10 + 1 * 1 • Weight is calculated as 10^position • Any decimal number can be represented this way. • 10 is called the base or radix of the number system. • We use notation ()rto represent the radix. • E.g., the decimal number 481 can also be written as: (481)10

  10. Other bases • Octal • Hexadecimal • Does it matter that you can convert between them?

  11. ConversionChart

  12. Basics • Conversion • How do you do it? • What is 10 in binary? • What is 100 in binary?

  13. Adding Subtracting • What is binary 1 + binary 0 = ? • What about binary 1 + binary 1 = ? • Addition is similar to decimal addition. • remember though that the answer will only use one of two digits: 0 or 1. • How about subtraction?

  14. Subtraction • 101 – 011 =

  15. Subtraction • Subtraction introduces some challenges: • Answer maybe negative. How to represent negative binary numbers? • Subtraction isn’t easy: requires carry-ins… • Can we make it easier? What type of subtractions are easy to implement? • Can we use the same circuit for addition and subtraction. ?

  16. Limitations • TWO key limitations: • It only represents positive numbers. • How do we accommodate negative numbers? • What about numbers that have too many digits? • A computer is bound by its data bus in the number of digits it can handle. • E.g., a 32 bit data bus, implies, the computer can store upto 32 bits for a basic data such as a byte. • Ofcourse, integers can be represented as multiple bytes, but this decreases the speed of compuration. • Solution: Floating Point Representation. • Next: Representing negative numbers.

  17. Negative numbers • Our goal: • We want a representation of negative numbers such that: • Subtractions are as easy as additions: • Instead of subtraction we should be able to simply add. • Or • If it is a subtraction, there should be no carry. • We have some facts at our disposal. The number of bits you can use to represent any number in a computer is limited.

  18. Complement notation. • The invention of complements. • Assume our computer is limited to two digits. • Find x in the following equation (restricting answer to two digits): • 54 – 45 = 54 + x • Introducing 10’s complement • 10’s complement of 45 = 55 • 10’s complement of 99 = 1 • What is 54 + (10’s complement of 45) restricted to two digits ? • The 9’s complement for decimal digits: • 9’s complement for 45 = 99 – 45 = 54 • 54 + 54 = 108 • 1 + 08 = 9 = 54 - 45

  19. One’s complement • Invert all positions in the number • To subtract, add the numbers • If there is a carry out, add it to the first number in the result • Done

  20. Question • In one’s complement what are the following numbers? • 000 • 111

  21. Questions • On 3-bit architecture, what are all the positive and negative numbers that can be represented if numbers are represented in one’s complement notation? • Write down the binary representations of all the numbers.

  22. Review • Binary • Numbering systems • Addition / Subtraction • One different way to represent them

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