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Fundamentals

Fundamentals. What kind of data do computers work with? Deep down inside, it’s all 1s and 0s What can you do with 1s and 0s? Boolean algebra operations These operations map directly to hardware circuits. Volts 5 4 3 2 1 0. High. Low. Volts 5 4 3 2 1 0. High. Medium. Low.

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Fundamentals

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  1. Fundamentals • What kind of data do computers work with? • Deep down inside, it’s all 1s and 0s • What can you do with 1s and 0s? • Boolean algebra operations • These operations map directly to hardware circuits CS231 Fundamentals

  2. Volts 5 4 3 2 1 0 High Low Volts 5 4 3 2 1 0 High Medium Low Computers are binary devices • Computers use voltages to represent information • Voltages are usually limited to 2.5-5.0V to minimize power consumption • This is only enough to represent two discrete (digital) signals • Low and High • 0 and 1 • False and True • Why? • To account for noise • To prevent transitional errors • How can we use these two signals to represent numbers? Good! Bad! CS231 Fundamentals

  3. Decimal review • Numbers consist of a bunch of digits, each with a weight • These weights are all powers of the base, which is 10. We can rewrite this: • To find the decimal value of a number, multiply each digit by its weight and sum the products. (1 x 102) + (6 x 101) + (2 x 100) + (3 x 10-1) + (7 x 10-2) + (5 x 10-3) = 162.375 CS231 Fundamentals

  4. Converting binary to decimal • We can use the same trick to convert binary, or base 2, numbers to decimal. This time, the weights are powers of 2. • Example: 1101.01 in binary • The decimal value is: (1 x 23) + (1 x 22) + (0 x 21) + (1 x 20) + (0 x 2-1) + (1 x 2-2) = 8 + 4 + 0 + 1 + 0 + 0.25 = 13.25 CS231 Fundamentals

  5. Opposite: converting decimal to binary • To convert an integer, keep dividing by 2 until the quotient is 0. Collect the remainders in reverse order. • To convert a fraction, keep multiplying the fractional part by 2 until it becomes 0. Collect the integer parts (in forward order). • This may not terminate! • Example: 162.375: • So, 162.37510 = 10100010.0112 162 / 2 = 81 rem 0 81 / 2 = 40 rem 1 40 / 2 = 20 rem 0 20 / 2 = 10 rem 0 10 / 2 = 5 rem 0 5 / 2 = 2 rem 1 2 / 2 = 1 rem 0 1 / 2 = 0 rem 1 0.375 x 2 = 0.750 0.750 x 2 = 1.500 0.500 x 2 = 1.000 CS231 Fundamentals

  6. Why does this work? • This works for converting from decimal to any base • Why? Think about converting 162.375 from decimal to decimal. • Each division “strips off” the rightmost digit (the remainder). The quotient represents the remaining digits in the number. • Similarly, each multiplication “strips off” the leftmost digit (the integer part). The fraction represents the remaining digits. 162 / 10 = 16 rem 2 16 / 10 = 1 rem 6 1 / 10 = 0 rem 1 0.375 x 10 = 3.750 0.750 x 10 = 7.500 0.500 x 10 = 5.000 CS231 Fundamentals

  7. Base 8 and base 16 are useful too • Octal (base 8) digits range from 0 to 7. Since 8 = 23, one octal digit is equivalent to 3 binary digits. • Hexadecimal (base 16) digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Since 16 = 24, one hex digit is equivalent to 4 binary digits. • We typically use octal and hex as a shorthand for those long messy binary numbers. CS231 Fundamentals

  8. Binary, octal, hexadecimal • Converting from octal (or hex) to binary is easy: just replace each octal (or hex) digit with the equivalent 3 (or 4) bit sequence • To convert from binary to octal (or hex): starting from the binary point, make groups of 3 (or 4) bits. Add 0s to the ends of the number if needed. Then, just convert each bit group to its corresponding octal (or hex) digit. 261.358 = 2 6 1 . 3 58 = 010 110 001 . 011 1012 261.3516 = 2 6 1 . 3 516 = 0010 0110 0001 . 0011 01012 These binary numbers are not equivalent! 10110100.0010112 = 010 110 100 . 001 0112 = 2 6 4 . 1 38 These numbers are equivalent! 10110100.0010112 = 1011 0100 . 0010 11002 = B 4 . 2 C16 CS231 Fundamentals

  9. Functions • A computer takes input and produces output... just like a math function! • We can express math functions in two ways • We can represent binary functions in two ways too: • As a Boolean expression • As a truth table. It shows all possibilities for an expression. As an expression As a function table f(x,y) = 2x + y = x + x + y = 2(x + y/2) = ... CS231 Fundamentals

  10. Basic Boolean operations • There are three basic operations. Each can be implemented in hardware using a primitive logic gate NOT (complement) on one input AND (product) of two inputs OR (sum) of two inputs Operation: Expression: xy x + y x’ Truth table: Logic gate: CS231 Fundamentals

  11. Boolean expressions • We can use these basic operations to form more complex expressions and equations • Some terminology and notation • f is the name of the function • (x,y,z) are the input variables. Each variable represents a 1 or 0. Listing input variables is optional; instead we often write • A literal is an occurrence of an input variable or its complement. The function above has four literals: x, y’, z, and x’. • Without parentheses, the NOT operation has the highest precedence, followed by AND, and finally OR. Fully parenthesized, this function is f(x,y,z) = (x + y’)z + x’ f = (x + y’)z + x’ f(x,y,z) = (((x +(y’))z) + x’) CS231 Fundamentals

  12. Larger circuits • We can build circuits for arbitrary expressions, using our basic gates f(x,y,z) = (x + y’)z + x’ (x + y’)z y’ (x + y’) x’ CS231 Fundamentals

  13. Truth tables • We can make a truth table too, by evaluating the function for all possible inputs. • In general, a function with n inputs will have 2n possible inputs. • The inputs are typically listed in binary order. f(x,y,z) = (x + y’)z + x’ f(0,0,0) = (0 + 1)0 + 1 = 1 f(0,0,1) = (0 + 1)1 + 1 = 1 f(0,1,0) = (0 + 0)0 + 1 = 1 f(0,1,1) = (0 + 0)1 + 1 = 1 f(1,0,0) = (1 + 1)0 + 0 = 0 f(1,0,1) = (1 + 1)1 + 0 = 1 f(1,1,0) = (1 + 0)0 + 0 = 0 f(1,1,1) = (1 + 0)1 + 0 = 1 CS231 Fundamentals

  14. Summary • Summary • To minimize analog voltage problems, computers are binary devices • That means we have to think in terms of base 2 • The basic operations on binary values include AND, OR, and NOT • These can be directly implemented in hardware (eventually we’ll show how to do useful stuff with these operations) • Expressions and truth tables are used to design and describe circuits • We’ve already seen some of the recurring themes of architecture: • We use 0 and 1 as abstractions for analog voltages • Gates are also an abstraction of the underlying technology • We showed how to represent numbers using just these two signals • Next time: • Using Boolean algebra to simplify expressions • This in turn will yield a simpler circuit CS231 Fundamentals

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