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ADD/SUB revisited

ADD/SUB revisited. Understand the examples again Overflow When two positive numbers added together or a negative number subtracted from a positive number yields negative Underflow When two negative numbers added together or a positive number subtracted from a negative number yields positive

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ADD/SUB revisited

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  1. ADD/SUB revisited • Understand the examples again • Overflow • When two positive numbers added together or a negative number subtracted from a positive number yields negative • Underflow • When two negative numbers added together or a positive number subtracted from a negative number yields positive C/B 0001 1111 0100 1000 1101 0000 1000 1100 A 0101 0101 0101 1001 0010 1011 0101 1011 B +0001 +1011 +0100 +1010 -0101 -1001 -1101 -0100 Res 0110 0000 1001 0011 1101 0010 1000 0111 Corr Corr Over Under Corr Corr Over Under

  2. Negation • To change sign of a number • In Sign and Magnitude • Just complement the sign • 1’s Complement • Complement all bits • 2’s Complement • Complement all bits and add 1 • Adding 1 is expensive operation (Example: Add 1 to 0111) • Alternate 2’s complement method • Scan the string from right • Retain all bits up to the first 1 • Then complement the remaining bits

  3. Converting negative number • Positive numbers are well understood • How to convert a negative decimal number to binary • Method 1: • The magnitude of number must be less than 2n-1 • Add -2n-1 to the number • Convert the magnitude of this number as an n-bit string • Example -4 + (-8) = -12 (decimal) = 1100 (binary) • Method 2 • Convert the magnitude to an n-bit string • Negate the number

  4. So what about logic functions • So far we discussed arithmetic (yes, no multiply/divide) • But what about a logic functions? • A binary variable can be used to represent two logic value • Answer to any question with two possible answers can be represented by a binary variable • Examples: • Does the second section of 210 meet at 10AM? Answer represented by variable A, value A=1 (yes), value A=0 (no) • Is it worth attending class at that time? Answer represented by variable B, value B=1 (yes), value B=0 (no)

  5. Functions of 1-bit variable • We can combine variables to get more complex functions • Functions of 1 variable X F1(X) F2(X) F3(X) F4(X) 0 0 1 0 1 1 0 1 1 0 • We have four possible operations upon 1 1-bit input value • F1 is always 0, F2 is always 1, F3 is always X and F4 is always complement of X • F4 is an important function and is called inversion • It is also called NOT operation • Inverse is denoted by X’ or X

  6. Functions of 2-bit variable X Y F0 F1 F2 F3 F4 F5 F6 F7 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 X Y F8 F9 F10 F11 F12 F13 F14 F15 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1

  7. Interpretation of function • F1 is called a logical AND denoted by X.Y • F6 is called an XOR denoted by X xor Y • F7 is called OR denoted by X + Y • F8 is NOR denoted by X + Y • F14 is NAND denoted by X.Y • Logic expressions allow us to implement function of binary inputs • For each logic expression there is a logic gate • See and practice logic circuit symbols for AND, OR, NAND, NOR, XOR, NOT…

  8. Combining logic circuits • Basic logic circuits can be combined to get a bigger function • Functions are written in truth table • Each row in a truth table is one combination of logic variable • Given the number of variables, n, we have 2n different entries • For example, for 1 variable there will 2 entries, for 4 variables there will be 16 entries, and so on • Most circuits may have many more than 4 inputs • However in class we will deal with only a few inputs • For example a function of four variables could be • F = (A+B).(C+D)

  9. An Example of Truth Table

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