Number Systems - Discrete Mathematical Structures - Lecture Slides, Slides of Discrete Mathematics

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Chapter 1

The Logic of Compound Statements

Section 1.

Number Systems and Circuits for Addition Karnaugh Maps

Binary Representation

• Binary numbers (base 2) are sequence of {0,1}digits. The value of the number depends on it’s location in the sequence. - Binary: 1101 => 1x8 + 1x4 + 0x2 + 1 - 1x2 3 + 1x2 2 + 1x2^1 + 1x2 0 - = (decimal) 8+4+0+1 = 13 (^10) - Binary: 10101 => 1x16 + 0x8 + 1x4 + 0x2 + 1 - 1x2 4 + 0x2 3 + 1x2^2 + 1x2 1 + 1x2 0 - = (decimal) 16+0+4+0+1 = 21 (^10)

Converting Binary to Decimal

• Converting from binary to decimal is performed by expanding the binary to a number sentence and summing over the number sentence. - Example: - 1101012 = 1x2^5 + 1x2^4 + 0x2^3 + 1x2^2 + 0x2^1 + 1x2^0 = 53 10

Example

• Convert 209 10 to binary
• Answer: 11010001

Binary Addition

• Binary addition is performed in the same manner as decimal addition.
• When and how do you perform a carry?
  • Carry is dependent on the base. Sum greater than the (base – 1) results in a carry.
  • Example:
    • 12 + 0 2 = 1 (no carry)
    • 02 + 1 2 = 1 (no carry)
    • 12 + 1 2 = 10 2 (carry)

Binary Subtraction

• Binary subtraction is performed in the same manner as decimal subtraction.
• When and how do you perform a borrow?
  • Borrow is dependent on the base. In decimal you borrow a 10’s value from the adjoining digit, however, in binary you borrow a 10 2 (2 10 ).
  • Example:
    • 1 2 - 0 2 = 1 (no borrow)
    • 1 2 - 1 2 = 0 (no borrow)
    • 10 2 - 1 2 = 01 2 (borrow from 2’s position)
    • 2 10 – 1 10 = 1 10 (check)

Examples

Addition Circuits

Half-Adder

Addition Circuits

Full-Adder

• A full adder requires three inputs, (P, Q) and a carry in (R).

Addition Circuits

Full-Adder

• Full-adder can be constructed from two half- adders.

2-

Unsigned Integers

• Non-positional notation
  • could represent a number (“5”) with a string of ones (“11111”)
  • problems?
• Weighted positional notation
  • like decimal numbers: “329”
  • “3” is worth 300, because of its position, while “9” is only worth 9

329 102 101 100

22 21 20 3x100 + 2x10 + 9x1 = 329 1x4 + 0x2 + 1x1 = 5

most significant least significant

2-

Signed Integers

• With n bits, we have 2 n^ distinct values.
  • assign about half to positive integers (1 through 2n-1) and about half to negative (- 2n-1^ through -1)
  • that leaves two values: one for 0, and one extra
• Positive integers
  • just like unsigned – zero in most significant (MS) bit 00101 = 5
• Negative integers
  • sign-magnitude – set MS bit to show negative, other bits are the same as unsigned 10101 = -
  • one’s complement – flip every bit to represent negative 11010 = -
  • in either case, MS bit indicates sign: 0=positive, 1=negative

2-

Two’s Complement

• Problems with sign-magnitude and 1’s complement
  • two representations of zero (+0 and –0)
  • arithmetic circuits are complex
    • How to add two sign-magnitude numbers? – e.g., try 2 + (-3)
    • How to add to one’s complement numbers?
      • e.g., try 4 + (-3)
• Two’s complement representation developed to make circuits easy for arithmetic.
  • for each positive number (X), assign value to its negative (-X), such that X + (-X) = 0 with “normal” addition, ignoring carry out 00101 (5) 01001 (9) + 11011 (-5) + (-9) 00000 (0) 00000 (0)