NET 104: Digital Logic Design - Lecture 1

NET 104: Digital Logic Design - Lecture 1

• The course is about digital logic design.
• Lecture 1 is an introduction.
• Digital systems use discrete amounts of data.
• Examples include letters in the alphabet (26) and decimal digits (10).
• Larger quantities of data can be built from discrete values (e.g., words made of letters or numbers).
• Computers use binary values (0 and 1).
• Binary values are easily represented electrically using voltages and currents.
• Digital circuits are the building blocks of modern computers.

Understanding Decimal Numbers

• Decimal numbers are made of decimal digits (0-9) with a base of 10.
• Decimal numbers represent items by their positional values (e.g., 8653 = 8 x 10^3 + 6 x 10^2 + 5 x 10^1 + 3 x 10^0).
• Decimal numbers can also include fractions (e.g., 97654.35).

Understanding Binary Numbers

• Binary numbers use binary digits (bits) which are either 0 or 1.
• Binary numbers represent items by their positional values (e.g., 1011₂ = 1 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2⁰ = 11₁₀).
• Binary numbers include fractions.
• Groups of 8 bits are called a byte.
• Groups of 4 bits are called a nibble.

Understanding Octal Numbers

• Octal numbers use octal digits (0-7) with a base of 8.
• Octal numbers represent items by their positional values ((4536)₈ = 4 x 8³ + 5 x 8² + 3 x 8¹ + 6 x 8⁰ = (2398)₁₀).
• Octal numbers do not include digits 8 or 9.

Understanding Hexadecimal Numbers

• Hexadecimal numbers use hexadecimal digits (0-9, A-F) with a base of 16.
• Hexadecimal numbers represent items by their positional values.
• Hexadecimal numbers can be converted to binary easily by representing each hex digit as 4 bits (e.g., (3A9F)₁₆ = (0011101010011111)₂).

Why Use Binary Numbers?

• Binary numbers are easy to represent using electrical signals (0 or 1).
• Binary systems can tolerate noise.
• Binary data is easy to transmit.
• Binary circuits are easy to build.

Converting Between Bases

• Converting integers from decimal to another base (e.g., binary): Divide the decimal number by the target base and record the remainders.
• Converting fractions from decimal to another base: Multiply the fraction by the target base and record the integers.

Binary Coded Decimal (BCD)

• BCD encodes each decimal digit using 4 bits.
• Example: 329₁₀ = (0011 0010 1001)BCD.
• BCD is NOT the same as binary.

The Growth of Binary Numbers

• Binary numbers grow exponentially (e.g., 2⁰=1,2¹=2,2²=4...).
• Common units based on binary powers are kilo, mega, giga, and tera.

Putting it all together (Conversion examples)

• A table showing conversions between decimal, binary, octal, hexadecimal, and BCD.

Binary Addition

• Binary addition is straightforward and follows the same rules as decimal addition, but with a carry operation.

Representing Signed Numbers

• Computers represent positive and negative numbers using different methods:
  • Signed magnitude
  • 1's complement
  • 2's complement

2's Complement Shortcuts

• Two algorithms to calculate 2's complements:
  • First: Flip all bits and add 1.
  • Second: Starting from the rightmost bit, copy all 0s until you encounter the first 1, then flip all the remaining bits.

2's Complement Addition and Subtraction

• Operations with 2's complement numbers use the same techniques as with unsigned numbers, with carries being ignored when appropriate.

Gray Code

• Gray code is a binary numeral system where only one bit changes between consecutive numbers.
• Gray code helps reduce errors during data transmission.
• Conversion between gray code and binary involves bit-by-bit manipulation with carries being discarded during conversion.