Intro

1. Decimal Number System: 10 digits (0—9)
2. Binary Number System: 2 digits (0—1)
   • Physical voltage-based switches
3. Octal
4. Hexadecimal

Remember: Mathematically, number systems can represent any value, it is the hardware that has limits.

Number Systems

Decimal

• Base: 10
  • aka: radix
• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Positional Notation: 1234 = 1\times10^3 + 2\times10^2 + 3\times10^1 + 4\times10^0
  • The position of digits relative to each other determines value.
    • e.g., the “2” in “1234” represents “2\times10^2” (200) because of its position relative to the decimal point

More on Positional Notation:

• Positional notation is important for converting ASCII input into a numeric value, among other things.

Example: Converting “123” into a decimal number, digit-by-digit ((1 \times 10) + 2) \times 10 + 3

Example: Positional notation with decimal points 1234.567 = 1 \times 10^3 + 2 \times 10^2 + 3 \times 10^1 + 4 \times 10^0 + 5 \times 10^{-1} + 6 \times 10^{-2} + 7 \times 10^{-3}

Binary

• Base: 2
• Digits: 0, 1
• Bit (Binary Digit): Smallest unit of information.
  • Nibble: 4 bits.
• Byte: Group of 8 consecutive bits.
  • Smallest addressable unit of information in most computers
  • Every byte has an address
  • Groupings:
    • Word: 4 consecutive bytes
    • Half-Word: 2 consecutive bytes
    • Double-Word: 8 consecutive bytes

More on Bytes:

• Anatomy: b_7 , b_6 , b_5 , b_4 , b_3 , b_2 , b_1 , b_0
  • MSBit: Most significant byte.
    • b_7
  • LSBit: Least significant byte.
    • b_0

Range: \text{Range: } 2^n - 1

• e.g., Range of values a byte can store is 0—255 (2^8 - 1 = 255)

Octal

• Base: 8
• Digits: 0, 1, 2, 3, 4, 5, 6, 7
• Octal Digit: Group of 3 consecutive bits.

Hexadecimal

• Base: 16
• Digits: 0, 1, 2, …, 9, A, B, C, D, E, F
• Hexadecimal Digit: Group of 4 consecutive bits.
  • Thus, two hexadecimal digits can represent one byte.
• Notation: Prefixed with 0x

Conversion

A. Decimal to Binary

Repeated Division by 2:

1. Divide the decimal by two. The remainder will be a bit (0 or 1)
2. Do step one again, until we we can no longer divide the decimal.
3. The last remainder is the most significant bit, add up the bits in reverse order we derived them.
   • One way to program this is recursion.

B. Binary to Decimal

Power Table:

• 2^n + ... + 2^2 + 2^1 + 2^0

C. Binary to Octal

1. Begin grouping triplets of bits from right-to-left
   • Pad any trailing starting triplet with 0’s
2. Convert each triplet of bits into octal

Example:

• 10 101 010_2 \to 252_8

D. Binary to Hexadecimal

1. Begin groups 4 groups of bits from right-to-left.
   • Pad any trailing starting group with 0s.

Example:

• 1010 0010_2 \to A2_{16}

Exercise