Introduction: Introduction & Learning Goals

This lesson explains how computers use different number systems such as decimal, binary, octal, and hexadecimal. It also teaches how to convert numbers between systems and how to perform binary arithmetic operations.

Learning Outcomes

• Distinguish number systems and data representation
• Perform number conversions
• Solve binary addition and subtraction
• Understand complements and signed numbers

Number: Number Systems

Decimal Number System Base 10

• Uses digits 0-9
• Place values: ones, tens, hundreds…
• Example:
  
  (1*1000) + (2*100) + (3*10) + (4*1)
  
  (1*10^3) + (2*10^2) + (3*10^1) + (4*10^0)
  
  1000 + 200 + 30 + 1
  
  1234⌄10

Binary Number System Base 2

• Uses only 0 and 1
• Used by computers
• ON = 1, OFF = 0
• Place values are powers of 2

Octal Number System Base 8

• Uses digits 0-7
• Place values are powers of 8

Hexadecimal Number System Base 16

• Digits: 0-9 and A-F
• A=10, B=11, C=12, D=13, E=14, F=15
• Used in memory addresses and programming

Conversion: Converting Between Number Systems

Number Conversion

• A. Binary → Decimal

Example:

101012 = (1 *2^4) + (0 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0)

= (1x16) + (1* 8) + (1*4) + (0*2) + (1*1)

= 16 + 0 + 4 + 0 + 1

= 21⌄10



• B. Octal → Decimal

Example:

1258 = (1* 8^2) + (2 * 8^1) + (5 * 8^0)

= (1 * 64) + (2 * 8) + (5 * 1)

= 64 + 16 + 5

= 85⌄10



• C. Hexadecimal → Decimal

Hex        Decimal

0-9         0-9

A           10

B           11

C           12

D           13

E           14

F           15




Example:

E = 14 ; 8 = 8 ;B = 11

E8B⌄16 = (14 * 16^2) + (8 * 16^1) + (1 * 16^0)

= (14 * 256) + (8 * 16) + (11 * 1)

= 3,584 + 128 + 11

=3723⌄10



• D. Decimal → Binary
• Divide by 2
• Write remainders
• Read bottom to top

Example:

17⌄10 = 10001⌄2

Example:

127⌄10 = 177⌄8

Example:

960⌄10 = 3C0⌄16

Base-to-Base Conversions:

Base

• Conversion         Method

Binary → Octal       Binary → Decimal → Octal

Octal → Binary       Octal → Decimal → Binary

Hex → Binary       Hex → Decimal → Binary

Hex → Octal       Hex → Binary → Octal

Example:

1010101⌄2 = 125⌄8

A2B⌄16 = 101000101011⌄2

Arithmetic

Binary Addition Rules

Rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (carry 1)

Example:

101 + 101 = 1010

1's Complement
To get 1's complement, change all 0s to 1s and all 1s to 0s.

Example:
10101110 → 01010001

Purpose:

• Used to represent negative numbers

Adding Signed Numbers
Include three cases:

1. Positive + Negative (positive bigger)
2. Positive + Negative (negative bigger)
3. Negative + Negative

Example:
1101 + (-1001) = 0101

Binary Subtraction

Rules:

• 0 - 1 = 1 (borrow)
• 1 - 1 = 0

Example:
1010 - 101 = 0101