Binary (Base-2) Basics

  • Binary uses two digits: 0 and 1 (OFF/ON states).
  • Bits: Each digit in binary.
  • Why Binary? Reliable for storing/transmitting data — all data (text, images, sound) are just long binary sequences.

Comparing Number Systems

Number Decimal (Base 10) Binary (Base 2) Octal (Base 8) Hexadecimal (Base 16)
0 0 0 0 0
1 1 1 1 1
2 2 10 2 2
4 4 100 4 4
8 8 1000 10 8
15 15 1111 17 F
  • Octal: Early computing.
  • Hexadecimal: Memory addresses, color codes.

Binary Conversions

  • Decimal → Binary: Divide by 2, record remainders bottom-up.
  • Binary → Decimal: Multiply each bit by 2ⁿ based on position and sum.

Example:
13 → 1101

Binary Arithmetic

  • Addition Rules:
    • 0 + 0 = 0
    • 0 + 1 = 1
    • 1 + 0 = 1
    • 1 + 1 = 10 (carry 1)
  • Subtraction Rules:
    • 0 - 0 = 0
    • 1 - 0 = 1
    • 1 - 1 = 0
    • 0 - 1 = 1 (borrow 1)

Two’s Complement (Negative Numbers)

  1. Invert all bits.
  2. Add 1.
    • Example: 5 → 00000101 → flip → 11111010 → add 1 → 11111011 (−5)

Bit Shifts

  • Left Shift: Multiply by 2.
  • Right Shift: Divide by 2.

Real-World Applications

  • Storage: 1 Byte = 8 Bits, 1 KB = 1024 Bytes, etc.
  • Networking: IP addresses in binary (e.g., 192.168.1.1).

Popcorn Hack 1: Identifying Binary

Example 1

Number: 101010

  • ✅ Only contains 0s and 1s → This is a valid binary number

Example 2

Number: 12301

  • ❌ Contains digits other than 0 and 1 → This is NOT a valid binary number

Example 3

Number: 11001

  • ✅ Only contains 0s and 1s → This is a valid binary number

Popcorn Hack 2: Examples for Adding and Subtracting Binary

Example 1 (Adding):

Binary Numbers: 101 + 110
✅ Answer: 1011

Example 2 (Subtracting):

Binary Numbers: 1101 - 1011
✅ Answer: 010

Example 3 (Adding):

Binary Numbers: 111 + 1001
✅ Answer: 1110

Popcorn Hacks and Homework Hacks Completed

Popcorn Hack 1

Problem:
What is the result of:
True or False and False

Answer:

  • False and False → False
  • True or False → True

✅ Final Answer: True

Popcorn Hack 2

Problem:
What is the result of:
not True and False

Answer:

  • not True → False
  • False and False → False

✅ Final Answer: False

Popcorn Hack 3

Problem:
What is the result of:
True or False and not False

Answer:

  • not False → True
  • False and True → False
  • True or False → True

✅ Final Answer: True

Homework Hack 1: Binary Converter

# Decimal to Binary
def decimal_to_binary(n):
    if n == 0:
        return "0"
    binary = ""
    while n > 0:
        binary = str(n % 2) + binary
        n = n // 2
    return binary

# Binary to Decimal
def binary_to_decimal(b):
    return int(b, 2)

# Example Tests
print("Decimal 10 to Binary:", decimal_to_binary(10))  # Output: 1010
print("Binary 1010 to Decimal:", binary_to_decimal("1010"))  # Output: 10

Homework Hack 2: Difficulty Level Checker

import time

difficulty = input("Enter difficulty (easy, medium, hard): ").lower().strip()

while difficulty not in ["easy", "medium", "hard"]:
    print("Please enter a valid difficulty level.")
    difficulty = input("Enter difficulty (easy, medium, hard): ").lower().strip()
    time.sleep(0.5)

print("Difficulty set to:", difficulty)