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  • Click on "browse channels"
  • Search for "coding"
  • Click the green "Join" button on the right

Learning Objectives

CollegeBoard Requirements for Binary

DAT-1.A: Representing Data with Bits

Basic Information

  • Bit is short for binary digit, and represents a value of either 0 or 1.
    • A byte is 8 bits.
  • Sequences of bits are used to represent different things.
    • Representing data with sequences of bits is called abstraction.

Practice Questions:

  1. How many bits are in 3 bytes?
  • There are 24 bits in 3 bytes.
  1. What digital information can be represented by bits?
  • It can represent strings, numbers, images, etc. It depends on how you store the bits.
  1. Are bits an analog or digital form of storing data? What is the difference between the two?
  • Bits are a digital form of storing data. Analog is constant, while bits are either 1s or 0s.

Examples

  • Boolean variables (true or false) are the easiest way to visualize binary.
    • 0 = False
    • 1 = True
import random

def example(runs):
    # Repeat code for the amount of runs given
    while runs > 0:
        # Assigns variable boolean to either True or False based on random binary number 0 or 1.
        boolean = False if random.randint(0, 1) == 0 else True 

        # If the number was 1 (True), it prints "awesome."
        if boolean:
            print("binary is awesome")
            
        # If the number was 2 (False), it prints "cool."
        else:
            print("binary is cool")
            
        runs -= 1
     
# Change the parameter to how many times to run the function.   
example(10)

binary is cool
binary is awesome
binary is cool
binary is awesome
binary is cool
binary is cool
binary is cool
binary is cool
binary is awesome
binary is awesome

DAT-1.B: The Consequences of Using Bits to Represent Data

Basic Information

  • Integers are represented by a fixed number of bits, this limits the range of integer values. This limitation can result in overflow or other errors.
  • Other programming languages allow for abstraction only limited by the computers memory.
  • Fixed number of bits are used to represent real numbers/limits

Practice Questions:

  1. What is the largest number can be represented by 5 bits?
  • 31.
  1. One programing language can only use 16 bits to represent non-negative numbers, while a second language uses 56 bits to represent numbers. How many times as many unique numbers can be represented by the second language?
  • 2^40 times.
  1. 5 bits are used to represent both positive and negative numbers, what is the largest number that can be represented by these bits? (hint: different thatn question 1)
  • 15

Examples 

import math

def exponent(base, power):
    # Print the operation performed, turning the parameters into strings to properly concatenate with the symbols "^" and "=".
    print(str(base) + "^" + str(power) + " = " + str(math.pow(base, power)))

# How can function become a problem? (Hint: what happens if you set both base and power equal to high numbers?)
exponent(5, 2)

5^2 = 25.0

DAT-1.C: Binary Math

Basic Information

  • Binary is Base 2, meaning each digit can only represent values of 0 and 1.
  • Decimal is Base 10, meaning eacht digit can represent values from 0 to 9.
  • Conversion between sequences of binary to decimal depend on how many binary numbers there are, their values and their positions.

Practice Questions:

  1. What values can each digit of a Base 5 system represent?
  • 0 to 4.
  1. What base is Hexadecimal? What range of values can each digit of Hexadecimal represent?
  • Hexadecimal is 16. The range of values in hexadecimal is from 0 to 15.
  1. When using a base above 10, letters can be used to represent numbers past 9. These letters start from A and continue onwards. For example, the decimal number 10 is represented by the letter A in Hexadecimal. What letter would be used to represent the Base 10 number 23 in a Base 30 system? What about in a Base 50 system?
  • N in base 30 system. Also N in base 50 system.

Examples

  • Using 6 bits, we can represent 64 numbers, from 0 to 63, as 2^6 = 64.
  • The numbers in a sequence of binary go from right to left, increasing by powers of two from 0 to the total amount of bits. The whole number represented is the sum of these bits. For example:
    1. 111111
    2. 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0
    3. 32 + 16 + 8 + 4 + 2 + 1
    4. 63

  • Fill in the blanks (convert to decimal)

    1. 001010 = 10
    2. 11100010 = 226
    3. 10 = 2

  • Fill in the blanks (convert to binary)

    1. 12 = 1100
    2. 35 = 100011
    3. 256 = 100000000

Hacks & Grading (Due SUNDAY NIGHT 4/23)

  • Complete all of the popcorn hacks (Fill in the blanks + run code cells and interact + Answer ALL questions) [0.3 or nothing]
  • Create a program to conduct basic mathematical operations with binary sequences (addition, subtraction, multiplication, division) [0.6 or nothing]
    • For bonus, program must be able to conduct mathematical operations on binary sequences of varying bits (for example: 101 + 1001 would return decimal 14.) [0.1 or nothing]
def add(x, y):
    # Convert binary sequences x and y to integers, add them, and convert back to binary sequence
    return bin(int(x, 2) + int(y, 2))[2:]

def subtract(x, y):
    # Convert binary sequences x and y to integers, subtract them, and convert back to binary sequence
    return bin(int(x, 2) - int(y, 2))[2:]

def multiply(x, y):
    # Convert binary sequences x and y to integers, multiply them, and convert back to binary sequence
    return bin(int(x, 2) * int(y, 2))[2:]

def divide(x, y):
    quotient = bin(int(x, 2) // int(y, 2))[2:]
    remainder = bin(int(x, 2) % int(y, 2))[2:]
    return quotient + " remainder: " + remainder


def binary_menu():
    select = input("Enter the binary arithmetic operation (+, -, *, /): ")
    x_num = input("Enter the first binary value: ")
    y_num = input("Enter the second binary value: ")

    if select == '+':
        # Perform binary addition
        return add(x_num, y_num)
    elif select == '-':
        # Perform binary subtraction
        return subtract(x_num, y_num)
    elif select == '*':
        # Perform binary multiplication
        return multiply(x_num, y_num)
    elif select == '/':
        # Perform binary division
        return divide(x_num, y_num)
    else:
        # Invalid operator
        return "Invalid operator!"

result = binary_menu()

print(f'Result: {result}')

Result: 1001111111011000