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import random
title = "Negative Binary Numbers"
description = "This module covers negative binary numbers and their representation."
def generate_question():
bit_lengths = [2, 3, 4, 5, 6, 7, 8]
num_bits = random.choice(bit_lengths)
def generate_binary_number(length):
return ''.join(random.choice('01') for _ in range(length))
number = generate_binary_number(num_bits)
def calculate_negative_binary(binary_str):
# Interpret the binary string as a signed number
if binary_str[0] == '1':
# Two's complement
ones_complement = ''.join('1' if bit == '0' else '0' for bit in binary_str)
twos_complement = bin(int(ones_complement, 2) + 1)[2:].zfill(len(binary_str))
decimal_value = -int(twos_complement, 2)
else:
decimal_value = int(binary_str, 2)
return decimal_value
def decimal_to_binary(value, length):
# Convert a decimal number to binary with a given length
if value < 0:
value = (1 << length) + value
return bin(value)[2:].zfill(length)
correct_decimal = calculate_negative_binary(number)
correct_binary = decimal_to_binary(correct_decimal, num_bits)
options = [correct_decimal]
# Generate incorrect answers
while len(options) < 4:
random_decimal = random.randint(-2**(num_bits-1), 2**(num_bits-1)-1)
if random_decimal != correct_decimal:
options.append(random_decimal)
random.shuffle(options)
question = f"What is the decimal value of the signed {num_bits}-bit binary number {number}?"
explanation = f"The decimal value of the signed {num_bits}-bit binary number {number} is {correct_decimal}."
step_by_step_solution = [
f"Step 1: Identify if the binary number is positive or negative. The leftmost bit indicates the sign.",
f"Step 2: If the leftmost bit is '1', calculate the two's complement to find the magnitude of the negative number.",
f"Step 3: Convert the binary number to decimal considering its sign.",
f"Step 4: The decimal value of the signed {num_bits}-bit binary number {number} is {correct_decimal}."
]
return {
"question": question,
"options": options,
"correct_answer": correct_decimal,
"explanation": explanation,
"step_by_step_solution": step_by_step_solution
}
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