Check whether product of integers from a to b is positive, negative or zero in Python

When we need to determine the sign of a product of integers in a range [a, b], we don't need to calculate the actual product. Instead, we can use mathematical rules to determine if the result will be positive, negative, or zero.

So, if the input is like a = -8, b = -2, then the output will be Negative, as the values in that range are [-8, -7, -6, -5, -4, -3, -2], and the product is -40320 which is negative.

Algorithm

To solve this efficiently, we follow these steps:

• If both a and b are positive, the product is always positive
• If a is negative and b is positive (range contains zero), the product is zero
• If both a and b are negative, count the negative numbers:
  • If count is even, product is positive
  • If count is odd, product is negative

Example

def check_product_sign(a, b):
    if a > 0 and b > 0:
        return "Positive"
    elif a <= 0 and b >= 0:
        return "Zero"
    else: 
        # Both are negative
        count = abs(a - b) + 1
        if count % 2 == 0:
            return "Positive"
        return "Negative"

# Test case 1: Range with negative numbers
a = -8
b = -2
result = check_product_sign(a, b)
print(f"Range [{a}, {b}]: {result}")

# Test case 2: Range including zero
a = -3
b = 2
result = check_product_sign(a, b)
print(f"Range [{a}, {b}]: {result}")

# Test case 3: Range with positive numbers
a = 2
b = 5
result = check_product_sign(a, b)
print(f"Range [{a}, {b}]: {result}")

Range [-8, -2]: Negative
Range [-3, 2]: Zero
Range [2, 5]: Positive

How It Works

The algorithm works based on these mathematical principles:

• Positive numbers: Product of positive numbers is always positive
• Zero included: Any product containing zero equals zero
• Negative numbers: Product of even count of negatives is positive, odd count is negative

Alternative Implementation

def determine_product_sign(start, end):
    # Check if range includes zero
    if start <= 0 <= end:
        return "Zero"
    
    # All positive numbers
    if start > 0:
        return "Positive"
    
    # All negative numbers
    negative_count = end - start + 1
    return "Positive" if negative_count % 2 == 0 else "Negative"

# Test the function
test_cases = [(-8, -2), (-3, 2), (2, 5), (-5, -1)]

for start, end in test_cases:
    result = determine_product_sign(start, end)
    print(f"Range [{start}, {end}]: {result}")

Range [-8, -2]: Negative
Range [-3, 2]: Zero
Range [2, 5]: Positive
Range [-5, -1]: Positive

Conclusion

This approach efficiently determines the sign of a product without calculating the actual value. The key insight is using the mathematical rule that an even number of negative factors produces a positive result, while an odd number produces a negative result.