Question

If \(m\) and \(n\) are each a number between \(-1\) and 0 , exclusive, what can you say about \(m n\) ?

Short answer

The product \(m n\) is a positive number.

Step-by-step solution

Understand the Range

It is given that \(m\) and \(n\) are both negatives as they fall between \(-1\) and 0. They are exclusive of \(-1\), meaning they are no smaller than \(-1\), and also are exclusive of 0, meaning they are less than 0, but not equal to 0.

Identify the Properties of Negative Number Multiplication

The multiplication of two negative numbers yields a positive number.

Conclusion

As both \(m\) and \(n\) are negatives, their product \(m n\) must be a positive number.

Key concepts

Negative Numbers

When it comes to negative numbers, it's important to understand where they sit on the number line. Negative numbers are to the left of zero and are less than any positive number. In this case, we have numbers ranging from - -1 to 0 which are negative. Importantly, we're dealing with numbers exclusive of - -1 and 0.
This means that both - \(m\) and \(n\) sit in this small slice of the number line, making them negative but not reaching the endpoints of -1 or 0.

• These numbers are less than zero.
• They are greater than - -1, so they are in the range {-0.99, -0.98, ..., -0.01}.
• They don't touch the definite end-points, staying strictly between -1 and 0.

Focusing on these properties will help us as we move to understand how they behave in multiplication.

Multiplication Properties

Multiplication is one of the basic operations in mathematics and understanding its properties is crucial, especially with different types of numbers like negatives.
When multiplying two numbers, two things matter: the sign and the magnitude.Understanding how these work with negative numbers reveals one of the foundational rules:

• Multiplying two negative numbers yields a positive product.
• Thus, when both \(m\) and \(n\) are negative, their product \(m \cdot n\) will be positive.

This happens because each number is less than zero, and when they are combined through multiplication, their effects on sign cancel each other out, resulting in a positive number.
So, when both numbers are negative - as is the case with our exercise-- their multiplication always gives a positive outcome.

Range of Numbers

The concept of a number range helps to set boundaries for numbers we are focusing on. In our context, this affects understanding how the numbers interact, specifically in multiplication scenarios.
When considering numbers in a range:

• You must always respect their defined boundaries (e.g., between -1 and 0).
• Exclusivity at 0 and -1 means values like 0, -0.5, etc., are not included if they are endpoints.

In terms of multiplication, since both numbers \(m\) and \(n\) are confined in this specific range of negative numbers, they will follow multiplication rules unique to negative numbers.
This idea becomes pivotal in predicting outcomes of combined operations. Here, because both \(m\) and \(n\) reside in the range between -1 and 0, the multiplication rule states that their product \(m \cdot n\) will not just be positive, but its magnitude would also be small, as negative numbers smaller than negative one are getting multiplied.