Understanding the Product of Positive and Negative Numbers
Understanding why the product of two negative numbers is positive and why the product of a positive number and a negative number is negative is a fundamental concept in mathematics. This article will provide a detailed explanation using the distributive property and the definition of negative numbers.
Proof for the Product of Two Negative Numbers
Let us start by understanding the definitions and properties that we will use to prove that the product of two negative numbers is positive.
Definitions:
Let a and b be positive numbers. Define -a as the additive inverse of a, i.e., a - a 0. Similarly, -b is the additive inverse of b.We want to show that -a - b ab.
We start with the expression:
0 a - a
Now multiply both sides by -b:
0 · (-b) (a - a)(-b)
By the property of multiplication, the left side is still 0:
0 a(-b) - a(-b)
Rearranging the equation:
Since a - b is negative because a is positive and -b is negative, we can express this as:
0 -ab - a(-b)
Solving for -a - b:
Rearranging gives us:
-a - b ab
This shows that the product of two negative numbers is positive.
Proof for the Product of a Positive Number and a Negative Number
Now let's consider the product of a positive number and a negative number. Again, we will use the distributive property.
Let a be a positive number and b be a positive number. We want to show that a - b -ab.
We can start with:
0 b - b
Multiply both sides by a:
0 · a (b - b) · a
Which simplifies to:
0 ab - ba
Rearranging the equation:
From this, we can rearrange:
a - b -ab
This shows that the product of a positive number and a negative number is negative.
Summary
The product of two negative numbers is positive: -a - b ab.
The product of a positive number and a negative number is negative: a - b -ab.
These results are consistent with our understanding of the number line and the properties of numbers.
Additional Considerations
Some readers might argue that if the product of two negative numbers is assumed to be negative, it would lead to a contradiction. Let's examine this:
Suppose -a and -b are negative numbers where a and b are positive and non-zero. Let ab z where z is some positive number. So, -a - b -z according to the assumption. This would imply:
-1 - 1 -1 (since ab z)
Removing the common factor -1, we have:
-1 1
Clearly, -1 / -1 1 is true for all non-zero complex numbers. This shows that the assumption that the product of two negative numbers is negative leads to a contradiction.
A similar contradiction arises if we assume that a positive number multiplied by a negative number gives a positive result. Let -a be a negative number and y be a positive number, and let ay z where z is positive. Then:
-ay z
-1 1 (since ay z)
Again, this is a contradiction.
I hope this explanation clarifies why the product of two negative numbers is positive and why the product of a positive and a negative number is negative.