In mathematics, the fact that multiplying two negative numbers results in a positive number, namely:
\[ (-1)\times (-1) = 1 \]
is a concept that many find counterintuitive. In this article, we will demonstrate how this equation is derived through a mechanical approach based on mathematical rules, and we will also explore its historical background.
Proof: Deriving the Equation from Basic Arithmetic Rules
To begin with, we will focus solely on deriving the equation using rules like the distributive property, without delving into philosophical discussions about the nature of positive and negative numbers.
Utilizing the Distributive Property
The distributive property in mathematics is expressed as follows: \[ a\times (b+c) = a\times b + a\times c. \] Using this property, let us examine the following relationships:
- From the basic definition of arithmetic, we know that \( 1 + (-1) = 0 \).
- Using this relationship, we calculate: \[ 1\times \{1+(−1)\}=1\times 0=0. \]
Applying the distributive property here yields: \[ 1\times 1+1\times (−1)=0. \]
Simplifying this equation gives: \[ 1\times (−1)=−1. \]
Expanding Using the Zero Product Rule
Next, we use the property that multiplying zero by any number results in zero: \[ 0=0\times (−1). \]
We rewrite this zero as follows: \[ 0=\{1+(−1)\}\times (−1).\]
Applying the distributive property again: \[ 1\times (−1)+(−1)\times (−1)=0. \]
Since we already established that \( 1 \times (-1) = -1 \), we substitute it into the equation: \[ −1+(−1)\times (−1)=0. \]
Simplifying gives: \[ (−1)\times (−1)=1. \]
Historical Background
It is said that it was only in the 18th century that it became possible to freely operate on negative numbers that appear in mathematical formulas. However, this was not justified at that time.
Establishing Mathematical Justification
Later, the 19th century British mathematician George Peacock (1791 – 1858) clarified in his book A Treatise on Algebra (1830) that the rules for arithmetic of negative numbers are derived from the associative law, the commutative law, the distributive law, etc.
Peacock introduced the principle of symbolical generalization, or the “Principle of Permanence.” This principle states that “propositions true for positive numbers should also hold for negative numbers.” By adopting this principle, the rules of operations with negative numbers became naturally consistent.
Conclusion
As demonstrated above, the reason why multiplying two negative numbers results in a positive number is rooted in fundamental mathematical rules. In mathematics, even seemingly complex operations are governed by consistent principles that build upon basic laws.
It is also worth noting the historical journey behind this understanding. The treatment of negative numbers, as we know it today, is the result of centuries of work by mathematicians who established its foundational principles.