Prove It If You Can — “-1 x -1 = +1”

A Visual Approach for Everyone, Even Secondary School Students

Have you ever wondered about the reasoning behind “-1 x -1 = +1”? Does it seem like magic or a mere mathematical rule to be memorized? While online explanations exist, they often come across as dull and overly abstract, making them challenging for regular students to grasp.

If you consult the renowned A.I. ChatGPT, you’ll discover some engaging explanations. However, considering it’s an abstract concept, comprehending it might pose a challenge for a thirteen-year-old student, don’t you agree?

In this blog post, I’ll present a straightforward and swift graphical solution to this renowned equation, eschewing the abstract narrative. The method involves just five simple steps!

Step 1) Area of a Rectangle

I’m confident that everyone here comprehends the concept that the area of a rectangle equals the product of its length and width. The aggregate area of the larger rectangle above results from the addition of the areas of its four smaller constituent rectangles.

Step 2) Substitute the b & d with a Negative Value

Substituting negative values for both “b” and “d” translates into a tangible deduction of the lengths at both “a” and “c” in the depicted diagram.

Our current focus pertains to the diminished area highlighted in yellow.

Step 3) Mathematical Form

Following the substitution of “-b” and “-d” into the equation outlined in Step 1), we can reshape the equation as follows:

Step 4) Graphical Form

Let’s analyze the graphical interpretation of the derived equation in a step-by-step manner:

The First part “ac”:

The value of “ac” corresponds to the area of the green rectangle, which we can regard as the initial or original rectangle.

The Second part “-bc”:

The significance of “-bc” lies in representing the subtraction of the area highlighted in orange within the diagram below.

The Third part “-ad”:

The interpretation of “-ad” pertains to the deduction of the area highlighted in blue within the following diagram.

Please keep in mind that both the orange and blue areas encompass the bottom-right corner portion. This results in a situation of double counting for this specific area.

Step 5) Comparing Mathematical Form Vs Graphical Form

It’s understood that both the mathematical expression depicted in Step 3 and the graphical representation demonstrated in Step 4 are both indicative of the area of interest.

Hence, they ultimately equate to one another.

We can equate the mathematical form and the graphical form as shown below. By cancelling out the common terms “ac,” “-bc,” and “-ad,” we can arrive at the realization that “(-1)(-1) bd” equals “bd”.

Ultimately, this deduction leads us to conclude that “-1 x -1 = +1”!

Conclusion

In conclusion, the seemingly puzzling result of “-1 x -1 = +1” becomes clear when we explore both its mathematical and graphical interpretations. By breaking down the components step by step and considering the areas represented by the terms, we uncover the rationale behind this concept.

The mathematical equation and the graphical representation converge to reveal the same truth: that the product of two negative numbers is indeed positive.

Through this exploration, we gain a deeper understanding of this fundamental principle in mathematics, bridging the gap between abstraction and visual representation.

Hope you enjoy this blog post! See you next time

My bio:

I’m a Mobile Lead at REAL Corporation Limited in Hong Kong, overseeing in-house app development for iOS and Android. Passionate about technology and user experiences, I’m dedicated to innovation. In my free time, I delve into subjects like Firebase, Google Cloud, and A.I., constantly improving my skills and knowledge.

Outside of work, I’m excited about freelancing in native app development. As I grow my skills, I’m open to collaborating with like-minded individuals and clients who value impactful mobile experiences. Reach out for comments, inquiries, or collaboration opportunities. Let’s explore together!

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