Why is the product of negative numbers equal to positive?

Source image


One of the most asked questions in an introduction to algebra is, why (-1)*(-1)=1? Our intuition guides us to think the result should be also negative instead of positive. But there is an intuitive explanation with geometry that answers why.

What would you learn?

  • A vector intuition for numbers
  • An intuition for complex numbers

@Alfredo Maussa

What is behind the most beautiful equation?


Recovered from Medium

The intuition

Let’s see negative numbers as vectors, where the negative stands for the opposite direction to one. This concretely is having a positive one with a rotation angle of 180°.

So, multiplying a negative number twice is equivalent to making half a revolution twice, hence it results in an entire revolution giving again a positive number direction.

The proof

This is like a trick, but it is mathematically valid? Yes, thanks to Euler’s identity. Any product of complex numbers, including positive and negatives, can be described as a product of Euler’s formulas, where making use of its exponential properties allows us the addition of angles (see “recall” annotation on the image):

While this works for the product, addition and subtraction are ruled by head to the tail method of vectors.

Conclusion context

In general, multiplying any number is the process of multiplying the absolute value and adding the angles… This is true for negative numbers whose absolute value’s product is 1 and adding the angles (twice 180°) is 360° which is the positive direction again.

If a kid asks you, why? You could answer by associating negatives number with half revolution. Turning them back twice would be a funny analogy.

This even works as a complex number introduction as the imaginary number i is one with an angle of 90°, and it just works the same as negative numbers with 180°. 🙌