Permalink Submitted by Anonymous on February 12, 2015

Very nice article! Was looking for some material to inspire high school student to solve quadratics, and this is super helpful!

One comment though --- I don't quite agree with the portayal of imaginary numbers. There is no "cheating" involved in defining a number system where i = sqrt(-1). The only reason x^2 > 0 for real x is because we have _defined_ multiplication that way. We ourselves make these rules about number systems, and we're free to make new ones; hence, the imaginary numbers are not more strange or "illegal" than anything else in mathematics. At some point in time, people were also uncomfortable with the idea of subtracting a larger number from a smaller one, because it was felt that negative numbers "don't exist". And the same can be said for irrational numbers.

Personally, I think the terminology "real" and "imaginary" numbers is unfortunate. The truth is of course that *all* numbers are imaginary; they exist only in our imagination!

## Imaginary numbers

Very nice article! Was looking for some material to inspire high school student to solve quadratics, and this is super helpful!

One comment though --- I don't quite agree with the portayal of imaginary numbers. There is no "cheating" involved in defining a number system where i = sqrt(-1). The only reason x^2 > 0 for real x is because we have _defined_ multiplication that way. We ourselves make these rules about number systems, and we're free to make new ones; hence, the imaginary numbers are not more strange or "illegal" than anything else in mathematics. At some point in time, people were also uncomfortable with the idea of subtracting a larger number from a smaller one, because it was felt that negative numbers "don't exist". And the same can be said for irrational numbers.

Personally, I think the terminology "real" and "imaginary" numbers is unfortunate. The truth is of course that *all* numbers are imaginary; they exist only in our imagination!