Mathematics without Negatives

The word “algebra” comes from the title of a book from around AD 800 by Muhammad Al-Khwarizmi. Despite this, the symbols that we associate with modern algebra (particularly, the use of single letter variable names) don’t appear in the book. Also, the conceptual field of mathematics called algebra came several centuries before: Al-Khwarizmi’s book is historically significant, but it built on previous work and the modern symbolism didn’t occur until long after.

One limitation of the book is that Al-Khwarizmi didn’t use negative numbers. This was typical of mathematicians of the era: Negative numbers were in use, but were heavily resisted by many.

He begins his book by showing three geometrical solutions to what we now call a quadratic equation, \(ax^2 + bx + c = 0\). He needs three because the limitation to positive numbers means he can’t use negative coefficients. So, rather, he shows how to solve the following:

  1. \(ax^2 + bx = c\)
  2. \(ax^2 + c = bx\)
  3. \(bx + c = ax^2\)

Likewise, he can’t solve for negative roots; the only time there are two solutions is when both solutions are positive.

This might seem odd to modern students, but it’s important to remember that he was providing geometric solutions. There are no negatives in geometry proper: All measurements are positive. From this standpoint, his approach makes perfect sense.

If you’d like to read more, my presentation of his first chapter is available on my other blog: First post; second post.

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