x^{n}

no historical likeness exists

c. 953 – c. 1029Baghdad, Abbasid Caliphate

al-Karaji

Powers are a single family ( $x^{n}$ indexed by $n$ ), not a list of separate ideas.

A short story

A Persian mathematician working in Baghdad a thousand years ago. Before him, mathematicians treated $x$ , $x^{2}$ , $x^{3}$ as a list of separate ideas: squaring, cubing, multiplying-by-itself-four-times. al-Karaji saw them as a single, indexed family ( $x^{n}$ for any positive whole $n$ ) and worked out their algebra: how to multiply them, divide them, sort them by degree. The systematic treatment of exponents in modern algebra starts here.

In their own words

Let the powers of $x$ be ranged in order: $x, x^{2}, x^{3}, x^{4}, \dots$ . Each is one factor of $x$ further than the one before.

Paraphrased: from al-Karaji's al-Fakhri, c. 1000 CE, on the systematic treatment of algebraic powers.

The lab their idea turned into

Square Lab

Multiplication stacked. Squares undone.

Open Square Lab →