Takakazu Seki Kowa (1642- 1708)

Seki, born in Japan, strikes a romantic figure: born to a samurai warrior family, adopted by a nobleman, self-taught child prodigy. On top of his storybook background, he was also a very good mathematician.

Seki studied the mathematics of ancient China but he was able to go further as he invented a new number notation system. He introduced Chinese ideograms to represent unknowns and variables in equations. This had the effect of liberating the mathematics of Japan from dependence on counting rods.

We know he set up a school in Japan for teaching mathematics but the details are sketchy and the Samurai code demanded that people like Seki were modest and not too forward about their achievements. His contributions to mathematics were lost over time and others did not build on his work, but fitting words were written on his tombstone: ‘the Arithmetical Sage’.

Seki’s mathematics

Seki anticipated many of the discoveries of Western mathematics. Seki was the first person to study determinants in 1683. A determinant is a critical aspect of a matrix and it was introduced as a means to solve simultaneous equations. For example, the equations:

\qquad ax+by=c
\qquad px+qy=r

can be written in the matrix form,

\bigl({a\atop p}{b\atop q}\bigr)\bigl({x\atop y}\bigr)=\bigl({c\atop r}\bigr)

The determinant of this matrix is the quantity, aq-pb and so long as this is non-zero, the equations may be solved. Simultaneous equations in any number of variables lead to matrices of a higher order; they too have a determinant.

Some ten years later Leibniz would independently use determinants to solve simultaneous equations. Seki’s version was the more general.

Seki also discovered Bernoulli numbers before Jacob Bernoulli. These are crucial in finding the sums of powers of the integers. For example,

\qquad \sum\limits_{k=1}^n k^0=1+1+1+\ldots 1={n\over{1}};
\qquad \sum\limits_{k=1}^n k^1=1+2+3+\ldots n={{n+n^2}\over{2}};
\qquad \sum\limits_{k=1}^n k^2=1^2+2^2+3^2+\ldots n^2={{n+3n^2+n^3}\over{6}};
\qquad \sum\limits_{k=1}^n k^3=1^3+2^3+3^3+\ldots n^3={{n^2+2n^3+n^4}\over{6}};

In each case the formula is based on a new Bernoulli number - which is the coefficient of n - and the ones appearing previously. So these numbers are, \left\{1,{1\over{2}},{1\over{6}},0,\ldots\right\}

He also investigated Magic Squares based on some ideas that Chinese mathematicians had developed, and developed a number of ways to solve equations involving powers greater than two. For example, he could solve the cubic equation

x^3+5x^2-14x-30=0

a hundred years before the same solution was discovered in the west; at the same time he was able to solve equations by iterative methods.

He was also interested in Diophantine equations and found the solution to the equation,

px+qy=1

in which all the terms are whole numbers.