# A Pegagogical Study of Geometric Relations between the Sums of Cubic Integer Numbers and the Sums of Square Integer Numbers with the Sums of Odd Numbers

## Authors

• Yukio Kobayashi Soka University

## Keywords:

square numbers, cubic number, series, matrix multiplication, pictorial proof, visual thinking

## Abstract

The geometric meaning of the sums of cubic and square numbers can be explained by expressing them as sums of consecutive odd numbers. By illustrating not only the equations but also simple diagrams, we can understand how these sums can be obtained. Geometrically viewing the relationship between series, rather than merely learning how to calculate them, helps students
understand these series systematically. If natural numbers are expressed as sums of only 1, the sums of square numbers and of cubic numbers can be related to the multiplication of a vertical vector with a horizontal one whose components are all 1’s. This view is educational in showing that matrix multiplication is related to the calculation of series. The notion that concepts as
different as series and matrices are fundamentally and closely related is an important perspective in mathematics education.